Applications of Pattern Recognition and Learning Controllers to Power System Analysis and Control.

Transcription

1 Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1977 Applications of Pattern Recognition and Learning Controllers to Power System Analysis and Control. Ronald Jerry Lacarna Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: Recommended Citation Lacarna, Ronald Jerry, "Applications of Pattern Recognition and Learning Controllers to Power System Analysis and Control." (1977). LSU Historical Dissertations and Theses This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact

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3 77-28,686 LaCARNA, Ronald Je rry, APPLICATIONS OF PATTERN RECOGNITION AND LEARNING CONTROLLERS TO POWER SYSTEM ANALYSIS AND CONTROL. The Louisiana State U niversity and A gricultural and Mechanical College, Ph.D., 1977 Engineering, electro n ics and e le c tric a l Xerox University Microfilms t Ann Arbor, Michigan 48106

4 APPLICATIONS OF PATTERN RECOGNITION AND LEARNING CONTROLLERS TO POWER SYSTEM ANALYSIS AND CONTROL A D i s s e r t a t i o n S u b m itte d to th e G rad u ate F a c u lty of the L o u is ia n a S t a t e U n iv e r s ity and A g r i c u l t u r a l and M echanical C ollege in p a r t i a l f u l f i l l m e n t o f th e re q u ire m e n ts f o r th e deg ree of D octor of P h ilo so p h y in The D epartm ent of E l e c t r i c a l E n g in e e rin g by Ronald J e r r y L ac am a B.S., L o u is ia n a S t a t e U n iv e rs ity, 1966 M.S., L o u is ia n a S t a t e U n iv e rs ity, 1970 A ugust 1977

5 ACKNOWLEDGEMENT The a u th o r w ish e s to e x p re s s h i s a p p r e c i a ti o n to Dr. Johnny R. Johnson f o r h i s in v a lu a b le a s s i s t a n c e in th e p r e p a r a tio n of t h i s d i s s e r t a t i o n and f o r h is f r i e n d s h i p and gu id an ce th ro u g h o u t h i s p e rio d of g ra d u a te s tu d y. He i s a ls o g r a t e f u l to h i s exam ining com m ittee f o r t h e i r p a tie n c e and a s s i s t a n c e in h e lp in g him com plete th e d i s s e r t a t i o n b e fo re th e sch ed u led d e a d lin e. He would a ls o lik e to th an k Dr. Leonard C. Adams f o r h i s c o n fid e n c e and su p p o rt d u rin g h i s p e rio d of g ra d u a te stu d y and work a s an i n s t r u c t o r. The a u th o r i s g r a t e f u l to Dr. P a u l M. J u l i c h f o r g a in in g him a c c e ss to a com puter te rm in a l w hich e n a b le d t h i s r e s e a rc h to be com pleted in a re a s o n a b le tim e. The a u th o r i s g r a t e f u l to s e v e r a l members o f th e LSU System Network Computer C e n te r f o r t h e i r a s s i s t a n c e. The a u th o r would l i k e to e x p re s s h i s most s in c e r e a p p r e c i a ti o n to many f r i e n d s who encouraged him in th e p r e p a r a tio n o f t h i s d i s s e r t a t i o n. In p a r t i c u l a r, he wou?d l ik e to thank Mr. Ray Brew in, Mr. Rafe D avison, and Dr. James F o n te n o t. A deep a p p r e c i a ti o n i s a f fo r d e d to h i s p a r e n ts, w ith o u t whose lo v e and d e v o tio n, n o th in g would be p o s s ib le.

6 TABLE OP CONTENTS PAGE ACKNOWLEDGEMENT... i i LIST OF FIG U R E S... v ABSTRACT v i CHAPTERS I. INTRODUCTION... 1 I I. PATTERN RECOGNITION AND LEARNING CONTROL (G e n e ra l C o n c e p ts)... 5 A. P a t t e r n R e c o g n itio n... 5 B. L e a rn in g S ystem s C. P a t t e r n R e c o g n itio n, L e a rn in g C o n tro l S ystem s and Power S ystem s I I I. APPLICATION OF A LEARNING CONTROLLER TO LOAD FREQUENCY CONTROL OF AN INTERCONNECTED POWER SYSTEM 20 A. I n t r o d u c t i o n B. S ta te -M o d e l and O ptim al C o n tr o l C. P a t t e r n R e c o g n itio n and L e a rn in g A p p l i c a t i o n D. Two-Area S t a t e Model E. L e a rn in g A lg o rith m... 3 F. S e l e c t i o n o f L e a rn in g P a rm e te rs... ^8 G. S im u la tio n R e s u lts and C o n c lu s io n s IV. OPTIMAL CORRECTIVE RESCHEDULING FOR STEADY-STATE POWER SYSTEM SECURITY VIA PATTERN RECOGNITION AND SEARCH METHODS A. I n t r o d u c t i o n B. P roblem F o rm u la tio n C. R e s u lts and C o n c l u s i o n s... 7^ i i i

7 V. CONCLUSIONS BIBLIOGRAPHY... 8? APPENDIX I APPENDIX I I...I 12 V I T A iv

8 LIST OF FIGURES FIGURE PAGE 2.1 P a t t e r n C l a s s i f i e r D e c isio n S u rfa c e s L ea rn in g C o n tro l System L ea rn in g C o n tro l System I n t e r p o l a t i n g H yperplane L in e a r M a c h i n e *4- N eighborhood o f c u r r e n t s t a t e F lo w ch art o f L e a rn in g C o n t r o ll e r I n t e r p o l a t i o n V a l id i t y Check... 5^ 3.7 Response o f 2 - a r e a system to change in T^2 w ith o u t l e a r n in g c o n t r o l l e r Response o f 2 - a r e a system to change in T12 w ith le a r n in g c o n t r o l l e r The P a c i f i c N orthw est System F a t t e r n C a te g o r iz e r fo rm u la tio n F lo w ch art (O ptim al R e sc h e d u lin g ) v

9 ABSTRACT P a t te r n r e c o g n itio n and le a r n in g c o n tr o l te c h n iq u e s were a p p lie d to two power system pro b lem s. A le a r n in g c o n t r o l l e r u sin g an i n t e r p o l a t i o n te c h n iq u e and i n i t i a l l y t r a i n e d by an o p tim a l feedback g a in m a trix ( te a c h e r ) was used to c o n tr o l th e re sp o n se of a 2 - a r e a in te r c o n n e c te d power sy stem. The system was s u b je c te d to s te p lo a d d is tu r b a n c e s and to a v a r i a t i o n in th e system t i e - l i n e c o n s ta n t. The le a r n in g c o n t r o l l e r s u c c e s s f u ll y a d a p te d to th e s e d is tu r b a n c e s. The problem o f o p tim a l r e s c h e d u lin g o f g e n e r a tio n and demand to m a in ta in s e c u r i t y o f a power system netw ork was re fo rm u la te d a s a p a t t e r n r e c o g n itio n problem and a s e a rc h te c h n iq u e used to s o lv e f o r th e o p tim a l re s c h e d u le based on a c o s t f u n c tio n t h a t in c lu d e d a l l e q u a l i t y and i n e q u a l i t y c o n s t r a i n t s. The method proved to be s im p le r th a n th e u s u a l l i n e a r program m ing a p p ro a c h. The f e a s i b i l i t y o f u s in g p a t t e r n r e c o g n itio n and l e a r n in g a lg o rith m s to so lv e problem s in pow er system a n a l y s i s and c o n tr o l was d e m o n stra te d.

10 CHAPTER I INTRODUCTION D uring th e 's and e a r l y 's, the a re a s of p a t t e r n r e c o g n itio n and le a r n in g c o n tr o l system s were e x te n s iv e ly d e v elo p ed, y e t th e f i e l d i s s t i l l in i t s i n fan cy. V a rio u s fo rm al and h e u r i s t i c a lg o rith m s and th e o r e t i c a l i n t e r p r e t a t i o n s were d e sig n ed to t r y and so lv e a d iv e rs e s e t o f problem s in such d iv e r s e a re a s as c h a r a c t e r r e c o g n itio n, speech r e c o g n i ti o n, rem ote s e n s in g, w e a th e r f o r e c a s t i n g, crop a n a l y s i s, m edical d ia g n o s is, image p r o c e s s in g, o p tim a l d i g i t a l c o n tr o l of complex system s w ith tim e -v a ry in g p a ra m e te rs, e tc. By 1973* many s c i e n t i s t s and e n g in e e rs in t h i s f i e l d r e a l i z e d t h a t th e e x i s t i n g p a t t e r n r e c o g n itio n and l e a r n in g t o o l s a lre a d y f a r exceeded r e a l w orld a p p l i c a t i o n s. Hence, v a r io u s p r o f e s s i o n a l e n g in e e rin g and com puter s o c i e t i e s c re a te d th e F i r s t I n t e r n a t i o n a l J o i n t C onference on P a t t e r n R e c o g n itio n h e ld in W ashington, D. C. on O cto b er 30 and November 1, w hich was su p p o rte d by th e N a tio n a l S c ien c e F o u n d a tio n. There were fo u r hundred a t t e n d e e from o v e r s ix te e n c o u n tr i e s. The theme was " B rid g in g th e gap betw een th e o ry and im p le m e n ta tio n in p a t t e r n r e c o g n itio n and problem s in p a t t e r n r e c o g n itio n r e s e a r c h." In one o f th e c h a irm a n 's sum m aries was th e 1

11 s t a te m e n t, Ht h e r e i s a n eed t o move away from th e p r e o c c u p a tio n w ith d e v e lo p in g 'n e w ' a lg o r ith m s, to a pro b lem o r i e n t e d u sa g e o f a l r e a d y e x i s t i n g t o o l s, " w h ile a t th e same tim e " c o n tin u e d s u p p o r t i s needed to d e v e lo p b r o a d e r t h e o r i e s t h a t w ould e x te n d th e t o o l s and th e th e o r y in th e c o n te x t o f t h e i r r e le v a n c e to p r a c t i c a l p ro b le m s." [f 2[ * T h is c o n fe re n c e h as b een a s tim u lu s to a p p lie d r e s e a r c h in p a t t e r n r e c o g n i t i o n and l e a r n i n g sy s te m s w hich 1 f e e l w i l l g iv e f u t u r e t h e o r e t i c a l f o r m u la tio n s a se n se o f d i r e c t i o n. A rev ie w o f some g e n e r a l c o n c e p ts b e h in d p a t t e r n r e c o g n i t i o n and l e a r n i n g c o n t r o l sy ste m s a re p r e s e n te d in C h a p te r I I o f t h i s d i s s e r t a t i o n. F o r a d e t a i l e d e x p o s i t i o n o f th e te rm in o lo g y found i n th e l i t e r a t u r e on l e a r n in g c o n t r o l sy ste m s and an o r g a n i z a t i o n o f e x i s t i n g l e a r n i n g s t r u c t u r e s p r i o r to 1 970, I r e f e r th e r e a d e r to my M a s t e r 's t h e s i s [LacJ. The r e a d e r w i l l a ls o fir.d some e a r l y t e x t s [nj, rj u s e f u l. More r e c e n t t e x t s o f i n t e r e s t a re (mfj, [tg], b, [aj, Jcij. The p u rp o se o f t h i s d i s s e r t a t i o n r e s e a r c h was to a d a p t e x i s t i n g t o o l s o f p a t t e r n r e c o g n i t i o n and l e a r n i n g c o n tr o l sy stem s th e o r y so a s to a p p ly them to c e r t a i n p r a c t i c a l pow er sy ste m s p ro b le m s. I chose pow er sy stem s as a " t e s t i n g g round " b e ca u se ( to my b e s t know ledge) th e a c t u a l u se o f p a t t e r n r e c o g n i t i o n and l e a r n i n g th e o r y in power sy ste m a n a l y s i s and c o n t r o l h a s been a p p lie d o n ly R e f e re n c e s d e n o te d by b r a c k e t s a re l i s t e d in th e b i b l i o g ra p h y.

12 3 i n th r e e p a p e rs j V l ], [ i s j, ^BeaJ and a l s o b e c a u se th e c h a r a c t e r i s t i c s and c o m p le x ity o f pow er sy stem s show pro m ise i n m aking t h i s a p p ro a c h q u i t e u s e f u l. I t i s hoped t h a t t h i s d i s s e r t a t i o n w i l l e n c o u rag e e n g in e e r s ( i n c l u d i n g th e a u th o r ) to c o n tin u e r e s e a r c h in th e a p p l i c a t i o n of p a t t e r n r e c o g n i t i o n and l e a r n i n g c o n t r o l sy ste m s to p ro b lem s in pow er sy ste m s and o t h e r f i e l d s. In C h a p te r I I, some g e n e r a l c o n c e p ts o f p a t t e r n r e c o g n i ti o n and l e a r n i n g c o n t r o l sy ste m s p e r t i n e n t to t h i s d i s s e r t a t i o n a r e in tr o d u c e d a lo n g w ith an exam ple a p p l i c a t io n to a d i r e c t d i g i t a l c o n t r o l l e d p r o c e s s. The r e a s o n s f o r a p p ly in g l e a r n i n g a lg o r ith m s to power sy ste m s a re a ls o d is c u s s e d a lo n g w ith th e t h r e e p r e v io u s a p p l i c a t io n s. In C h a p te r I I I, th e p ro b lem o f n e a r - o p tim a l m egaw att lo a d - f r e q u e n c y c o n tr o l o f an in te r c o n n e c t e d power sy stem i s fo rm u la te d w ith an o n - l in e l e a r n i n g c o n t r o l l e r a s an a l t e r n a t i v e to fo rm a l o f f - l i n e " R i c a t t i e q u a tio n " ty p e o f f o r m u la tio n s efj, [fej, [RedJ. S t u d i e s a re made w ith and w ith o u t th e t r u e o p tim a l s o l u t i o n a s a " te a c h e r " in th e i n i t i a l l e a r n i n g s t a g e s. P l a n t p a ra m e te rs a r e v a r i e d to show th e o n - l in e a d a p t i v i t y and g e n e r a l i z a t i o n c a p a b i l i t y o f th e l e a r n i n g sy ste m, w hich i s n o t p o s s e s s e d by th e more fo rm a l a p p ro a c h, to y i e l d a n e a r - o p tim a l ( i. e., s a t i s f a c t o r y ) c o n t r o l. The c o n tr o l f u n c tio n s (o f th e s t a t e ) a re d e fin e d by a s e t o f h y p e r s u r f a c e s in r e g io n s o f s t a t e sp a c e.

13 4 These s u r f a c e s (most l i k e l y n o n lin e a r ) a re ap p ro x im ated by a sequence of p ie c e w is e l i n e a r h y p e rp la n e s each d e t e r mined by a le a r n in g p ro c e s s s i m i l a r to t h a t used in my MS t h e s i s bacj which was p a tte r n e d a f t e r W altz and Fu*s o n - lin e le a r n in g c o n tr o l system wfj and u s in g the method of sam ple s e t c o n s tr u c tio n d e v ise d by S e b e sty e n [se]. U n lik e th e "bang-bang" c o n tr o l (two v a lu e s ) used in my MS t h e s i s, th e c o n tr o l c o n s i s t s of a sequence of s te p f u n c tio n s (w ith a c o n s ta n t tim e i n t e r v a l ) whose v a lu e s a re c a l c u l a t e d by i n t e r p o l a t i o n m ethods su g g e ste d by Hammer [hlg]. In C h a p te r IV, th e c o n c e p t o f power system s e c u r i t y i s d e fin e d and e x i s t i n g m ethods f o r s e c u r i t y e v a lu a tio n a re d is c u s s e d. The sim p le pow er flow netw ork model and r e s u l t i n g s e t o f i n e q u a l i t y c o n s t r a i n t s f o r s e c u re o p e ra t i o n fo rm u la te d by K a lte n b a c h and Had ju a re p r e s e n te d Jkh]. They used an o f f - l i n e l i n e a r program m ing (sim p le x ) method to d e te rm in e an system s e c u r i t y. o p tim a l c o r r e c t i v e re s c h e d u le f o r power I used th e t o o ls o f p a t t e r n re c o g n itio n and s e a rc h as opposed to a l i n e a r program m ing approach to d e te rm in e c o r r e c t i v e a c tio n to b r in g a "n o rm al", b u t v u ln e r a b le, system in to a s e c u re o p e ra tin g s t a t e ( i. e., i f th e system i s s u b je c te d to a s e t of d e fin e d d i s t u r b a n c e s, would th e system rem ain in th e norm al o p e ra tin g s t a t e? ). In C h a p te r IV, o v e r a l l c o n c lu s io n s and recommend a tio n s f o r f u tu r e r e s e a rc h a re p re s e n te d.

14 CHAPTER I I PATTERN RECOGNITION AND LEARNING CONTROL (G en eral C o n cep ts) The p u rp o se of t h i s c h a p te r i s to d e fin e p a t t e r n r e c o g n itio n and l e a r n in g c o n tr o l, d is c u s s th e r e l a t i o n s h i p of th e s e c o n c e p ts to power system a n a l y s i s and c o n tr o l, and review th e works o f p re v io u s a u th o rs who have a p p lie d p a t t e r n r e c o g n itio n and le a r n in g th e o ry to power sy ste m s. Pattern Recognition The b a s ic co n ce p t o f p a t t e r n r e c o g n itio n i s a mapping o f a g iv en s e t o f " p a t t e r n s o r " f e a t u r e s " to a s e t o f " c a t e g o r i e s ". T his m apping i s u n iq u e, b u t in g e n e ra l i s n o t o n e -to -o n e i t h a t i s, s e v e r a l p a t t e r n s may be a s s o c i a te d w ith th e same c a te g o ry. Suppose t h a t th e s e t o f p a t t e r n s c o n ta in s handw r i t t e n l e t t e r s o f th e E n g lis h a lp h a b e t, such as p =\<l, a- #,<&; } The f i r s t fo u r elem en ts of th e p a t t e r n s e t P a re a s s o c i a te d w ith th e same u nique c a te g o ry, " th e E n g lis h l e t t e r A", which i s c o n ta in e d in th e s e t o f c a te g o r ie s _ f th e l e t t e r s o f th e E n g lis h A lphabet [in th e p ic a s t y l e type fo n t A s s o c ia te d w ith each p a t t e r n i s a s e t of num bers r e p r e s e n tin g th e f e a t u r e s o f th e p a t t e r n (such as s i z e, number of p o in ts o f in te r c o n n e c tio n o f l i n e s, number o f open sp a c e s, e t c. ). The human " e y e - b ra in " e a s i l y p erform s th e

15 6 m apping from P to C by r e c o g n iz in g th e e s s e n t i a l f e a t u r e s embedded w ith in th e p a t t e r n s. A p a t t e r n r e c o g n itio n m achine ( c a l l e d a p a t t e r n c l a s s i f i e r ) would c o n ta in a f e a t u r e e x t r a c t o r and a d e c is io n -m a k e r c a lle d th e c a t e - g o r l z e r. The f e a tu r e e x t r a c t i o n p ro c e s s i s o fte n th e most d i f f i c u l t to im plem ent. One method o f d e te rm in in g th e e s s e n t i a l f e a t u r e s would be to remove n o n - e s s e n ti a l ones. In th e h a n d w ritte n c h a r a c t e r exam ple, th e o v e r a l l s iz e o f th e c h a r a c t e r has no b e a r in g on th e c a te g o ry to w hich i t b e lo n g s. Hence th e f e a t u r e e x t r a c t o r could a d ju s t th e c h a r a c t e r s i z e s to some " s ta n d a rd " s i z e. D efine a p a t t e r n as a v e c t o r o f n r e a l num bers, A = (x1, x2, X^,... xn ) w hich r e p r e s e n ts a s e t of d a ta to be c l a s s i f i e d. A p a t t e r n c l a s s i f i e r i s th u s a d e v ic e which s o r t s p a t t e r n s, A, in to c a t e g o r i e s, u, as shown in f ig u r e 2.1. A in p u t p a t t e r n P a t t e r n C l a s s i f i e r u o u tp u t c a te g o ry F ig u re 2.1 P a t t e r n C l a s s i f i e r A ll e le m e n ts o f th e p a t t e r n, j, a re a p p lie d s im u lta n e o u s ly to th e c l a s s i f i e r. The o u tp u t, u, i s th e c o r r e c t c a te g o ry c o rre s p o n d in g to th e g iv en in p u t p a t t e r n and b e lo n g s to th e s e t o f p number of c a t e g o r i e s, c 2,..., Cp^ I f we assume t h a t th e e s s e n t i a l p r o p e r t i e s o f th e p a t t e r n ( x ^, x2,... xn ) a re l i n e a r l y in d e p e n d e n t, then we can l e t th e v e c t o r a g e n e ra te a sp ace X of in p u t p a t t e r n s. From a g e o m e tric v ie w p o in t, we can c o n s id e r th e p

16 7 p o s s ib le o u tp u t c a t e g o r i e s a s re g io n s o f X. The s u r f a c e s w hich s e p a r a te th e s e r e g io n s o r p o in t s e t s a r e c a l le d d e c is io n s u r f a c e s and a re i m p l i c i t l y d e fin e d in th e p a t t e r n c l a s s i f i e r by a s e t o f s c a l a r, s in g le - v a lu e d fu n c tio n s j f j (jt). f 2 0t)*... F ig u re 2.2 shows a g e o m e tric i n t e r p r e t a t i o n when n=2, p=^, and q=3 0 *- x^ 0 F ig u re 2.2 D e c isio n S u r f a c e s 0 We can c o n s id e r th e v e c t o r f u n c tio n X(l ) =...Iq (is)) as a means by w hich to map p a t t e r n s in to t h e i r p ro p e r o u tp u t c a te g o r y. I f X w ere n o t s in g le - v a lu e d (hence th e r e i s o n ly one s u r f a c e X=.Q ) th e r e m ight be more th a n one v a l i d o u tp u t " d e c i s i o n M f o r a g iv e n in p u t p a t t e r n jj. Hence th e m achine would have to make a g u ess w hich would d e f e a t th e p u rp o se o f th e d e c is io n s u r f a c e s. One may c o n s id e r th e a u to m a tic c o n tr o l problem as a p a t t e r n r e c o g n i ti o n problem by c h o o sin g th e s y s te m 's s t a t e sp ace X a s th e s e t o f p a t t e r n s and a s e t o f feed b ack g a in s o r c o n tr o l c h o ic e s a s th e c a te g o ry s e t U. The m apping o f X in to U could be chosen so as to s a t i s f y some g o a l o r index o f p e rfo rm an c e.

17 8 L e t c o n tr o l s i t u a t i o n s be d e fin e d as r e g io n s in X f o r w hich a s in g le c o n tr o l c h o ic e o r s e t o f feed b ack g a in s r e s u l t s in a s a t i s f a c t o r y p erfo rm an ce (may be s u b -o p tim a l) f o r a l l p o i n ts in t h a t r e g io n. C o n tro l s i t u a t i o n s may be d ecid ed a p r i o r i by p r e - g r id d in g th e e n t i r e space X in to p re d e te rm in e d "h s^ ts such a s h y p ercu b es J#len, cubesj o r th ey may be c o n s tr u c te d as c o n tr o l s i t u a t i o n s a re needed in th e p o r tio n o f X t h a t i s u se d. By p a r t i t i o n i n g X in to c o n tr o l s i t u a t i o n s, we a re assum ing t h a t p o in ts w ith in a sm a ll enough n eig h b o rh o o d o f p o in t X, which has th e known s a t i s f a c t o r y c o n tr o l c^ U, sh o u ld have th e same o r a " s i m il a r " c o n tr o l c h o ic e. S u c c e s s f u l c o n tr o l s i t u a t i o n s a re n o t a r b i t r a r y b ecau se we c an n o t be a ssu re d t h a t a l l p o in ts w ith in an a r b i t r a r y r e g io n w i l l have a c o n tr o l t h a t r e s u l t s in a s a t i s f a c t o r y p e rfo rm an c e. C o n tro l s i t u a t i o n s s im p lif y o p tim iz a tio n problem s in t h a t th e y l o c a l i z e th e problem to re g io n s in X and th u s reduce memory and tim e re q u ire m e n ts o f c o m p u tatio n c o n s i d e r a b ly. L e a rn in g C o n tro l System s To im prove upon a p a t t e r n c a t e g o r i z e r, memory i s u p d a te d. T h at i s, th e c o n tr o l c h o ic e t h a t h as g iv e n th e most s u c c e s s f u l r e s u l t f o r each p a r t i c u l a r c o n tr o l s i t u a t i o n i s s t o r e d. T h is sy stem becomes a le a r n in g system i f we d e v is e a le a r n in g h e u r i s t i c ; t h a t i s, a m ethod so t h a t th e system can g e n e r a liz e on p a s t e x p e rie n c e by u sin g d e c is io n m ethods w hich p o s i t i v e l y r e i n f o r c e s u c c e s s f u l d e c is io n s and n e g a tiv e ly r e i n f o r c e u n s u c c e s s fu l d e c is io n s,

18 9 a s in p s y c h o l o g ic a l m odels o f l e a r n i n g [bm]. I t sh o u ld be em p h asized t h a t th e l e a r n i n g sy ste m m ust be a b le to g e n e r a l i z e by c o p in g w ith s t a t e s t h a t i t h as n o t y e t e x p e r ie n c e d, b u t a r e s i m i l a r ( in some way) to i t s p a s t e x p e r i e n c e Note t h a t in a p a t t e r n c l a s s i f i e r w ith o u t a l e a r n i n g h e u r i s t i c, th e d e c i s i o n s u r f a c e s m ust be known a p r i o r i w h e rea s in th e l e a r n i n g sy stem th e y a re " le a r n e d " j i.e., th e s u c c e s s f u l c o n t r o l c h o ic e f o r each c o n tr o l s i t u a t i o n i s p o s i t i v e l y r e i n f o r c e d and th e u n s u c c e s s f u l c h o ic e s a re n e g a t i v e l y r e i n f o r c e d. I n i t i a l l y, we may make a c ru d e s e a rc h f o r a c o n t r o l c h o ic e f o r e a c h s i t u a t i o n and a f t e r s u f f i c i e n t r e i n f o r c e m ent, o b t a i n th e c o r r e c t c o n t r o l c h o ic e f o r t h a t s i t u a t i o n. A n o th e r m ethod i s to have a " te a c h e r " c a l l e d th e t r a i n i n g s e t w hich i s a r e p r e s e n t a t i v e s e t o f p a t t e r n s a lo n g w ith t h e i r c o r r e c t c o n tr o l c h o ic e s from w hich th e l e a r n i n g c o n t r o l sy ste m can g e n e r a l i z e. The i n i t i a l p r o b a b i l i t y o f u s in g a c o n t r o l from t h i s t r a i n i n g s e t w ould o f c o u rs e be e q u a l to one. I f th e sy ste m p a ra m e te rs ch ange, new c o n t r o l s w ould be s e l e c t e d b u t a t f i r s t w ould be " s i m i l a r " to th e t r a i n i n g s e t. I f an a d e q u a te t r a i n i n g s e t i s n o t a v a i l a b l e ( p o s s ib l y due to a la c k o f u n d e rs ta n d in g o f th e sy ste m o p e r a t i o n ), s e a r c h m ethods m ust be used i n s t e a d of th e t e a c h e r. In c e r t a i n c a s e s, one may d e s i r e to u se a c o m b in a tio n o f t e a c h e r and s e a r c h. O fte n th e t r a i n i n g s e t i s ta k e n from a d a ta

19 10 acquisition device that monitors the performance of a Bystem that is being controlled by a human operator. When a sufficient number of control situations have been constructed and the reinforcement learning algorithms have sufficiently converged n], the learning system "takes over". This process is analogous to teaching a child the alphabet and from this "training set", he learns to easily generalize to large sets of handwritten characters (such as the set P on page 5) he hab not previously experienced, but which possess features similar to the training set. A le a r n in g system i s a d a p tiv e in th e se n se t h a t i t has a s h o r t- te r m memory f o r f a c i l i t a t i n g th e u p d a tin g of th e p r o b a b i l i t i e s o f c h o o sin g c o n tr o l s w ith in a c o n tr o l s i t u a t i o n based on s h o r t- te r m in fo rm a tio n t i.e., " d id th e l a s t c o n tr o l c h o ic e c o rre sp o n d to an im provem ent in system p erfo rm an ce?" An a d a p tiv e system i s d e sig n ed to m odify i t s e l f when p r e s e n te d w ith a new e n v iro n m e n ta l s i t u a t i o n. A lo n g -te rm memory i s n e c e s s a r y to s t o r e the b e s t l a s t computed v a lu e s f o r th e p r o b a b i l i t i e s when th e system le a v e s th e c o n tr o l s i t u a t i o n in o rd e r t h a t th e y may be r e c a lle d f o r f u t u r e t r i a l s when th e same c o n tr o l s i t u a t i o n a g a in o c c u rs. The above c o n c e p ts a re d i r e c t l y an alo g o u s to p s y c h o lo g ic a l m odels of le a r n in g a s d e t a i l e d by Bush and M o s te lle r [ H In f a c t, some o f th e le a r n in g h e u r i s t i c s used f o r le a r n in g c o n tr o l system s p a r a l l e l th e s e m odels. An a d a p tiv e system i s d e sig n ed to m odify i t s e l f when

20 11 p r e s e n te d w ith a new s i t u a t i o n such t h a t i t s perfo rm an ce i s o p tim ize d to t h i s new s i t u a t i o n i f th e changes in th e environm ent a re slow enough to e n a b le a s e a rc h f o r th e optimum. A le a r n in g c o n tr o l system may be c o n s id e re d to re c o g n iz e f a m i l i a r p a t t e r n s in a new e n v iro n m e n ta l s i t u a t i o n and based on p a s t e x p e rie n c e, m odify i t s s t r u c t u r e so as to r e a c t in an optimum m anner. Thus a le a r n in g system encom passes th e f u n c tio n s o f p a t t e r n r e c o g n itio n, memory, and a d a p tiv e m o d if ic a tio n. A le a r n in g c o n tr o l system becomes j u s t i f i a b l e when p la n t p a ra m e te rs o r th e environm ent v a ry so f a s t t h a t a p u re ly a d a p tiv e sy stem can n o t m a in ta in a near-optim um p e rfo rm an c e. S in c e th e le a r n in g sy stem a c q u ir e s more in fo rm a tio n a s tim e p r o g r e s s e s, t h e o r e t i c a l l y, i t can r e a c t p r o p e r ly to more r a p id p la n t- e n v ir o n m e n ta l ch an g es. A lso, th e l e a r n in g sy stem sh o u ld approach th e optimum more c lo s e ly th an a p u re ly a d a p tiv e system b ecause i t p r o p e r ly u t i l i z e s s to r e d e x p e rie n c e. A le a r n in g system has th e d is a d v a n ta g e o f b e in g more complex th a n an a d a p tiv e sy stem, and a s t a b i l i t y problem e x i s t s in a le a r n in g system when attem p s a re made to speed up th e l e a r n in g p ro c e s s. A le a r n in g sy stem i s a b le to g e n e r a liz e by e x t r a p o l a t i n g th e s to r e d c o n t r o l l e r param e t e r s to c o v e r new s i t u a t i o n s, b u t an e x tr a p o la t io n beyond a " re a s o n a b le " range may r e s u l t in a n o n -o p tim al c o n t r o l l e r and may have s e v e re consequences in a p r a c t i c a l s i t u a t i o n. The g r e a t e r th e accum ulated in fo rm a tio n ab o u t th e system,

21 12 th e b e t t e r a re th e e x t r a p o l a t i o n s. In a s i m i l a r m anner, th e human "m achine" can make e rro n e o u s d e c is io n s as a r e s u l t of g ro s s g e n e r a l i z a t i o n s. These problem s have been c o n tr o l le d by a d j u s t i n g p a ra m e te rs in the le a r n in g a lg o rith m s (such as s iz e o f c o n tr o l s i t u a t i o n s, le a r n in g r a t e, etc.) to fit th e system b e in g c o n tr o l le d. In summary, we may say t h a t a le a r n in g system i s a c o n tr o l sy stem which m o d ifie s i t s c o n tr o l p a ra m e te rs to m a in ta in and im prove th e o v e r a l l system p erform ance based on i t s p a s t e x p e rie n c e even i f fac ed w ith u n p r e d ic ta b le e n v iro n m e n ta l ch an g es. T h is d e f i n i t i o n i s p a r a l l e l to t h a t used in th e l i t e r a t u r e [say], A g e n e ra l s t r u c t u r e f o r a l e a r n in g c o n tro l system i s p ic t u r e d in f ig u r e 2.3. J Memory -T V G o a l C i r c u i t P a t t e r n C l a s s i f i e r R e in fo rcem en t L e a rn in g Network F ig u re 2.3 Environm ent n z z P ro c e s s L e a rn in g C o n tro l System O utput

22 13 L e a rn in g system s have been d e v is e d to c o n tr o l p ro c e s s e s c u r r e n t l y u n d e r D ir e c t D i g i t a l C o n tro l (DDC). L e a rn in g c o n t r o l l e r s a re p a r t i c u l a r l y v a lu a b le in th e c o n tr o l o f complex p ro c e s s e s s in c e t y p i c a l p ro c e s s tim e c o n s ta n ts a re g r e a t e r th a n one se co n d, a llo w in g more th a n a d e q u a te tim e f o r t r a i n i n g in th e l e a r n in g p h ase and th e f a c t t h a t many i n d u s t r i a l p ro c e s s e s a re n o t w e ll u n d e rsto o d and s u i t a b l e m odels t h a t can be a n a ly z e d may be in a d e q u a te o r u n a v a ila b le, In one o f th e e a r l i e s t exam p les, G arden h a s a p p lie d a l e a r n in g c o n t r o l l e r to a DDC sy stem to a c t u a t e v a lv e s in a p r o c e s s, w here c o n v e n tio n a l fee d b ac k c o n tr o l o f c e r t a i n ty p e s o f th e s e v a lv e a c t u a t o r s i s d i f f i c u l t becau se o f t h e i r v a ry in g c h a r a c t e r i s t i c s. G arden demons t r a t e d t h a t a l e a r n i n g sy stem w hich le a r n s th e a c t u a t o r s i g n a l s b ased on th e a c t u a t o r 's v a ry in g c h a r a c t e r i s t i c s (which a re unknown a p r i o r i ) p ro v id e s a c o n tin u o u s o n - lin e a d a p ta tio n to th o se c h a r a c t e r i s t i c s in c lu d in g th o s e t h a t c an n o t be m easured (a p o s s ib le c o n c lu s io n could be t h a t th e le a r n in g system a ls o p o ss e s se d some o f th e c h a r a c t e r i s t i c s o f a 3 t a t e o b s e r v e r ). F u rth e rm o re, th e le a r n in g c o n t r o l l e r p ro v id e d b e t t e r p erfo rm an ce th a n t h a t a v a i la b l e from th e c o n v e n tio n a l c o n tr o l t h a t was based on an o v e r - s im p lif ie d m odel. An added b e n e f i t o b ta in e d by u s in g th e le a r n in g sy stem was a re d u c tio n in th e m onetary c o s t, b ecau se l e s s s t a b l e d e v ic e s were re q u ire d s in c e changing c h a r a c t e r i s t i c s w ere c o n tin u o u s ly le a rn e d. An im proved r e l i a b i l i t y was s a id to r e s u l t because of

23 hardw are s i m p l i f i c a t i o n, and a d a p ta tio n and d ia g n o s is o f d e t e r i o r a t i n g p a r t s. The p r i n c i p l e s o f p a t t e r n r e c o g n itio n and le a r n in g system s have been a p p lie d to d iv e r s e a re a s jc ij o v er th e l a s t f i f t e e n y e a r s such a s space v e h ic le c o n tr o l jlflen2j, com m unications, m edicine lj, a u to m a tic re a d in g m achines, etc. The r e s u l t s of th e s e d iv e r s e s t u d i e s in t h i s r e l a t i v e l y new f i e l d a re q u i t e e n c o u ra g in g. P a t t e r n Re c o g n i t i on. L e a rn in g and Power System s Power sy stem s a re c h a r a c t e r i z e d by a la r g e number o f v a r i a b l e s, u n c e r t a i n t i e s in th e env iro n m en t (lo a d, w e a th e r, ecconom ics, etc.), u n c e r t a i n t i e s in th e sy stem p a ra m e te rs, and a re q u ire m e n t f o r an e f f i c i e n t o n - lin e c o n tr o l. A le a r n in g c o n t r o l l e r, by d e f i n i t i o n, a d a p ts to a new enviro nm ent and to changes in th e system p a ra m e te rs and im proves th e b e h a v io u r o f th e system based on a s e t of g o a ls and th e c o n t r o l l e r 's p a s t p e rfo rm an c e. Fixed c o n t r o l l e r s b ased on a presum ed e x a c t knowledge o f p l a n t param e t e r s must be r e c a l c u l a t e d O f f - l in e to accommodate u n p re d i c t a b l e c h an g e s. The le a r n in g c o n t r o l l e r u se s i t s p a s t e x p e rie n c e to d e te rm in e o n - lin e (w ith m inim al co m p u tatio n compared w ith an o n - l in e o p tim a l c o n t r o l l e r ) a n e a r o p t i mal c o n tr o l f o r th e c u r r e n t s i t u a t i o n. P a t t e r n r e c o g n itio n was s u g g e s te d as a p o s s ib le alternative method o f d e te rm in in g th e Bteady-state s e c u r i t y o f power system s by T. E. DyLiacco In 1973, Pang, K oivo, and E l-a b iad s u c e s s f u l l y a p p lie d

24 15 p a t t e r n r e c o g n itio n and l e a r n i n g th e o ry to s t e a d y - s t a t e H [p2] and t r a n s i e n t M s e c u r i t y e v a lu a tio n o f pow er sy ste m s. Pang s e l e c te d a s e t o f f e a t u r e s (zj,z2 * *»zn ) - - a s u b s e t o f th e s t a t e o f th e sy ste m t h a t seemed to be e s s e n t i a l in d e te rm in in g w h e th er th e system was se c u re (sy stem would rem ain in th e norm al o p e ra tin g s t a t e when s u b je c te d to a s p e c i f i e d s e t of d i s t u r b a n c e s ). A d e c is io n f u n c t i o n, s(a) = wo + niw. z i i i= l where th e w e ig h tin g c o e f f i c i e n t s,, were d e te rm in e d by a le a r n in g (re in fo rc e m e n t) scheme wa3 c a l c u l a t e d such t h a t S(i)>0 i f was a s e c u re p a t t e r n and S ( i ) < 0 i f i was i n s e c u r e. Second o r d e r p o ly n o m in a ls were a ls o used as d e c is io n f u n c tio n s to b e t t e r a p p ro x im ate th e p ro b a b le "c u rv e d d e c is io n s u r f a c e. A p a tt e r n - r e c o g n i z i n g s te a d y s t a t e s e c u r i t y e v a lu a to r was t e s t e d o n - lin e on a s im u la tio n o f th e CIGRE 225 kv sy stem. The t r a i n i n g s e t c o n s is te d of 179 p a t t e r n s o f w hich 75 were s e c u re and 104 in s e c u r e. The t r a i n i n g s e t was used to d e te rm in e th e w e ig h ts,. T his t e s t showed t h a t " a c c u r a te " ( l e s s th an 10# w o rst e r r o r ) s e c u r i t y f u n c tio n s a re r e a l i z a b l e by p a t t e r n r e c o g n itio n te c h n iq u e s. Dynamic s e c u r i t y e v a lu a tio n ( r e l a t i n g to th e system re sp o n se in the o r d e r of a few m in u te s) i s a more complex problem and p a t t e r n r e c o g n itio n has n o t y e t been s u c c e s s f u l l y a p p lie d. In d e v e lo p in g th e dynamic s e c u r i t y

25 16 e v a l u a t o r, one m ust c o n s id e r th e s t o c h a s t i c n a tu r e o f load f o r e c a s t i n g. Hence a s t a t i s t i c a l p a t t e r n r e c o g n itio n ap p ro ach i s n e c e s s a r y compared to th e d e t e r m i n i s t i c one used by Pang in s t e a d y - s t a t e and t r a n s i e n t ( s h o r t- te r m ) s e c u r i t y e v a lu a tio n. The n a tu r e o f th e system te n d s to make a s t a t i s t i c a l app ro ach d i f f i c u l t. Though th e f o r e c a s t of th e lo a d s can be assumed n o rm a lly d i s t r i b u t e d, th e o th e r v a r i a b l e s such as v o lta g e m ag n itu d es and a n g le s a re coupled to th e lo a d s v i a th e power e q u a tio n s w hich a re h ig h ly n o n lin e a r ( h y s t e r e s i s effects, etc.) r e s u l t i n g in "abnorm al" d i s t r i b u t i o n f o r th e system s t a t e as a s t o c h a s t i c p r o c e s s. The r e q u ir e d t r a i n i n g s e t would be cons i d e r a b l y l a r g e r th a n th e s t e a d y - s t a t e s e c u r i t y e v a lu a tio n (tim e f a c t o r s a re now in v o lv e d ) and more d i f f i c u l t to o b ta in b ecau se th e fre q u e n c y of o c c u rre n c e of a d y n a m ica lly in s e c u r e ( a l e r t ) s t a t e in a w e ll-d e s ig n e d system i s v e ry sm a ll compared to t h a t of th e se c u re ( p r e f e r r e d ) s t a t e. F or a d eq u a te t r a i n i n g th e r e sh o u ld be an e q u a l number of sam ples f o r each s e c u r i t y c o n d itio n c o rre sp o n d in g to a r e p r e s e n t a t i v e sam ple o f th e sy stem b e h a v io u r. S im u la tio n s t u d i e s o f l i n e a r m odels w ere used to o b ta in t r a i n i n g sam ples f o r t r a n s i e n t s e c u r i t y, b u t would n o t be a d e q u a te f o r th e dynamic re s p o n se on th e o rd e r o f s e v e r a l m in u tes (the e r r o r would grow and d iv e rg e from th e tr u e system o p e r a t i o n ). L. A. B e a ttie a ls o i n v e s ti g a t e d th e use of p a t t e r n r e c o g n itio n f o r power system s e c u r i t y e v a lu a tio n Beal.

26 17 B e a t t i e u sed o v e r a l l a c c u ra c y o f c l a s s i f i c a t i o n a s a p e r form ance c r i t e r i o n and a g a u s s ia n a s w e ll a s d e t e r m i n i s t i c p a t t e r n c l a s s i f i c a t i o n schem es f o r s t e a d y - s t a t e s e c u r i t y. He deemed u n a c c e p ta b le a t e r m i n a l a c c u ra c y l e s s th a n 95^* B e a t t i e found t h a t w ith h i g h e r d im e n sio n sy s te m s (sim u lated), an o v e r a l l a c c u ra c y b e t t e r th a n 95?^ was n o t f e a s i b l e w ith o u t h i g h e r - o r d e r a p p ro x im a tio n s t h a t w ould r e q u i r e to o much co m p u ter B to ra g e. S in c e o f f - l i n e te c h n iq u e s a r e e x a c t in s e c u r i t y a s s e s s m e n t (assu m in g th e d a ta i s valid), B e a t t i e c o n c lu d e d t h a t " o v e r a l l h ig h a c c u ra c y i s n o t a r e a s o n a b le g o a l f o r power sy ste m s e c u r i t y a s s e s s m e n t u s in g p a t t e r n r e c o g n i t i o n tec h n iq u e s'* [Bea]. H is s t r i n g e n t re q u ire m e n t i s b a sed on th e n e c e s s i t y f o r e x a c t s e c u r i t y e v a l u a t i o n in a r e a l power sy ste m. But th e o f f - l i n e t e c h n iq u e s, w h ile a c c u r a t e w ith r e s p e c t to a s e t o f p a s t d a t a, may be i n a c c u r a t e i f sy ste m ch an g es o c c u r d u r in g th e c o m p u ta tio n tim e. S ystem o p e r a t o r s have a n eed f o r o n - l i n e s e c u r i t y m o n ito rin g in e v e n t o f e m e rg e n c ie s. P a t t e r n r e c o g n i t i o n i s an a tte m p t o f an o n - l i n e ( w ith in sy ste m tim e c o n s t a n t s ) te c h n iq u e. One w ould n o t t o t a l l y r e l y on a p a t t e r n r e c o g n i z i n g a lg o r ith m (a lth o u g h i t " l e a r n s " from m i s ta k e s ), b u t w ould u se a c o m b in a tio n o f human o p e r a t o r, o n - l i n e p a t t e r n r e c o g n i t i o n, and o f f - l i n e te c h n iq u e s. I t may be true, how ever, t h a t th e a c c u ra c y o f th e p a t t e r n r e c o g n i t i o n te c h n iq u e - - w i th known l e a r n in g a lg o rith m s --m a y be u n a c c e p ta b le in a c t u a l, h ig h d im e n sio n

27 18 system s w ith s t i f f re q u ire m e n ts and p o s s ib ly t h a t a p u re ly o n - lin e te c h n iq u e may n o t be f e a s i b l e f o r pow er system s e c u r i t y - - e s p e c i a l l y dynam ic e v a lu a tio n p ro b lem s. The only o th e r a p p l i c a t i o n o f p a t t e r n r e c o g n itio n and le a r n in g c o n tr o l system s t h a t I co u ld fin d th ro u g h an e x te n s iv e s e a rc h of m ajor a p p lic a b le jo u rn a ls* e n g in e e rin g in d e x e s, a b s t r a c t s, b i b l i o g r a p h i e s, etc. was th e work of Lee and Schweppe [i^j. Lee and Schweppe d e v is e d a method of u s in g p a t t e r n r e c o g n itio n to red u ce th e c o m p le x ity o f a h e a v ily in te r c o n n e c te d power system a s an a id to t r a n s i e n t s t a b i l i t y s t u d i e s. H e a v ily in te r c o n n e c te d system s can n o t be t r e a t e d as i s o l a t e d sy ste m s. Due to th e i n o r d in a te dim ension o f th e e n t i r e sy stem, i t i s n o t ecco n o m ical o r m a th e m a tic a lly d e s i r a b l e to r e p r e s e n t th e e n t i r e system in d e t a i l in o rd e r to conduct s t a b i l i t y s t u d i e s. Sim ple e q u iv a le n ts " t h a t model th e t r a n s i e n t re sp o n se o f g e n e r a to r s f a r removed from th e " c e n te r o f th e system " a s changes o c cu r in th e c e n te r would h e lp in sy stem r e d u c tio n. W ith in an e q u iv a le n t, i t i s u s u a lly assumed t h a t th e g e n e r a to r s a re c o h e re n t [ca]. The boundary betw een the " e q u iv a liz e d " a re a s and th e a re a m odeled in d e t a i l where th e f a u l t or d is tu r b a n c e o c c u rs had p r e v io u s ly been l e f t up to th e judgm ent of system p la n n e r s. Lee and Schweppe d e v ise d a method (and w ro te two com puter program s) in c o r p o r a tin g d is ta n c e m easures to d e te rm in e th e s e b o u n d a rie s and used p a t t e r n r e c o g n itio n to i d e n t i f y c o h e re n t g e n e r a to r s w ith in

28 19 th e same d ista n c e measure. to aid in s t a b i l i t y s tu d ie s. This inform ation was then used T heir techniques were ap p lied to a sample Bystem c o n s is tin g of 128 buses, 253 l in e s, and 31 g e n e ra to rs. This sample study dem onstrated the e ffe c tiv e n e s s of the p a tte r n re c o g n itio n approach to th is problem.

29 CHAPTER I I I APPLICATION OF A LEARNING CONTROLLER TO LOAD FREQUENCY CONTROL OF AN INTERCONNECTED POWER SYSTEM The e l e c t r i c u t i l i t i e s of the United S ta te s and some Canadian are as are grouped in to fiv e major pools some of which are a lso in te rco n n e cte d. Other c o u n trie s such as G reat B rita in [krij have s im ila r types of in te rco n n e cte d power a re a s. Each in terco n n ected pool or group is divided in to a number of "co n tro l a re a s each of which o p erates more or le s s independently of the rem ainder of the in te rc o n n e c tio n. A c o n tro l area may be p a rt of, an e n tir e, or a group of u t i l i t y companies. Each c o n tro l area has the r e s p o n s ib ility of a d ju s tin g i t s own g en eratio n in response to i t s own load changes (d istu rb a n c e s), but must m aintain scheduled in terch an g es over i n t e r - t i e s w ith o th er c o n tro l areas in the i n t e r connection cohnj. Great B rita in has a N ational C ontrol Center and a c e n tra l computer i n s t a l l a t i o n along w ith d iv erse in stru m e n tatio n and communication networks to a s s i s t in co o rd in atio n of c o n tro l between areas kn]. Power g en eratio n and flow in the in terco n n ected system must be c o n tro lle d so as to generate s u f f i c i e n t power to supply the t o t a l load of the in te rc o n n e c tio n with 20

30 21 optimum economy w ithout lo ss of the s e c u rity of the sta b le s ta te of the system or s ig n if ic a n t changes in frequency or v o lta g e s. An in crease in load (d istu rb an c e) r e s u lts in a decrease in system speed or frequency. Most g en erato rs are equipped w ith speed governors to d e te c t the frequency changes and a d ju s t g e n e ra to r output to minimize f u r th e r changes in frequency. C h a r a c te r is tic a lly, th is frequency change (uncorrected) is about.01 cycle fo r each 0.2# change in system load [cohnj. The "area net interchange t i e - l i n e b ia s c o n tro l" is a control method which causes each a r e a s interchange of power schedule to be a fu n ctio n of system frequency. When the system is in i t s normal o p era tin g s t a te (A F=0, e t c. ), the area net interchanges w ill m aintain the normal schedule. A decrease in frequency in d ic a te s th a t one or more areas are experiencing a lo ss of g en eratio n and a l l the o th er a re as w ill a l t e r t h e i r interchange schedules to d e liv e r more net outgoing power (from t h e i r re se rv e s) to a s s i s t the d istu rb ed a re as. I f the frequency is in creased, the opposite occurs. One or more areas have too much g en eratio n and the load is shared w ith the d istu rb ed a re a s. C la s sic a l c o n tro l methods have been used to implement c o n tro ls, but as power ra tin g s and i n e r t i a co n stan ts in crease markedly in fu tu re g en eratin g s ta tio n s, the c la s s ic a l methods may not y ie ld a secure operatin g system w ith adequate s t a b i l i t y margins or the most optimum

31 22 perform ance. q u ite cle arly * E lgerd s ta te d the need f o r new approaches "C onsidering the many inadequacies th a t c h a ra c te riz e the c la s s ic a l design approach and co n sid erin g the complexity of the c o n tro l problem, i t is su p risin g how w ell our in terco n n ected systems do fu n c tio n." [^lgj Elgerd fu rth e r p re se n ts o p e ra tin g problems th a t have occurred r e s u ltin g from the inadequacies of the p re se n t design procedures and models. The need fo r design improvements is exem plified by Elgerd and Fosha s a p p lic a tio n of modem optim al c o n tro l theory to the load frequency (PF) c o n tro l problem. They proved th a t a b e t t e r response and w ider s t a b i l i t y margins are o b tain ab le by other t i e - l i n e b ias s e ttin g s than th a t c u rre n tly in widespread use. [fej The o b ject of t h is ch ap ter is to develop a le a rn in g c o n tr o lle r as an a lte r n a tiv e to the c o n tro l of the load frequency (PF) problem. Although a lea rn in g algorithm is applied to a lin e a r model of a tw o-area system, the r e s u lts in d ic a te the f e a s i b i l t y of the method. Much more o n -lin e study and work w ill have to be done before implem entation of any modern c o n tr o lle r can be made to a ctu al power system s. State-M odel and Optimal Control Elgerd EFj chose the follow ing s ta te v a ria b le s fo r the lin e a riz e d model of a sin g le area of a m u ltiarea system.

32 23 = j A P dt in te g r a l of t i e - l i n e power d ev iatio n = I A F dt in te g r a l of frequency d e v ia tio n J (d e v ia tio n in phase angle/2tt) x 3 = A F frequency d e v ia tio n x^ = A P incremental change in turbine generator output x, = A P increm ental change in governor ^ valve p o s itio n The system is subjected to a step load disturbance v ~ A P d and is c o n tro lle d by varying the speed changer p o s itio n u = A P. r c For a tw o-area in terco n n ected system, th ere would be nine s t a te v a ria b le s, since AP^ = ~a\2^k^2 * That the t i e - l i n e power d e v ia tio n of the f i r s t area would be p ro p o rtio n a l to the t i e - l i n e power d ev ia tio n of the second area by the constant a ^ th a t depends upon the in terco n n ectio n. There would be two c o n tro ls. The problem is to determine controls for each area so as to sa tisfy some system response c rite ria. This is centralized control. Another approach is to optimize system response by a control action only in the area con* fronted with the disturbance. Minimum requirements for response to a step d istu r bance set forth by the North American Power Systems Interconnection Committee (NAPSIC) arei 1. The steady-state frequency deviation must be zero, The steady-state change in the tie -lin e newer must, be zero.

33 2k 3. The frequency deviation should not exceed Hz during the transient period. k. The time error (integral of frequency deviation) should be less than or equal to +3 seconds. Based on the above c rite ria and state model, Elgerd used well-known methods of v aria tio n a l calculus to solve for the matrix of feedback gains K such that u - K x would result in an optimal control (^Fe. The K matrix is found by solving for the steady-state solution (backwards in time) of the matrix d iffe r e n tia l equation known as the R icatti equation (in fin ite time problem) [af]. The sta te model of Elgerd was reformulate d by Heddoch, cl. al. jrcuil j. Reddoch chose the tie -lin e power interchange AP and the integral of the area con- / trol error (IACE) as state variables instead of A P dt and A F dt. The area control error (ACE) is given by ACE =- AP + BAF where R is the "frequency bias" specified by NAPS1C. Thus an attempt to minimize the state TACK would directly minimize the total ACE deviations over the tran sitn t period and would guarantee a "reset action" of the ACE. Based on a reformulated set of cost functionals, Reddoch also determined an optimum controller u K x for this state model. The resu lts indicated a better transient response (less overshoot over a shorter sett,ling time) and the control was capable of forcing the ACE to zero in the a ssistin g area while maintaining the proper ratio between frequency and power deviation

34 25 set by the power industry. Elgerd's model did not have these c ap a b ilities. Pattern Recognition and Learning Application The feedback gain matrix is calculated o ff-lin e and is, of course, dependent upon an accurate knowledge of the system parameters and must be recalculated o ff-lin e if the basic parameters of the interconnected system are perturbed. The "system parameters" used to calculate the "optimal control'* are based on a linearized model of a basically non-linear system. If the deviations, result, in "large signal dynamics," the governor valve rate and/or position lim its may be reached and the "small signal" linearized model may become inadequate in determining the optimal control. A learning control system would depend upon a measurement, of the sta te and not on an exact knowledge of the system parameters or the system function; i.e., the learning controller would implement a control based on the state of the actual system and not on the state of a linearized model. I propose that an approach via learning systems would result, in an on-line controller that would be adaptive to changes in power area parameters and would modify the controller in the best manner based on its past experience. This is an a 1tornative control method which would be of most value when used in conjunction with the o ff-line "optimal" controller1 as a teacher. The feedback gains

35 26 could be given to the learning controller in the i n it i a l stage. When su fficien t" learning has occurred, the learning system would enter the "operate phase" and take over control of the system. When changes occur in the system that would invalidate the teacher's in itia l instructions, the learning system would s t i l l control the system on-line (due to minimum complexity of computation in the operate phase) while the teacher must identify and measure the perturbed system parameters and recalculate the new optimal gain matrix o ff-lin e. An efficient, on-line controller may help keep the system in a near optimal secure operating state even if an unexpected contingency should occur. The time saved in determining the best control for the new situ atio n could prevent disastrous consequences. A periodic interaction between the learning controller, the "optimal" teacher, and the human operators (supervisors) would seem to be a good scheme. The in te r play between these three possible controllers and goal monitors would be quite in terestin g. Perhaps another learning system could be devised at a higher level to observe this interplay and "optimize" the coordination. Two-Area S tat e Model The best way to te s t any c o n tro ller is to implement it to the actual, real world, system. In the absence of the,1 a v a ila b ility of such a system, one tests his control algorithm on a simulation. Often the simulation is.

36 2? o v e rs im p lifie d, but n o n eth eless i t serves as a v e h ic le fo r te s tin g the f e a s i b i l i t y of the c o n tro l method. The follow ing tw o-area s ta te model and cost fu n ctio n form ulated by Reddoch was used fo r t e s ti n g the le a rn in g system. For a comprehensive d e riv a tio n and j u s t i f i c a t i o n of th is model and cost fu n c tio n a ls, the read er is re fe rre d to Dr. Reddoch's PhD d is s e r t a t io n jredj. The values of the system param eters are the same as th a t used by Elgerd ^FEj. 2] S ta te Dynamicsi v, where A F X A P gl A p v1 A f 2 A p g 2 A P v2 A p 12 ia c e 1 ia c e The power q u a n titie s are expressed in p e r-u n it of the ra te d power of th a t a r e a, Pri Control Vector* _ u = V [A p c r _U2 A P CZ_ (speed changer p o s iti on v a ria tio n ) Tie-line power flowi ^ ptiel = ^ Pl; A P + - o = -a. AP, 0 tie Area Control E r r o r i - p i. A p 12 r2 = AP*, t i e l + B1 ^ F1 y2 = ^ Ptie 2 + B^AFn where is the area " frequency-bias** sp e c ifie d by in d u stry stan d ard s. Cost F unctional (shown fo r m-area model)i J - / m, \ ACEi2 + IACEi 2 + RGC2 + u / i~l d t where RGC = (A P V> - ^ Pg i ^ Tgi 13 the rate generation

37 28 change, the c o e f f ic ie n ts << ^ a r e w eighting param eters, and T ^ is a system param eter. S u b s titu tin g the s ta te v a ria b le n o ta tio n fo r the q u a n titie s ACE^, IACE^ A P., and A P., we o btain the eq u iv alen t fo rm u la tio n i V1 it,00 J = 41 it + utn dt The r e s u lta n t m atrix,, is p o s itiv e d e f in ite, but has "o ff diagnonal" elem ents. The m atrix A, and can be determined from the s ta te equations* *1 = X1 + X2 " x? V1. 1 _1_ 2 Tgl X2 + Tgl X3 4 = _ 1 v - - v + 1 u 3 Tv1R1 X1 Tvl 3 + Tvl U1, _ ^2 ^ 1_ A f l 2 1_ **4- ~ M2 M2 X5 M2 X7 M2 V2 ^6 T R T x6 + T U2 o v2 2 * v2 v2 ^ x? _ T12 X1 T12 *8 = B1 X1 + x7 ~ ^2 x4 ~ a12 x7

38 29 The cost m atrix is given byi o<1b " l B l ~ ^ 2 T g l 0 T 2 g l rp c g i - h Jfi 2 0 rp 2 0 Ag l 0 0 o * 2b *2B * 2 rp 2 g ' ^2 T 2 g 2 - & rp 2 g 2 * 2 T 2 g V i 0 0 -*2b2 0 0 (Wl + * 2 a l 2 2 ) A l * System Param eters (equal a re a s )i H^=H2=5 seconds ( p e r-u n it i n e r t i a constant) D^=D2= 8.33x10 ^ pu MW/Hz (load-frequency co nstant) Tg l_tg2 * 08 seconds (prime mover time constant) TV - T V2=0.3 seconds (governor time constant) R1=R2= 2 H z / p u MW Pr l =Pr2 =2000 MW rated area power T12~ * 5^5 pu MW A P d l.001 pu MW, A P d2=0. (load demand change)

39 30 Fq = 60 Hz (nominal frequency) Mi = 2 H^/Fq (" e ffe c tiv e " pu i n e r t i a constant) E>i = Di + (1/R i) (frequency b ia s) a 12 = Pr l / Pr2 Teacher Param eters (feedback gains) RedJ For the cost param eters 1 ^ i= ^ i= 1, ^ = 0 fo r i= l,2 _ *4-2* *4-2* L earning Algorithm The le a rn in g c o n tr o lle r w ill o btain a sample of the s ta te a t time t= (k -l)t and determ ine the c o n tro l to be ap p lied over the sample in te r v a l (k-l)t, ktj, where T is the sample time and k is the sample number. For n o ta tio n a l convenience, the sample time T is om itted from the argument of the d is c r e te fu n ctio n s of time. In the development of a le a rn in g c o n tr o lle r fo r an m-area in terco n n ected power system, i t w ill be assumed th a t the system param eters and equations are unknown to the c o n tr o lle r. The process is thus described by the dynamic equation* i ( k ) = a [* (k -l), u ( k - l )j where M = unknown process fu n ctio n (of dimension n) jt(k) = s ta te v e c to r (of dimension n) u(k) = c o n tro l v e c to r (of dimension m) The c o n tro l i s thus a sequence of step fu n ctio n s. We s h a ll define the outputs of the system as the ACE of each a re a which can be e a s ily c a lc u la te d from the s ta te s corresponding to AF and A P.

40 31 y1 = ACEt = A P 12 + B1A F 1 y2 = ACE2 = - a 12A P A ^2 The co st fu n c tio n a l i s described byi oo J = h T(k) S x(k) + u ( k - l ) T (k-1 )J K«I + K * ' where the m atrix is the same as th a t described previously. Since the le a rn in g c o n tr o lle r must e x h ib it a d a p tiv ity, I a lso d efine a subgoal {joj fo r the le a rn in g process as* I (k) = jtt (k) $ ^(k) + iit(k -l) u (k -l) The problem is to determ ine a c o n tro l law,yo (k -l) th a t minimizes I(k ) fo r each ste p k. That i s, determ ine the v e c to r fu n ctio n ^ where \,i0 (k -l) = i [it(k-l), p ast experience] such th a t a minimum (over a l l p o ssib le c o n tro l choices) subgoal i s obtained fo r each k. The le a rn in g system is i l l u s t r a t e d in fig u re 3«i Unknown Process w ith measurable and c o n tro lla b le s t a te x Output Equation ACE Learning C o n tro lle r Subgoal Memory Figure 3.1 L earning C ontrol System

41 32 G eom etrically, we may describ e the c o n tro l law by a s e t of m number of hy p ersu rfaces in an n+m dim ensional space, where is the value of an m-dimensional v e cto r fu n ctio n of the n-dim ensional s ta te 2. These hypersu rfa ce s are approxim ated by a number of piecew ise lin e a r hyperplanes whose equations can be found by in te rp o la tio n of p o in ts th a t are se le c te d based on p ast experience. A fte r an i n i t i a l tr a in in g period (with a teach er or a search method) an e v alu atio n is made to determ ine i f s u f f ic ie n t inform ation (d ata on c o n tro l choices th a t re s u lte d in s a t is f a c to r y perform ance) has been sto re d in the v i c i n i ty of the s t a te Jt(k). I f so, a minimum-distance c l a s s i f e r JaJ, [VJ is used to s e le c t a s e t of i n t e r p o la tin g p o in ts. Define I 1 1 as t ^ie ae't p rev io u sly J sto red s t a t e s along w ith the c o n tro l u asso c iated with the s ta te and the r e s u lta n t subgoal. These p o in ts w ill be c a lle d c o n tro l s itu a tio n s as prev io u sly discussed in chapter I I. Define the weighted d istan ce between j (k) and ^ as where the le a rn in g param eter A is a constant p o s itiv e d e fin ite m atrix to be se le c te d by the u ser. The m atrix A is a f e a tu r e - e x tr a c to r fo r the p a tte r n v e c to r With a proper choice of the elem ents of A, one can force the s t a te s th a t have l i t t l e e ffe c t on the subgoal to have a minimal e f f e c t on the le a rn in g process and the re s u lta n t

42 33 mapping to the category ^. The d istan ce measure i s a lso weighted by the subgoal to minimize the e f f e c t of sto red p o in ts th a t had a h is to r y of m arginal perform ance. The use of in te rp o la tio n in a le a rn in g c o n tro l Bcheme had been p rev io u sly applied by Hammer, e t. a l. and is also used fo r t h i s c o n tr o lle r. The sto red p o in ts corresponding to the n+1 sm a llest d istan c es d^ are defined as in te r p o la tio n p o i n ts. A se t of n+1 d i s t i n c t p o in ts d efin es a unique lin e a r hyperplane. The in te r p o la tin g hyperplane fo r n ~ 2, m=l is i l l u s t r a t e d in fig u re 3*2. Figure 3,2 In te rp o la tin g Hyperplane In te rp o la tio n Procedure Each hyperplane r approxim ates i t s corresponding hypersurface which d efin es the c o n tro l law fo r Up, p = l, 2,..., m. We s h a ll assume th a t near-optim al c o n tro ls Up(k) l i e on the corresponding hyperplane. Thus the p a irs [x(k), u (k) m p=l s a t i s f y the s e t of hyperplane

43 3 4 eq u atio n si Wp. X j(k) + Wp(n+1 = up (k) p = l,2... J=i where w p = l,2,..,,m and j = l, 2,..., n are the hyperplane d e fin in g co n stan ts determ ined by the se t of i n t e r p o la tio n p o in ts. Define the weight v e c to r Wp= *^p2...n+1) T s* T and the augmented s ta te v e c to r x = (x ^,x^..., 1). Then we have the more compact equation a T X Wp = up fo r each p = l,2 m This form ulation is id e n tic a l to th a t of the sim p lest p a tte rn c l a s s i f i e r, the l in e a r m achine. where the c o e f f i c ie n ts W*. re p re se n t "weights'* th a t are to be learned from p a s t experience. The hyperplane equations are the d isc rim in an t fu n ctio n s of the p a tte rn c a te g o riz e r. I f the hyperplanes prove too crude an approxim ation of the hy p ersu rfaces, one could consider h ig h e r-o rd e r d iscrim in an t fu n c tio n s, but a t the r is k of in creased complexity both in memory and r e a l time c a lc u la tio n s. L inear machines in conjunction w ith a maximum s e le c to r as a d e cisio n mechanism have been used p rim a rily when the s e t of p o ssib le output c a te g o rie s is f i n i t e such as in c h a ra c te r re c o g n itio n, rec o g n itio n of binary sig n a ls am idst noise, and in the sy n th e sis of sw itching lin e s fo r relay c o n tro l system s. In my M. S. th e s is [kacj, reinforcem ent algorithm s were used to tra in a learn in g

44 35 system to implement a bang-bang c o n tro l fo r a second-order system which dem onstrated the same performance ( a f t e r an i n i t i a l le a rn in g period) as th a t of a bang-bang c o n tr o lle r c a lc u la te d v ia Pontryagin*s minimum p rin c ip le (excluding s in g u la r c o n tro ls ). In o rd er to o b tain the "best" c o n tro l u(k) fo r the c o n tro l s i t u a t i o n (lc), l in e a r in te r p o la tio n is p re fe rre d in ste a d of a s e le c tio n from a s e t of a p r i o r i quantized c o n tro ls. Each of the in te r p o la tio n p o in ts s a t i s f i e s the equation* [j^1] T = U p1 r i l T r i i i where ys J = [x^, x2,... xn, lj F u rth e r, d efin e Up = * up^ ' * * * Upn+^J ^ anc* l e t D = (jt1 * T (an n+1 by n+1 m atrix ). We can then express the s e t of n+1 sim ultaneous equations in n+1 unknowns JjTp a si Up There are m of these s e ts of equations (one fo r each component of the c o n tro l v e c to r). I f the in te r p o la tio n p o in ts are se le c te d to re p re se n t n+1 d i s t i n c t, lin e a r ly independent s t a te s, then Up = p i p re p re se n ts a s e t of n+1 lin e a r ly independent equations fo r each p and hence i s a n o n sin g u lar m atrix. Thus Mp = fi'1 Up and the d e sired c o n tro l i s i Up(k) = IT 1 Up p = l, 2,..., m. Thus we are so lv in g a set of n+1 equations in n+1 unknowns, m number of tim es. However, i t i s not as complex as i t

45 36 may seem. The c o e f f ic ie n t m atrix {js(k)] ^ is common to each s e t of eq u atio n s. Hence only one inverse need be c a lc u la te d. Using a hybrid computer, the w eights W.. could be s e t *J by se rv o -se t p o ten tio m eters (or banks of p a r a l le l e le c tro n ic m u ltip lie rs and summers) c o n tro lle d by a m inicomputer. The w eights could a lso be sto red d i g i t a l l y in RAM's. The lin e a r machine diagram is shown in fig u re 3.3. In k) k) ' j ( n ( k) + 1 mn m m.n+l Figure 3.3 L inear Machine

46 37 Once the w eights are s e t hy a tr a in in g procedure, the computation of the c o n tro l.y(k) in the operate phase i s very f a s t. The in te rp o la tio n procedure should be used to c a lc u la te u only i f the follow ing co n d itio n s are meti 1. There are a t le a s t n+1 sto re d p o in ts in the " v i c i n i ty of the c u rre n t s ta te i(k ), and i f 2. The in te rp o la tio n p o in ts from the previous c o n tro l in te r v a l re s u lte d in a s a tis f a c to r y co n tro l choice. These two requirem ents w ill now be expounded upon. Define the le a rn in g param eter D as the rad iu s of a hypersphere w ith jfc(k) as i t s c e n te r. Only those p o in ts th a t l i e w ith in th is n-dim ensional hypersphere would be considered e lig ib le as p o ssib le in te rp o la tio n p o in ts. There must be a t le a s t n+1 of these p o in ts to use the in te rp o la tio n algorithm. This requirem ent is i l l u s t r a t e d in fig u re 3*^ fo r n=3. * 2 7 Not enough p o in ts fo r in te rp o la tio n There are the needed n+1 p o in ts for in te rp o la tio n Figure 3.^ Neighborhood of c u rren t s ta te

47 38 To sim p lify the com putation, a sim pler ab so lu te d iffe re n c e d ista n c e measure could be used in ste a d of the E uclidean norm. That i s, l e t G<^Q be the number of sto red «p o in ts fo r which jj1 s a t i s f i e s x j D fo r a l l j = l, 2,..., n (where Q is the to ta l number of sto re d p o in ts a t time t=kt). I f t h i s measure i s used, the neighborhoods become hypercubes about the s t a te jt(k). I f G ^ n+1, then the p o in ts jj1 corresponding to the n+1 sm allest w eighted d is ta n c e s, d^ (defined on page 32 )* are s e le c te d as in te r p o la tio n p o in ts. I f D i s chosen too la rg e, then the in te rp o la tin g hyperplane may be a poor approxim ation to the hypersurface c o n tro l law. I f D is choben too sm all, in te r p o la tio n may never occur. I f in te r p o la tio n is to be used again in the same v i c i n i ty of s t a te space, the in te r p o la tio n p o in ts from the previous c o n tro l in te r v a l should have re s u lte d in a s a tis f a c to r y c o n tro l choice. That i s, i f lit,) -.loa-l) / c I ( k - lj ^ q then in te rp o la tio n can be used to determ ine.y(k). The le a rn in g param eter,, 0<CCq<C 1, should depend on p a st experience. I f there are only a few sto re d p o in ts, le s s s trin g e n t requirem ents on s a tis f y in g the subgoal should be imposed, othenvise in te rp o la tio n may never occur. I f th ere a re a larg e number of sto re d p o in ts, the i n t e r p o la tio n in the previous in te r v a l should have been accurate and a more s trin g e n t requirem ent should be placed on the subgoal c r i t e r i o n. A large in crease in the subgoal would

48 39 have in d ic a te d a p a r t i c u l a r l y poor c o n tro l choice and in te rp o la tio n should not be used in the next in te r v a l (the te a ch e r or a search would be used). The read er may ask, why not make i f Q is larg e? This would allow in te r p o la tio n to be used only i f the subgoal is decreased. The answer is th a t one wants to choose a c o n tro l to o b tain the sm a llest subgoal in each in te r v a l so th a t the accum ulation over the e n tire time of operatio n r e s u lts in the minimum performance index J. This does not preclude the p o s s i b i l i t y of a good c o n tro l choice r e s u ltin g in a s l i g h t l y la rg e r subgoal than the preceding in te r v a l. What the le a rn in g param eter prevents i s the occurrence of a s ig n if ic a n t in cre ase in the subgoal in d ic a tin g poor perform ance. I f the two co n d itio n s fo r in te r p o la tio n are not met, in te r p o la tio n i s not used and e it h e r feedback gain d ata from the updated optim al ( o f f - lin e ) so lu tio n of the lin e a riz e d system is used as a te a ch e r or a search alg o rith m is employed. As le a rn in g p ro g resse s, the number of sto red p o in ts in cre ase to a p oint where in te rp o la tio n is p rim a rily used. I t should be noted that since only those c o n tro ls th a t r e s u l t in a s a tis f a c to r y are sto re d, too small a value of C w ill decrease the q le a rn in g r a te since p o in ts w ill be more slowly sto re d. In order to complete the le a rn in g algorithm, 1. the method of s to rin g new p o in ts must be s p e c ifie d, and 2, the reinforcem ent scheme fo r removing p o in ts a sso c ia te d with rep eated ly unsuccessful c o n tro l

49 4-0 choices must a lso be s p e c ifie d. Storage of flew Points I f the previous s t a te i ( k - l ) is " s u f f ic ie n tly fa r" from a l l the p rev io u sly sto red c o n tro l s itu a tio n s and the c o n tro l u ( k - l ) th a t drove the system to the s ta te jj(k) from js (k -l) was a "su c ce ssfu l" c o n tro l choice, then jjl(k -l), u ( k - l), I(k )j would be sto re d as a new c o n tro l s itu a tio n. Define the follow ingi s Z tk - l ) - * 1 i t k - D - z 1 minimum... minimum jttk-lj-jj1 */ a ) ~ 1 r-l,j - rtl Q; where j * re p re se n ts the E uclidean norm. and re p re se n t the sto re d c o n tro l s itu a tio n s n e a re s t to and next n e a re st to j (k), re s p e c tiv e ly. Compute the d ista n c e si D, i ( k - l ) - AB B *A *B Define " is o la te d regions" as those regions of s t a te space where DAg New P ^n-ts are be e sta b lis h e d only in these is o la te d reg io n s. This would insure th a t the previous s t a te i s sto re d only i f i t is s u f f i c ie n tly f a r from the c lo s e s t p rev io u sly sto red point (and consequently, a l l o th er stored p o in ts ). The lea rn in g param eter should be c a re fu lly chosen. I f is close to zero and the sample time is very small

50 compared to the system time co n stan t, p o in ts may be Btored very close to each o th e r. For com putational purposes, the p o in ts may not be s u f f i c i e n tl y d i s t i n c t and the m atrix in v erse used to c a lc u la te the new c o n tro l v ia in te r p o la tio n may not be a cc u ra te. I f i s chosen too la rg e, the c o n tro l s itu a tio n s may be spread q u ite f a r a p a rt and the hyperplanes r e s u ltin g from in te rp o la tio n may not be an accu rate approxim ation of the c o n tro l h y p e rsu rfac es. The choice of C^, sampling tim e, and o th e r le a rn in g param eters re q u ire s a f a m ilia r ity w ith the system to be c o n tro lle d and many " t r i a l runs" (at l e a s t w ith a sim ulation of the system ). The fin a l so lu tio n does not depend upon the exact value of the le a rn in g param eters, but one must determ ine lim its of the param eters fo r s a tis f a c to r y le a rn in g r a t e, le a rn in g s t a b i l i t y, e tc. Since a choice of one le a rn in g param eter may a f f e c t the choice of an o th er, an experim entation phase (lik e a t e s t p i l o t of a new a i r c r a f t ) is n ecessary. I f both Da > C ddab and < c ( i. e., u (k -l) was a good c o n tro l choice), then the p o in t j"i (k-l), u ( k - l ), I ( k ) j may be sto red as a new c o n tro l s itu a tio n. An upper bound should be se t on the number of sto re d c o n tro l s itu a tio n s (e s p e c ia lly i f a small computer system i s used). be sto re d. Let H be the maximum number of p o in ts th a t can I f Q=H, a d d itio n a l p o in ts can be sto red only

51 42 i f a p rev io u sly sto re d p o in t ib erased from memory. The p o in t th a t i s f a r t h e r e s t from the previous s ta te i s erased, Let be the p o in t to be erased. Then, I Las (lc-1) - j(h. = maximum I ) Ir A i r 1 H? M where xx- y i + x2- y 2 ] + Xn'ynl is the norm used in order to sim p lify computation and.j is the a b so lu te value fu n ctio n. The above choice of the p o in t to be erased is the one th a t would le a s t a f f e c t the equatio ns of the in te r p o la tin g hyperplanes in the neighborhood of the c u rre n t s ta te ^ ( k ). Reinforcem ent Scheme A ssociate a number 0 ^ p^ *C 1 to each sto re d p o in t, (i=l,2,...,q ). The i n i t i a l value of p^ is s e t to 1 x i_ when the i c o n tro l s itu a tio n is e s ta b lis h e d. I f.u(k-l) has been c a lc u la te d by in te r p o la tio n and the r e s u ltin g I (k ) is too la rg e, then the p^ a sso c ia te d w ith the in te r p o la tin g p o in ts are m odifiedi "new" pi = CN* ( "old** pi ) i=l, 2...n+1 K Q, where is the negative reinforcem ent le a rn in g param eter o p e ra to r proposed by Bush and M o ste lle r ^BmJ in t h e i r s tu d ie s of m athem atical lea rn in g models f o r use in psychology. / *\ Let p^ XL be the r e s u lt of the j a p p lic a tio n of the o p e ra to r CN to p^. Then p - ^ ^ = > where is the i n t i t a l value of p^.

52 43 lim p. ^ ^ = lim P j/ ^ = 0 Thus a re p e ate d application of applleb n e g ativ e rein fo rcem en t to the p r o b a b i l it y t h a t the c o n tro l u1 -* should be used ag ain in a v i c i n i t y of j Removal of C o n tro l S itu a tio n s I f a sto re d p o in t has an a s s o c ia te d p r o b a b ility p^ t h a t i s sm a lle r than a given d efin e d th re s h o ld, t h a t p o in t would be la b e le d an u n su c c e ss fu l c o n tro l s i t u a t i o n and would be removed from memory. L et s be the number of tim es the subgoal i s "too l a r g e." P o in ts x1 t h a t were used to c a lc u la te u t h a t y ie ld e d an u n s a tis f a c to r y subgoal are e ra se d i f P^<C C^8. I f s is chosen too la rg e, a la rg e number of u n d e sira b le c o n tro l s i t u a t i o n s can be s to re d. I f s i s too sm a ll, i t i s hard f o r the le a rn in g c o n t r o l l e r to keep enough p o in ts sto re d and a v a ila b le f o r i n te r p o la t io n. Replacem ent (U pdating) b f Old P o in ts I f the p rev io u s s t a t e js(k -l) is s u f f i c i e n t l y close to the n e a r e s t s to re d p o in t (D^ ^ and ^ subgoal IC k) r e s u l t i n g from u sin g ii(k -l) is sm a lle r than t h a t of the sto re d p o in t ^, then the s e t f x t k - l ), u ( k - l ), I(k )^ is s to re d in memory and XA e ra se d. A flo w ch a rt of the le a rn in g c o n tr o l le r a lg o rith m is p re se n te d in fig u re 3-5 ar*d the program l i s t i n g in Appendix I.

53 4 4 F igure 1.5 Flow chart of L earning C o n tro lle r START ENTER DATAi (1) L earning P aram eters T,D,0 ^,H,CD CN,s (2) T eacher feedback gain m atrix, (3) Subgoal m atrix S t a r t sample tim e co u n te r k=l tim e t= (k -l)t S to re the i n i t i a l memory p o in t {2=0,u=0, 1=0^ Q 1 (Q=number of s to re d c o n tro l s i t u a t i o n s ) Measure s t a t e jj(k) and output y (k )=ACE Compute Subgoal Is IlX)-I(li-l) ^ c I(k-i) v u ^ NO (Apply N egative R einforcem ent) YES (s to r e new memory p o in t)

54 45 F ig u re 3.5 (c o n tin u e d ) E s t a b l i s h o r r e p la c e memory p o i n ts used a < cdd Q=Q+1 = ( k - i ) R eplace memory p o in t XA w ith ),1 (k) u - u ( k l ) I - I (k) E ra se Memory p o in t f a r t h e r e s t from i ( k - l ) and r e p la c e w ith i Jif a r = a ( k - D Uf a r = ja (k -i) j f a r _ I ( k ) ^ f a r l., E. ->j Go To 4 U

55 k 6 F ig u r e 1,5 ( c o n tin u e d ) Remove memory p o i n t s i f n e c e s s a r y N e g a tiv e ly r e i n f o r c e a l l I n t e r p o l a t i n g p o i n t s u sed to c a l c u l a t e u ( k - l ) new P in te r p = CN (o ld P in te r p ) Remove i n t e r p o l a t i o n p o i n t from memory S e t Q = num ber o f re m a in in g memory p o i n ts Go To k (com pute c o n tr o l u ( k ) )

56 4? F ig u re 3.5 (c o n tin u e d ) Compute th e c o n tr o l NO C a lc u la te th e w e ig h te d d is ta n c e d 2 = [j (k ) - ^J T A [j (k ) - 1 ^ betw een th e s t a t e Jt(k) and a l l memory p o i n ts i = l, 2 f... Q t h a t a re s u f f i c i e n t l y c lo s e to jj( k ). S e t G=number o f such p o i n t s. NO F ind th e memory p o i n ts x c o rre s p o n d in g to th e n+1 s m a lle s t d i s t a n c e s d^. D efin e th e s e as th e i n t e r p o l a t i o n p o in ts i= l,2,...,n+l Use th e te a c h e r u=kx to compute u (k ) S olve f o r th e c o n tr o l v i a in te r p o la t io n * D - r o 1 ; 2 t 1-1! T M ~ _lt * It... J it r 1 2 n + li T " [ up,u p up J up (k) - [^ (^ )]T for p 1 ^ 2 i *i )m ^ In cre m en t sam ple number k -* -k + l Go To 1

57 48 S e le c tio n o f Learning Par&metara S u b je c tiv e c r i t e r i a f o r s e l e c t i n g th e l e a r n i n g p a ra m e te rs were p r e v io u s ly d is c u s s e d. W hile an e x a c t v a lu e f o r th e s e p a ra m e te rs i s n o t c r i t i c a l to th e f i n a l r e s u l t, th e y must be s e l e c te d w ith in m argins of s t a b i l i t y, e tc. S in c e th e l e a r n in g system i s c o n tin u o u s ly u p d a te d, th e c lo s e d -lo o p p o le s a re c o n tin u o u s ly "m oving". I t would be v e ry complex to t r y to p r e d i c t th e e x a c t co u rse o f th e s e p o le s a s th e l e a r n in g loop depends upon th e system p a ra m e te rs, le a r n in g p a ra m e te rs, and d is tu r b a n c e s to th e nom inal p l a n t p a ra m e te rs and p o s s ib le e n v iro n m ental i n f lu e n c e s. A f te r " s u f f i c i e n t " le a r n in g h a s o c c u rre d (and th e p l a n t r e c e iv e s no f u r t h e r d i s t u r b a n c e ), th e p o le s sh o u ld be c lo s e to th o s e r e s u l t i n g in th e o p tim a l p e rfo rm an c e. Too la r g e a sam ple tim e caused th e c o n tr o l s i t u a t i o n s to be c o ll e c t e d and u p d ated to o slo w ly. In a d d i t i o n, th e d i s c r e t e c o n t r o l l e r had d i f f i c u l t y in a p p ro x im a tin g a c o n tin u o u s o p tim a l c o n tr o l. The sam ple tim e sh o u ld be chosen so a s to g iv e th e le a r n in g c o n t r o l l e r com puter program tim e to p e rfo rm th e i n t e r p o l a t i o n c a l c u l a t i o n s. S in c e th e ch o ice o f each l e a r n in g p a ra m e te r depends, in p a r t, upon th e c h o ic e s o f th e o t h e r s, i t i s n o t alw ays an e asy ta s k to s e l e c t p a ra m e te rs t h a t y i e l d r e l a t i v e l y " f a s t " le a r n in g w ith o u t s t o r i n g e x c e s s iv e p o i n ts o r r e s u l t i n g in i n s t a b i l i t y in th e le a r n in g a lg o rith m. Once one has a " f e e l" f o r th e sy stem s c lo s e d - lo o p re sp o n se

58 49 to th e le a r n in g p a ra m e te r v a r i a t i o n s (v ia s i m u la t io n s ), one i s a b le to choose s p e c i f i c valueb which a re " b e s t" f o r th e sy stem b e in g c o n t r o l l e d. An e f f e c t i v e tim e -s h a r in g i n t e r a c t i v e system (o r a r e a l - t i m e h y b rid system ) to do th e s e s im u la tio n s t u d i e s i s a v i t a l n e c e s s i t y. The le a r n in g system f o r th e tw o -a re a c o n tr o l problem was s im u la te d on th e H oneyw ell 6000 tim e s h a rin g com puter system (TSS) and a f t e r s e v e r a l hundred ru n s and e x p e r i m e n ta tio n, th e fo llo w in g s e t o f l e a r n in g p a ra m e te rs were d e te rm in e d to g iv e th e "best** r e s u l ts * T = 1.5 s e c, (sam ple tim e) A = i d e n t i t y m a trix ( v a r i a t i o n s o f A were n o t t e s t e d ) D - any v a lu e l a r g e r th a n 0.1 C =.1 ( f ix e d v a lu e ) o r C = l/(l+ 2 Q ) H = 30 (maximum number o f s to r e d p o in ts ) CD = 0.5 ( i n i t i a l c y c l e ), CD=2 -(la te r c y c le s ) CN = 0.7 (n e g a tiv e re in fo rc e m e n t p a ra m e te r) s = 3 (number o f a p p l i c a t i o n s of CN b e fo re e r a s u r e ) The p a ra m e te r, D, i s u s e f u l o n ly i f H i s la r g e. I t s e rv e s to red u ce c o m p u tatio n tim e by com puting d is ta n c e s o n ly in a c o n fin e d r e g io n a b o u t th e c u r r e n t s t a t e when s e l e c t i n g i n t e r p o l a t i o n p o i n t s. For H = 30, th e p a ra m e te r D i s n o t needed and was a r b i t r a r i l y s e t to 100. For th e su b g o a l p erfo rm an ce c r i t e r i a,, a com binat i o n o f a fix e d and a v a r i a b l e s e t t i n g t h a t depended upon th e number o f p o in ts s to r e d was u sed. The a c t u a l v a lu e was n o t e s p e c i a l l y s e n s i t i v e. A v a lu e o f s l a r g e r th an 3 o r 4 ten d ed to r e s u l t in i n s t a b i l i t y of th e le a r n in g c o n t r o l l e r

59 50 b e c a u se c o n t r o l s i t u a t i o n s t h a t a r e a s s o c i a t e d w ith p o o r c o n t r o l c h o ic e s w ere n o t e r a s e d f a s t enough and w ere u se d to o o f t e n i n i n t e r p o l a t i o n c a l c u l a t i o n s f o r th e c o n t r o l. When i s ch o sen to o s m a ll, i t becom es d i f f i c u l t f o r th e c o n t r o l l e r to keep enough p o i n t s s t o r e d to u se i n t e r p o l a t i o n. The sy stem was a l s o s im u la te d on th e IBM 360/65 com puter f o r p l o t t i n g p u rp o s e s. The IBM l i s t i n g i s p r e s e n t e d i n A ppendix I (th e H oneyw ell TSS v e r s i o n h a s a few w r i t e s ta te m e n ts f o r d a ta in p u t p r o m p tin g ). The CPU tim e f o r th e i n t e r p o l a t i o n c a l c u l a t i o n s p e rfo rm ed in F o r t r a n IV on th e IBM com puter ( u s in g a " b e fo re " and " a f t e r " i n t e r p o l a t i o n c a l c u l a t i o n s c a l l to th e l i b r a r y s u b r o u tin e CPUTME) ran g e d from 133 "to I 83 m il l is e c o n d s w ith c a l c u l a t i o n s p e rfo rm ed in d o u b le p r e c i s i o n. T h is tim e w ould be re d u c ed f o r s i n g l e p r e c i s i o n c a l c u l a t i o n s, o r i f th e program w ere w r i t t e n in a s s e m b le r la n g u a g e. Thus I c o n c lu d e t h a t th e s e l e c t i o n o f 150 m il l is e c o n d s a s th e sa m p lin g tim e y i e l d s an o n - l i n e c o n t r o l l e r. The t r a i n i n g p ro c e d u re was to p r e s e n t th e sy ste m w ith a d i s t u r b a n c e, a llo w th e c o n t r o l l e r to d r iv e th e ACE to z e ro and th e n r e p e a t th e c y c le by r e s e t t i n g th e s t a t e to z e ro and a g a in a p p ly in g th e d i s t u r b a n c e. I f th e sam ple tim e i s in c r e a s e d by much l a r g e r th a n 0.2 se c o n d s, th e sy ste m ta k e s s e v e r a l c y c le s b e f o r e s i g n i f i c a n t im provem ent b e c a u se c o n t r o l s i t u a t i o n s a re c o l l e c t e d v e ry s lo w ly. I f th e sam ple tim e i s re d u c ed much below 0,1 s e c.,

60 51 t h e l e a r n i n g p a ra m e te r m ust he in c r e a s e d t o p r e v e n t a l a r g e num ber o f p o i n t s t o be B to re d to o q u i c k l y. I t was n o t u n u su a l to f in d o v e r 100 p o i n t s s t o r e d. C e r t a i n l y th e c o n t r o l l e r - - e s p e c i a l l y a f t e r h a v in g le a rn e d from an o p tim a l t e a c h e r and n o t from a c ru d e s e a r c h m ethod - - does n o t need t h i s l a r g e num ber o f p o i n t s to cope w ith a s i n g l e d i s t u r b a n c e. A lso, i f th e sa m p lin g tim e i s re d u c e d to o s m a ll, c a l c u l a t i o n s may n o t be co m p leted w i t h i n th e s h o r t sam ple i n t e r v a l and th e o n - l i n e n a tu r e o f th e c o n t r o l l e r i s d e f e a t e d. The p a ra m e te rs T,, H, and w ere found to be th e m ost s e n s i t i v e. The p a ra m e te r s l i s t e d w ere found to g iv e th e " b e s t" r e s u l t s f o r th e sy ste m. Many o t h e r c h o ic e s re a c h e d a s i m i l a r f i n a l s o l u t i o n, b u t n o t a s e f f i c i e n t l y ( tim e - w is e, m em ory-w ise, and s u b - g o a l p e rf o rm a n c e -w is e ). From th e above d i s c u s s i o n o f th e s e l e c t i o n o f th e l e a r n i n g p a ra m e te r s, i t sh o u ld be e v id e n t t h a t a l e a r n i n g sy ste m m ust be " c u s t o m - t a i l o r e d " to f i t th e p l a n t to be c o n t r o l l e d. The v a r i e t y and num ber o f l e a r n i n g param e t e r s, w h ile tim e consum ing to o p tim iz e, g iv e th e l e a r n i n g a lg o r ith m a c e r t a i n " g e n e r a l i t y " w hich makes i t a d a p ta b le to a v a r i e t y o f sy stem s and p r o c e s s e s The hyperolane approxim ation In f i g u r e 3.6, th e d i s t u r b a n c e v e c t o r v = J was a p p lie d d u r in g th e f i r s t c y c le and v = \ O^j was a p p lie d d u r in g th e second c y c le in th e s i m u la t io n o f th e tw o -a re a i n te r c o n n e c t e d pow er sy stem.

61 52 S in c e th e sy ste m c o n tin u e d t o o p e r a te i n an o p tim a l m anner (a s ta u g h t by th e t e a c h e r ), we can c o n clu d e t h a t th e s e t o f l i n e a r i n t e r p o l a t i n g h y p e rp la n e s r e s u l t e d i n a good a p p ro x im a tio n to th e o p tim a l c o n t r o l h y p e r s u r f a c e s. Had u n s a t i s f a c t o r y o v e r a l l p e rfo rm a n c e o c c u rr e d, a h ig h e r o r d e r a p p ro x im a tio n o r more s t o r e d p o i n t s ( i n c r e a s e H) w ould be n e ed e d. Thus a l i n e a r h y p e rp la n e te c h n iq u e i s s a t i s f a c t o r y f o r t h i s m odel o f th e tw o -a re a pow er sy ste m ( w hich i n d i c a t e s t h a t i t may be f e a s i b l e f o r th e a c t u a l s y s te m ), P l a n t P a ra m e te r D is tu rb a n c e Suppose t h a t a p a ra m e te r i n th e pow er sy stem i s a l t e r e d. The s o l u t i o n f o r th e o p tim a l c o n t r o l v i a R i c a t t i te c h n iq u e s r e q u i r e s an e x a c t v a lu e f o r th e sy stem p a ra m e te r s. Once th e p a ra m e te r c h an g es a r e l o c a te d and m e a s u re d. one m ust s o lv e f o r th e s t e a d y - s t a t e s o l u t i o n o f th e m a tr ix R i c a t t i e q u a tio n w hich i s a s e t o f n (n + 1 )/2 = 45 n o n l i n e a r d i f f e r e n t i a l e q u a tio n s. T hese e q u a tio n s a lo n e r e q u i r e s e v e r a l m in u te s o f co m p u ter e x e c u tio n tim e, a lth o u g h th e tim e can be re d u c e d by c a l c u l a t i n g a s u b o p tim a l c o n t r o l l e r ^RedJ D uring th e p e rio d o f t h i s o f f - l i n e c o m p u ta tio n, th e sy ste m may o r may n o t be o p e r a t i n g s a t i s f a c t o r i l y. A s a t i s f a c t o r y p e rfo rm a n c e may r e s u l t i f th e sy ste m i s r e l a t i v e l y i n s e n s i t i v e t o th e p a ra m e te r d i s t u r b e d. To p r o p e r l y t e s t th e l e a r n i n g c o n t r o l l e r, one m ust p e r t u r b a sy ste m p a ra m e te r t h a t h a s a m arked e f f e c t on sy ste m

62 53 p erfo rm an ce (a s seen by th e ACE c u r v e s ). In p e ru s in g th ro u g h th e o p tim a l ACE c u rv e s p l o t t e d by Reddoch Jkedj, i t i s se en t h a t th e t i e - l i n e c o n s ta n t T12 t h i s marked e f f e c t we a re lo o k in g f o r. In f i g u r e 3*7» th e v a lu e T^2=.5 ^ 5 was used f o r th e f i r s t c y c le and T^2 = was used f o r th e second. The numbers chosen a re th e same a s t h a t o f E lg e rd and Reddoch FEj, jredj. The i n t e r p o l a t i o n p ro c e d u re was n o t used to c a l c u l a t e th e c o n tr o l in t h i s f i g u r e. The o p tim a l feed b ack g a in s f o r T^2=.5^5 were used f o r th e second c y c le as w e ll, where T12= The r e s u l t was c l e a r l y an u n s a t i s f a c t o r y p erfo rm an ce (v ery la r g e o s c i l l a t o r y sw ings in th e ACE). T h is i s an i l l u s t r a t i o n o f th e system perfo rm an ce u n t i l th e new o f f - l i n e c a l c u l a t e d fe e d b a c k g a in s ( f o r T12" 2 -? 25) can im plem ented. In f i g u r e 3*8, th e same problem i s p re s e n te d to th e le a r n in g c o n t r o l l e r. D uring th e t h i r d c y c le, th e o s c i l l a t i o n o f th e ACE o f th e d is tu r b e d a r e a i s somewhat re d u c e d. The p re se n c e o f o s c i l l a t i o n in a system i s o f g r e a t b e n e f i t to a l e a r n in g c o n t r o l l e r s p e rfo rm an c e, because i t p a s s e s th ro u g h s i m i l a r s t a t e s many tim e s in th e t r a i n i n g c y c le and i s a b le to more q u ic k ly u p d a te th e memory p o i n t s. A c o n tin u e d im provem ent i s se en in th e f o u r th c y c le ( f ig u r e 3*8b) and in th e l a s t two c y c le s, th e le a r n in g system seems to have converged (no more memory p o i n ts s to r e d, e ra s e d, o r u p d a te d ). A com parison of th e ACE shows t h a t i t i s n e a r ly i n d i s t i n g u i s h a b l e from

63 54 F ig u re 1.6 I n t e r p o l a t i o n V a l i d i t y Check. Response o f 2 - a r e a sy ste m to change i n d is tu r b a n c e v e c t o r v from ( ) to ( ) o a s s i s t i n g a r e a c > CD r>t CD. I ; i. i «. > / j CD Ci U. V -PT CD m / '\ k I ID. MI ^ J* \ I 1if-'? T-* SEC. AREA CONTROL ERROR * 0 1to u. Ci ' * p a c d _ \ 1 \ i 4- t ro. m C3 1 INstu rb e d a re a 1 j f j a : CD CT* te a c h e r I i [ learning interpolation

64 55 F ig u re 3.7 Response o f 2 - a r e a system to change in T12 from.5^5 to L e a rn in g ms. n o t used. Feedback g a in s f o r b o th c y c le s a re th e o p tim a l g a in s f o r T^2=, 5^5 C ' p 1 C 'I / X a s s i s t i n g a r e a )i'.ini I11I ^. IJIJ SEC. AREA CONTROL ERROR I ' f' t 1 * 0 ' ( iii ; A I t 1 Cr' 1 ri i.01 r ' r-.i ;v -d is tu rb e d a r e a L1 ir

65 56 F ig u re (a) 3.8 Response of 2 - a r e a sy stem to change in T12 from.5^5 to The te a c h e r (fee d b ac k g a in s f o r waa UBe(i th e i n i t i a l s ta g e and l e a r n in g i n t e r p o l a t i o n used th ro u g h o u t +he r e s t o f th e re s p o n s e. o <0 o ' a s s i s t i n g a r e a o CO o 1 I [ / T ^ - A A / H cj. 0 U M.IJL' J* r ~ i C,! \ I SEC. C.1 Cl d is tu r b e d a re a o C ',J Cl LTl O i_n te a c h e r lea rn in g in te r p o la tio n

66 57 F ig u re 3.8 (c o n tin u e d ) (to) d is tu r b e d a re a lea r n in g in te r p o la tio n

67 F ig u re 3.8 (c o n tin u e d ) (c) j\ i 1 a s s i s t i n g T ^ a r e a C.1 0 ' C V AREA CONTROL ERROR Ci CJ.' I r I I! \ '! \ A, i f v, V* r h, MS 1 / L I I. I I U \;? r 1.-ft* ra ;, / * V f 1 c / i LI J >l c:» - -1 c 1 * 1 c i a f i I C ) l\j t.j Lf.' d is tu r b e d a r e a I**' If c * r.j T j i C ) III lea rn in g in te r p o la tio n

68 59 th e o p tim a l c o n tr o l p l o t t e d by Reddoch. I t sh o u ld be em phasized t h a t w h ile th e s e r e s u l t s show a marked im provement i n sy stem p e rfo rm a n c e, th e y a re based on th e c o n tr o l o f a sim p le s im u la te d model o f th e a c t u a l sybtem by a le a r n in g sy stem whose p a ra m e te rs were chosen (by b ru te fo rc e and l o g i c a l o b s e rv a tio n ) to y i e l d th e b e s t r e s u l t s. Thus I have shown th e f e a s i b i l i t y o f u s in g le a r n in g c o n tr o l h e u r i s t i c s in power sy stem c o n tr o l. In th e a c tu a l sy stem, th e le a r n in g p a ra m e te rs may need to be a l t e r e d and a b e t t e r model chosen as th e te a c h e r. In an a p p l i c a t i o n to th e r e a l- w o r ld pro b lem, th e le a r n in g a lg o rith m would have an a d v an tag e o v er feedback g a in s c a l c u l a t e d v i a a sim p le model b ecau se th e le a r n in g system would o b ta in i t s u p d a tin g in fo rm a tio n by a m easu rement of th e s t a t e o f th e a c tu a l system and n o t from th e p a ra m e te rs o f a s t a t e model used to c a l c u l a t e feedback g a in s. The i n i t i a l le a r n in g system would, o f c o u rs e, be based on th e te a c h e r (fee d b ac k g a in s ). The c o n tr o l could be found by o th e r h e u r i s t i c m ethods such a s g r a d ie n t se a rc h a lg o r ith m s, b u t i t would r e q u i r e a l a r g e r number o f c y c le s b e fo re convergence and would be j u s t a n o th e r o f f - l i n e c o n t r o l l e r lik e th e more d i r e c t R i c a t t i s o l u t io n. The p l o t s shown a re f o r th e "optimum" le a r n in g param e t e r s. For l a r g e r sam ple tim e s, d i f f e r i n g su b g o al c r i t e r i a, and empty re g io n c r i t e r i a, th e le a r n in g c o n t r o l l e r s p erform ance s i g n i f i c a n t l y s u f f e r e d and made slow changes

69 6o tow ard th e d e s ir e d c o n t r o l. F or some s e t s o f le a r n in g p a ra m e te rs, th e l e a r n in g system seemed to e n t e r a c y c l ic o s c i l l a t i o n betw een h e a d in g tow ard and d iv e r g in g from th e o p tim a l s o l u t i o n. But a s e r r o r s were made, memory p o in ts were n e g a tiv e ly r e in f o r c e d and new p o in ts c re a te d. The p e rfo rm a n c e, a lth o u g h n o t a s d ra m a tic as t h a t w ith th e " b e s t" le a r n in g p a ra m e te rs, s t i l l y ie ld e d b e t t e r r e s u l t s th a n t h a t u s in g th e wrong feed b ack g a in s. One can co n clu d e t h a t f o r th e s e exam ples, th e l e a r n in g sy stem o f f e r s an im proved c o n tr o l u n t i l th e new te a c h e r p a ra m e te rs can be c a l c u l a t e d o f f - l i n e.

70 CHAPTER IV OPTIMAL CORRECTIVE RESCHEDULING FOR STEADY STATE SYSTEM SECURITY VIA PATTERN RECOGNITION AND SEARCH METHODS A power system i s s a id to be in th e norm al o p e ra tin g s t a t e i f a l l lo ad demands a re met a t th e s p e c i f i e d fre q u e n c y and v o lta g e s and a l l sy stem com ponents a re lo ad ed w ith in a c c e p ta b le l i m i t s. The system o p e ra to r s must change c o n tr o l s ( o p e r a tin g s c h e d u le s ) to m a in ta in norm al o p e r a tio n. The o ccu ren ce o f c e r t a i n d is tu r b a n c e s may cau se th e system to e n t e r an em ergency o p e r a tin g s t a t e such a s o v e rlo a d in g o f l i n e s and v i o l a t i o n o f v o lta g e c o n s t r a i n t s. The o p e r a to r need s some s o r t o f m easure o f th e s e c u r i t y of th e norm al o p e r a tin g s t a t e so t h a t he can be a l e r t e d to th e need to c o r r e c t c o n tr o l p a ra m e te rs to p re v e n t a d is tu r b a n c e from c a u s in g an em ergency. Assume t h a t th e s e t of m ost p ro b a b le d is tu r b a n c e s ( c o n tin g e n c ie s ) i s s p e c i f i e d. A power system in th e norm al o p e r a tin g s t a t e i s th e n s u b je c te d to each d is tu r b a n c e (one a t a tim e ). I f f o r e v e ry s i n g l e d is tu r b a n c e in th e s p e c i f i e d s e t, th e Bystem rem ain s in th e norm al o p e ra tin g s t a t e (no c o n s t r a i n t s v i o l a t e d ), th e n th e system i s s a id to be s e c u r e. O th e rw ise, i t i s in s e c u r e. 61

71 62 A s i m p li f i e d a p p ro x im a tin g n etw o rk model [ kpg] o f power flow betw een g e n e r a tio n /lo a d a r e a s c o n n ec te d by tr a n s m is s io n l i n e s w i l l be u sed. Each o f th e a r e a s a r e r e p r e s e n te d by one node and th e s e t o f l i n e s c o n n e c tin g th e same p a i r o f nodes a re r e p r e s e n te d as a s in g le b ra n c h. The e l e c t r i c a l v a r i a b l e s a s s o c i a t e d w ith th e nodes c o rre sp o n d to an a v erag e o v e r th e a c t u a l p h y s ic a l "subnodes" t h a t com prise each node (" s u p e m o d e s " ) o f th e m odel. An exam ple o f such a system i s shown in f ig u r e *1.1. P uget Sound Mid- Colum bia Longview P o r tla n d ( L A 11 Snake R i v e r, Idaho Salem Eugene So. Oregon Figure **.1 The P a c i f i c N o rth w est System ( s i m p li f i e d model ) [kpg) Power system s e n g in e e rs have found t h a t r e a l power i s most s e n s i t i v e to n o d al p h ase a n g le s and r e a c t i v e power to v o lta g e m ag n itu d es. I t h as been shown [ja J t h a t th e d eco m p o sitio n of r e a l and r e a c t i v e e q u a tio n s r e s u l t in th e fo rm u la tio n o f two subproblem s. In a c t u a l p r a c t i c e, th e two subproblem s a re a l t e r n a t i v e l y so lv e d u n t i l th e d e s ir e d

72 63 a c c u ra c y i s o b ta in e d. To i l l u s t r a t e th e use o f a p a t t e r n r e c o g n i z e r and s e a r c h m eth o d s, pow er flo w i s c o n s id e r e d h e r e, o n ly th e p ro b lem o f r e a l th e m ethod e a s i l y e x te n d e d to r e a c t i v e pow er flo w. A b ra n c h o f a power sy ste m i s c o n s id e r e d to be o v e r lo a d e d w h en ev er th e m ag n itu d e o f th e p h a s e - a n g le d i f f e r e n c e b etw een i t s end n o d es i s g r e a t e r th a n a g iv e n s e t v a lu e. Assume t h a t t h e r e a re n nodes and b b ra n c h e s in th e c o n n e c te d g rap h o f th e pow er a r e a n e tw o rk. The n e t flow from node j to node i i s g iv e n by T.. < 0,-00 w here T.. i s -L J J J th e t i e - l i n e pow er flow c o n s t a n t and (0i~0^) i 0 'th d i f f e r e n c e i n th e p h a se a n g le s o f n o d es i and j. The n e t pow er i n j e c t e d, 1^, i n t o node i i s th u s th e sum o f a l l th e T.. (0,-0 )'s t h a t a r e e m a n atin g from th e J- J J- J n o d e s c o n n e c te d to i t. Hence we have th e e q u a tio n * w here i s d e fin e d a s th e s e t o f n o d es c o n n e c te d to node i. The t i e - b r a n c h c o n s ta n t i s g iv e n by w here s(k) i s th e num ber o f p a r a l l e l l i n e s c o n n e c te d b etw een n o d es i and j and fo rm in g b ran c h k, ^li^m J l i n e v o l t a g e, (X..) i s th e r e a c ta n c e o f l i n e m, and l j m i s th e m ag n itu d e o f th e im pedance o f l i n e m.

73 64 If we define I = [ij...l J T. g = [((j... and th e nxn m a tr ix such t h a t ' sum o f a l l, k. and i = j Ai j = \ ~Tij if J J i 1 ^ J, 0 i f j ^ J i ^Qd i / j th e n we have th e e q u a tio n i ^ = A D e fin e th e b ra n c h p h a se a n g le 8^ = 0^ - 0 ^, f o r k = l, 2... b, su ch t h a t k i s th e num ber o f th e b ra n c h c o n n e c tin g n o d es i and j flow from n o d es i to j. and w here pow er i s assum ed to The assum ed d i r e c t i o n s o f pow er flo w g e n e r a t e s an o r i e n t e d g ra p h r e p r e s e n te d by a rro w s on th e b ra n c h e s. Then we may w r i t e Q = 0. w here =,..., 0 J ^ 0 = \$\ * ** *&n\ T and i s a bxn m a tr ix r e p r e s e n t i n g th e to p o lo g y o f th e n e tw o rk. 0 i f b ra n c h k i s n o t c o n n e c te d to node j +1 i f pow er i s assum ed to le a v e node j v i a b ra n c h k -1 i f pow er i s assum ed to e n t e r node j v i a b ra n c h k L et P^ r e p r e s e n t th e n e t pow er g e n e r a te d and th e n e t lo a d consum ed a t node i. F u rth e rm o re, d e f i n e th e v e c t o r s p = [p^, P2»., P j ^ and i = [l ^,L2»... ***. Then J = P ~ L i t h a t i s, th e n e t pow er i n j e c t e d i n t o th e sy ste m a t node i i s th e d i f f e r e n c e betw een l o c a l g e n e r a tio n and l o c a l co n su m p tio n in a r e a i. S in c e th e

74 65 t o t a l power g e n e ra te d m ust e q u a l th e t o t a l pow er consumed, n 0 (eqn U.l) 1=1 The m a tr ix, A, h a s an in v e r s e i f and o n ly i f th e s e t o f e q u a tio n s, I = A ^ a re in d e p e n d e n t. S in ce th e sum of th e com ponents o f 2 a re zero by e q u a tio n **.!, th e e q u a tio n s a re no t in d ep e n d en t and th e in v e rs e o f A does n o t e x i s t. I f one node phase a n g le i s a r b i t r a r i l y s e l e c te d, say node m, th e n we have a s e t n-1 e q u a tio n s in n-1 unknowns which a re in d e p e n d e n t (from w ell-know n argum ents o f b a s ic ^ ^ Ok C Ok c i r c u i t s ). D efine t h i s s e t a s 2 = A where 2 an d JS a r formed by rem oving th e m^h elem en t from 2 ^ d re s p e c - S' Ok ^ t i v e l y. 2 and ft a re n-1 v e c t o r s, and & i s th e n-1 x n-1 th m a trix a f t e r rem oving th e m row and column from Aj ^ 1 ^. a o l, c Hence, J0 = A 2 and = ( A ) 2. where 5 i s th e b x n-1 m a trix a f t e r rem oving th e m column o f th e m a trix. z' A A _1 D efine th e b x n-1 m a trix 3 = 5 A A ^ S' Then 0 = 2 where r e l a t e s th e b - v e c to r of branch p h ase a n g le s 0^ to th e n-1 v e c to r o f in d ep e n d en t n e t node i n j e c t i o n s. I f we r e q u ir e th e branch phase a n g le s to be cons t r a i n e d, th e n we must s a t i s f y th e fo llo w in g i n e q u a l i t y c o n s tr a in t* ^ where ( i)i r e p r e s e n ts th e i^*1 elem ent in th e v e c to r (jat I ) and a re s p e c i f i e d c o n s t r a i n t s ( c o n s t a n t s ).

75 66 L et, i, and i r e p r e s e n t th e power g e n e ra te d ^ lo a d consumed, and power i n j e c t e d u n d e r norm al o p e ra tin g c o n d itio n s. T hat i s, th e i n e q u a l i t y c o n s t r a i n t s (4.2 ) a re s a t i s f i e d i (Q? )^( ^ f o r each i=l,...,b G e n e ra to r O utage and Load L o st C o n tin g e n c ie s I f g e n e r a tio n i s l o s t in a r e a m, o r i f th e lo ad i s changed in a re a p, th e n a c o r r e c t i v e a c t io n must be ta k e n to s a t i s f y th e e o u a l i t v c o n s t r a i n t s o f e q u a tio n 4.1. The sc h e d u le d in te r c h a n g e s m ust be r e s c h e d u le d. T h is r e q u ir e s t h a t th e g e n e r a tio n in "some" a re a s may need to be changed and p o s s ib l y th e lo a d in "some" a re a s c u r t a i l e d. O bviously (from th e c u s to m e r's s ta n d p o in t) a c u r ta ilm e n t o f lo ad would be a l a s t r e s o r t. I f an in c r e a s e in g e n e r a tio n r e s u l t s in a pow er flow and hence th e n e t i n j e c t e d pow ers i t i s p o s s ib l e t h a t th e phase a n g le i n e q u a l i t y c o n s t r a i n t s 4.2 may be v i o l a t e d. In t h i s c a se, "some" o f th e lo ad must be c u r t a i l e d to m a in ta in s e c u r i t y o f th e sy stem. L et A P ; r e p r e s e n t th e in c r e a s e in g e n e r a tio n a t J node j and P.* th e new g e n e r a tio n a t node j. Then tj P* = + A We s h a l l assume t h a t th e g e n e r a tio n o u tp u t i s c o n s tr a in e d a t each node. Thus, [A P j < - P j 0 j = l n (eqn 4.3 ) tnq Y where P. i s th e maximum p o s s ib le g e n e r a tio n a t node j. J L et A l. r e p r e s e n t th e d e c re a s e in th e load a t node j and tj

76 67 L j ' th e new lo a d a t node j. Then Lj* = L^0 - L^ and we s h a l l assume t h a t th e maximum a llo w a b le c u r ta ilm e n t of th e lo ad a t node j i s g iv e n by A L j 1 * = L^0 - Lmm^n where L,mln i s th e minimum lo ad a t node j. J th e i n e q u a l i t y c o n s t r a i n t! Thus we have 0 A A l. L, - L mln (eqn 4.4 ) J J J The o b je c t i s to d e te rm in e v a lu e s o f A P and A L such t h a t th e c o n s t r a i n t s 4.1, 4.2, 4.3, and 4.4 a re s a t i s f i e d f o r P*, L* and I = ' - '. Branch Outa g e C o n tin g en cy I f a b ran ch i s opened, o r th e t i e - l i n e p a ra m e te rs changed, th e m a trix i s changed. L et r e p r e s e n t th e new m a trix. We must f in d A P and A i such t h a t th e c o n s t r a i n t s 4.1, 4,2, 4.3, 4.4 a re s a t i s f i e d f o r P*., P '^ P '- L *, and th e new m a trix O ptim al S o lu tio n T here may e x i s t many p o s s ib le s o l u t i o n s to th e problem o f c h o o sin g A P and A i to s a t i s f y th e e q u a l it y and th e i n e q u a l i t y c o n s t r a i n t s. I f we assume t h a t th e system i s o p e ra tin g most e c o n o m ic a lly in th e norm al o p e ra tin g s t a t e (P,, ^ ), th e n we would w ant to m inim ize any change from t h i s s c h e d u le. We would a ls o l ik e to m inim ize any c u r ta ilm e n t o f lo a d. Hence a c o s t f u n c tio n must be d e v ise d to f in d an "o p tim a l" s o lu tio n f o r A P and A l.. The e q u a l it y c o n s t r a i n t 4.1 can be in c o rp o ra te d in to th e c o s t f u n c tio n,

77 68 S, such t h a t d r i v i n g S t o a minimum w ould c a u se th e e q u a l i t y c o n s t r a i n t ^.1 to be s a t i s f i e d. We s h a l l assum e t h a t r num ber o f g e n e r a t o r s can be r e s c h e d u le d and q num ber o f lo a d s can be c u r t a i l e d. I s h a l l choose th e f o llo w in g c o s t f u n c t i o n t o be m in im iz e d, w here, tr-, and ^ a r e p o s i t i v e c o n s t a n t s. The c o s t p a ra m e te r s and sh o u ld be ch o sen l e s s th a n 1 in o r d e r to em p h asize th e e q u a l i t y c o n s t r a i n t in th e c o s t. The p a ra m e te r cr sh o u ld be g r e a t e r th a n to e m p h asize th e lo a d c u r t a il m e n t more th a n th e change in pow er g e n e r a t i o n. The p ro b lem may be s t a t e d a s f o llo w s i D eterm in e v a lu e s f o r A f and f \L so a s to m inim ize S s u j e c t to th e i n e q u a l i t y c o n s t r a i n t s k.3 and ( th e e q u a l i t y c o n s t r a i n t and th e s e c u r i t y c o n s t r a i n t a re i n c o r p o r a te d i n t o th e c o s t ). I f th e above p ro b lem can be s o lv e d f o r each o f a g iv e n s e t o f p ro p o se d "m oat p ro b a b le " c o n tin g e n c ie s ( g e n e r a t o r, lo a d, and b ra n c h o u t a g e s ), th e n th e sy ste m i s s a i d t o be i n a s e c u r e o p e r a t in g s t a t e. o th e r w is e i t i s i n s e c u r e (w ith r e s p e c t to th e c o n tin g e n c y set). K a lte n b a c h and H adju (kh] and P ai and P a r a n j o t h i PP and o th e r s have s o lv e d t h i s p ro b lem w ith l i n e a r

78 69 program m ing m ethods by u s in g th e c o s t fu n c tio n * 4 F=^iApii+yy iali 1=i l-i and in c o r p o r a ti n g {w ith a n e c e s s a r y change o f v a r i a b l e s ) th e e q u a l it y and i n e q u a l i t y c o n s t r a i n t s in to th e l i n e a r program m ing sim plex ta b l e a u. Use o f p a t t e r n r e c o g n i ti o n to D eterm ine S e c u rix y We s h a l l assum e th e m a trix i s known and hence th e "w e ig h ts" o f th e p a t t e r n r e c o g n iz e r a re known. I f (J i s unknown o r th e d a ta e s p e c i a l l y n o is y, one may use a le a r n in g te c h n iq u e to d e te rm in e th e w e ig h ts and t r a i n th e system to s u c c e s s f u l l y d i s c r im i n a t e se c u re from in s e c u r e s t a t e s. T h is of c o u rs e r e q u i r e s a la r g e and d iv e r s e t r a i n i n g s e t. The new node i n j e c t i o n vector(2*> i s g iv en by 2*= P - = p + AP - L + A i The system i s s e c u re i f j( 1*)^ 5; f o r i=l,...,n and for each member of th e p o s s ib l e c o n tin g e n c y s e t. We can r e p r e s e n t t h i s d i s c r im i n a t io n v ia a p a t t e r n r e c o g n iz e r ( l i n e a r m ach in e). T h is i s i l l u s t r a t e d in f i g u r e ^.2. The new node i n j e c t i o n s 2* t h a t have been c a l c u l a t e d to s a t i s f y th e c o s t,s, a r e th e p a t t e r n v e c to r s and th e s e t o f c a t e g o r i e s c o n s i s t s o f th e two e le m e n tsi 2* i s a s e c u re p a t t e r n, 2' an in s e c u re p a t t e r n Hence a s ta n d a rd p a t t e r n d ic h o to m iz e r i s u sed. I f one d e s ir e d, a le a r n in g scheme co u ld be in c o rp o r a te d A to com pensate f o r m inor changes in The w e ig h ts, W,

79 th en Je Hi cf OJ7 m. U (* *r * \, Jy. % o ^ - Q. J *J U, a t t e*>7 C«t ( &l> *o *2C/ ti O/j ;

80 71 I n s t e a d o f th e l i n e a r program m ing te c h n iq u e, a s e a r c h m ethod c o u ld be em ployed t o s o lv e f o r th e c o n t r o l s A P and A i w ith c o s t f u n c t i o n S and s u b j e c t to th e i n e q u a l i t y c o n s t r a i n t s. The o v e r a l l scheme i s p r e s e n te d i n th e f lo w c h a r t in f i g u r e ^.3 I t sh o u ld be n o te d t h a t i f a s o l u t i o n c a n n o t be fo u n d, th e g e n e r a t i o n / l o a d c o n s t r a i n t s may n eed to be a l t e r e d, o t h e r g e n e r a t i o n / l o a d s be a llo w e d t o be a d j u s t e d, o r th e m a tr ix i t s e l f ch an g ed, p e rh a p s by a d d in g new l i n e s to " b e e f up" th e b ra n c h e s. The s e a r c h te c h n iq u e used was t h a t d e v is e d by P o w e ll {^P y] a n d in c o r p o r a te d i n t o th e F o r tr a n IV l i b r a r y on th e IBM 360/65 by P r o f. A. J. M cphate o f th e M e c h a n ic a l E n g in e e r in g D e p a rtm e n t, L o u is ia n a S t a t e U n i v e r s i ty. A program was w r i t t e n to r e s c h e d u le lo a d s and pow er g e n e r a t i o n to m a in ta in sy ste m s e c u r i t y. T h is F o r t r a n program i s l i s t e d i n A ppendix I I. D ata and r e s u l t s f o r th e P a c i f i c N o rth w e st sy stem ( s im u la te d ) a r e p r e s e n te d in th e t a b l e s and co m p u ter p r i n t o u t s on p a g es The and m a t r ic e s w ere e n te r e d i n t o a SPEAKEASY A program and B = A, th e m a tr ix, and th e r e s u l t a n t p h a se a n g le s f o r th e sy ste m i n th e norm al o p e r a t in g s t a t e w ere c a l c u l a t e d. * The m a tr ix was th e n fe d i n t o th e s e a rc h and s e c u r i t y e v a l u a t i o n program f o r g e n e r a t o r o u ta g e s.

81 72 F ig u re 4. 3 F lo w c h a rt (O ptim al R e sc h e d u lin g ) START : ' ^ INPUT DATAi Normal O p e ra tio n V alues* P.L, Number of nodes, n, and b ra n c h e s, b S e t o f most p ro b a b le c o n tin g e n c ie s Node num bers of g e n e r a to r s ( r of them) and lo a d s (q of them) t h a t can be r e s c h e d u le d. Chosen c o s t p a ra m e te rs (/ anc* p i max = maximum a llo w e d g e n e r a tio n a t node i j i = l, 2,..., r L min _ mi n i mum a llo w e d lo ad a t J node jt j= l, 2 q M, = maximum phase a n g le a llo w ed f o r b ran c h kt k = l, 2,.,., b G> Apply f i r s t c o n tin g e n c y ( g e n e r a tio n, lo a d, o r l in e o u tag e) S earch f o r A P ^ ( i = l r) to s a t i s f y c o s t S and i n e q u a l i t y c o n s t r a i n t s on A th e r e s o l u t io n Go To 3 C a lc u la te 1* = A + A l + - PATTERN RECOGNITION Map ' in to s e c u r i t y c a te g o ry " -~ r _: _ Go To 2 J

82 73 F ig u re 4.^ (c o n tin u e d ) f I s ^ P r i n t i r YES System i s a ~ - > s e c u r e f o r s e c u re t h i s.p attern * ^ c a te g o r y NO Has a S e a rc h f o r and AX o c c u rre d f o r t h i s c o n tin g e n c y YES System is insecure fo r th is category Have continbeen t r i e d? NO >f Apply n e x t c o n t i n gency f Go To 1 S e a rc h f o r A and AX to s a t is f y co st and in eq. co n stra in ts AS. and AX th e r e so lu tio n AS and AX YES i Co To m on Was sy ste m s e c u re f o r all c o n t i n g e n c ie s? S o l u t i o n, O v e r a ll S y stem i s I n s e c u r e L oosen C o n s t r a i n t s o r a llo w o t h e r gen. o r lo a d s to change YES P r i n t i O v e r a ll S ystem is S e c u re STOP IGo To 5

83 74 The r e s c h e d u l in g p ro b lem was s o lv e d f o r th e f o llo w in g d a ta f o r th e n e tw o rk in f i g u r e 4.1 u s in g th e d a ta in [KH] and [KPG] DATAi The r e a c ta n c e and im pedance o f each l i n e a re assum ed e q u a l. X*. =.06 (pu) and Zj.2=.OO36 (pu) J v B ranch Nodes L in e s 1 2 to KV, KV, KV 2 1 to KV, I - 23 OKV 3 2 to KV, KV 4 5 to KV, 2-115KV 5 6 to 5 I - 69KV, 1-115KV, 1-230KV 6 5 to KV 7 5 to KV 8 2 to KV, KV 9 2 to 5 I - 69KV, KV 10 6 to KV, 1-230KV 11 6 to KV, I - 23OKV 12 4 to KV, 1-115KV 13 7 to 8 I - 69KV, KV T i e - l i n e c o n s t a n t s. T ^ = T ^ (com puted) A ll T.. = 0. e x c e p t th e f o llo w in g i (in MW) tj T12 = T14 = 6133 T T25 = T26 - T23 4 q T 36 = 6133 T45 = r II cr^ II 6900 T57 = II 00 T78 " 23OO 8834

84 75 The A m a trix was c a l c u l a t e d from th e T^ j ' s and "the S' ft m a trix from th e netw ork to p o lo g y. Both m a tr ic e s A and were fe d in to a SPEAKEASY program and B=A, th e m a trix, and th e r e s u l t a n t b ran c h p h a se a n g le s, f o r th e system in i t s norm al o p e r a tin g s t a t e were c a l c u l a t e d. The com puter r e s u l t s a re th e fo llo w in g * K (A 7 BY 7 1 AT RI X) 1 ^ *'' ft -ftl 37 r, r ft R* 7207 ft ft ft ft ft -#>1 M - b i l l ft - 27 ft 0 ft r -ft n n ft n 7 r U 2'" 1-3H ^t* r- c -or 3U 11 1 id ft 11 BY 7 P U T R r n > ft ft 1 0 ft - 1 c 0 ft ft 1-1 ft ft 0 ft ft ft ft ft ft 0 ft ft 1 ft ft 0 0 ft n ft ft ft ft -1 ft ft 0 ft 1 ft ft ft. ft ft (ft -1 0 ft 1 0 ft ft ft ft 0 0 ft ft 0 ft ft ft ft 1-1

85 ?6 If if so itr vo * 9 vo sc it SO 9 m in Si m i r 1 1 i 1 1 t 1 t I i 1 t I 1 1 «i to u i t> U- u i ft- U1 U3 *1 W frl w 01 fr: P r - rr, o> s > r*> m m cr r - o> s m o tt n rsi r - m in fn vo r~ ro w m IN sc r - =r ft^ r*- c.* r - t r - ^ *- Csl O' 9 ro r- c» fn r- q o»- f_^ <> ro r- 00 if. o O «*- 0D CJ o if n. «* * # ft # * ft «t *- (? <N n *- T" rn r - rr fn r r- r - Si1 fn a sc t t 1 1 i 1 1 i i lti I f IT IT vc O' N \C s > in so O m m n s i in 1 * 1 1 i i t I i t 1 1 t 1 i i i 1 O' 1 U* [tl bu u to ft] u i ti ca tu *: til u. p U! i to or OC»n 00 r* r - IT o 00 m r cr cr VL cn r- o f\» oc r~ f - s > rr r- fn 1^ o sn IN rn» fn I f 9 m c cr r- X m s ) c cr c Lf sc m ( r- vo r- c CT in r- LP c < o I f r' zt st i f m fn ft * ft ft «ft *»» * <N * - r- m a * i t a in or r- ft r- ft r~ r -i r* 1 «t t t 1 t < 1 kt Lf m kn in n sc vr CT m m in vt vt in lt if is If Vi 1 1 i i i i 1 > «1 ( i I 1 U U= to to M to CO n ft- U- 1* & r > in ti Cl Lf a s i Csl IT cr a *- Zt Vi' so i f vt1 r- IT a n ac rr. m No o fn T Zf m 1C zt n> fn a r ON a rn r- or rn v i i 1 *- c sc sn si <r 3 rr m if r~ <n rr, ce (T ft r c s r a a»- n - LT o ft» «ft «* # * # ft * rr r - CN a or r ir fn CL a r- r* o o t^-? I 1 i i m m it n i/ in m m IT m m in m m sf «si 1 t ir 1 i I 1 i m i I t * «i i IT «i «u co CO M U-. I* tt; i UJ J (p to k. c> n p (IT g r~ a to n j a LT cr fn ft: O' tn cr IT cr c P Si st: f c s i rr- r in r- if m r rn r- in r- u C s i r- C O n c c o i i rr i f <N o si m ith n C 1 r r- r ft IT r, c O' m C " c O rr> C o s i o IT O m ft ft * * ft» ft # * «* ft * m 9 n : r fn rr^ CN * On r- rsi IN O e- u. Zf v i 1 1 t t 1 I t 1 lt a IT i r \ i si in & IT i/i sn if rt m so» i r t in 1 i 1 1 * 1 IT 1 i I in m i 1 t i P1 * U- t* ti & (* (*' 1 U bl r> # i i g H PJ b-' 0 vc i*-' c> CT r- Ik] si (*> Z*) rr * o r - rsi r- cr m cr m IT cr t OP fft- r- vt r o fn rn r cr cr a m rsi cr a 3 r- i rs* O' r - in (M r- o <N ir * m r* cr fn f r~ rsi r 4 or fsrf» * «*»» ft * # in * n r- *" rr r- r- fn r- CT fn m P4 f- IN n lt> if I f if' i f If' in m Lf' kf m I f m IT m m sc * t 1 i f X 1 * t 1 «i * i t i i m t >< p : [O i ft E t> N CO to to b U1 CO P to b ft] p. U*» i ft ' f u Cr m or O a. m fn cr gn X r - sb» m U j m a. (N rr r - s ) on fn If vr X fn so r*i Lf cr rt f - fn cr r t~ ft - m r- r- ^*- r t m a rr n Lf m» If kp fn r^, si m c CN r^i m ft VO fn i/i - si a- * *» *» ft ft * ft st If o ft- r - r - ft- fn C\. * rr P- r- rg rn r- 4» «U >- CO K =t ir IT it n i f m CC. if If i/s IT m If if Lf i; si' i f s > ex QQ 1 i 1 1 t 1 1 t i 1 1 m 1 t 1 i 1 to P P' Uft b U1 Us m Uh U: l>. W u to U4 I u Cm u ftu b > r- O ft- zt rt cr r- <CL ft- r^ r^s a r u r- r j q- c»~ C f 4 rg C c fn O' C r - r <N fn n t c CJ *-i tn rsi rr C r ft -r CD o m c f r - rsi fn X if%x ii m Lf i/s m ft sn M * ' O' n ' cr lt in \f' i t f c H rn a ft» < n \ * ft * * ft (Tj ft- si 9 Lf rn r j w~ o s f * Lf f-' ft- r~ D (T- c*~ 1 i 1 1

86 77 INPUT-. \ TP (A VECTOB UITH 7 COMPONENTS) (n«>j* i^rcttaas,) J H UO U PHI=$*TP.. PHI PHI (A VECTOP UITH 11 COMPONENTS). UU3 -. r n3l0(*e>. 2 3 U tiJU"*.1R99R P1C605 The (i m atrix was fed in to the search and s e c u rity e v a lu a tio n program where AP and AL were c a lc u la te d to solve the optim al problem. The computer r e s u l ts f o l low I 'JODES- ^ HP A VC H PS- 1 1 V ODE G E V E E A T T n V * ^. 1 ^ " E ' U 0. 6 n ^ E r U NODE LO AP- - q. e 7 t ' u h S t ^ *b r ^ r r* tp ~ r. * r. 5rfff "u *. 1 U o r ' y r, 1 1 f. F ' -J '. i ^ ' tl A Li INTIVOENCY 5 T'~- - NO. 1 V 3 o F = 1 N "V O IT _ - **r * * r NO. 2 NO I' > T ~ " f y 7 1 'J. = O r *'1 NO. "? V >0 F- 1 *- _ r v yw - ^ r r «r NO. u» " n r - u V E w i»;'. - #1 " ' T7 f " NO. 5 vnn f= t N r' 7'..»t ' * - *. Ur " * u NO. f. N'M' - t- * N F W SEN.- * 1 " T " u NO. 7 von I = T N pw SEN. - NO. P V o p P r H NEW G F. = r ^ ^ r ^ '1 r< r ^ * i? '

87 78 GENERATORS TO PE p ^SP HE DMLPD -- GEN. NO.? NAX. GFNFRA'r ION - M -F 'S GEN. NO. 6 HAX. GENERATION = n. Rr ^F " U LOADS TO PE RESCHEDULED-- L n A D N 0. 1 I ^» P - ' E ~ ^ LOAD NO. T - i r n i LOAD = '.?1«F * U NAXTNDF" ABSOLUTE Pll»Er ANGL^O^ ". G 1 ' E " ~ S L P ' _ r, ^ r t, 1 I.1 ' ' c, 1 ' r.- ' ' ^ f, y :? r r, i ' p r C r* S rt' PI UJN / 1>r -r---- n in - '. fi - ' * ^ j».;v= ^ i r ~~*\r- ^ r " 7 0 N T T N G L U C Y N O. 1 c n 7 t = 7 7r r r 1 G E N - ' R A T O :> C H A N G E O N L Y i r N. n o. r> n r > - ^. N - M P } T. V] P T 7 (.! ' * 1 * r V - s u r =, 7 h K - "? H ^ A N C M n H A G E S = ". S ' 1 P. *. S ~ " f ~. 1 < S R r - r r n r * r r '. ' 7 E - * 1 -. U S ' * ' E ' M ' i 7 E * p 1 T f I\ 1 Q 17 '.1«1r '.?p 1 r ^. 1 2 <i E - r 1 P H A S E A N G I. ^ I M I O P A L T T Y 0 O N S T r - A I N G E N, A N D E O A D l ' P A c V - ' -j ~. 7 c " U G P N. N O.? p u ^ f ( 1 1 l l V ' ' T G - N. N O. D D P - I. O A D \ o. 1 D L - ' C, 1 7 E- 7 1 V ^ 7 1 I > l o a d N O. g P L -, u n r " ^ r j ^ i ^ ^ r r B R A N C H P H A E E E -. U O R " C " -. < i 2 V - 1 T Q ' r * '. S S? r. 1 f r u r. T r r r.. ^ 7 1 E ^ " _ ( j V " r 7 ' r, E " 7 _ 7 ^ 7 q r t? _ - 7 Ort \ 1 U 7 t 7 Q Q T ^ A _ l i r r. L «. * i P H A S E A N G L E T, : O H M I " Y C o N S ~ p A I N

88 79 CONTINGENCY HO. 2 C OST = r. 7 u r>f "3 GENEPA70F CHANGE ONLY n r N. in, 7 no= ^. SP^E o * tr. GFN. NO. 6 n:* = " ' v * 7 GUY = - s.. 7P IF - r 2 HHANCH flihfes-- r. a m r ~ r -. 2 i H r - " 1 -, 7 f, u z - ' '.:>u'ip ' ^ 1? ft r * '. t 1 7 r ' " > r* r f r. n i ' ' t 1"U" ', 1 h ^ ^ - r i ^ 7 :: " '.? ( 0 ^ ~ I*r- ' > PHASE ANGI.F INT^'<'.!TV CONSTPAINT N<r F» ^ T T ^: "I* GEN. A ND LOAD c EAPCH COST" :;. 1 n 3 F f'-tl G P N. Nn. 2 ni>^ 32 " E - n- 6 G PH. NO. p pi - a U 0 3 F ' 3 LOAD NO. 1 p». * * 2' E-' P 1,0 111' n 1' r, 1 w. - * ry ' 7E 3.3 ur - 4 t ' 1 p --1a n c H P H a f 0 - ^ * ^ r. u P r * r ir- " 1 -." ' n *. 2 Q 1 * 11 Cf - ^ r r r r - ^ * U 7? f '. 1 ^ U r r r r 1 7ur ) r - ~ 1 1 ' U T- * i ;> t r* ' ii i r r PHUSF AN dl, p I N1- (.HI A!. I T Y C r' N S T P AT NT V r FATT^P:!^ r. o r O NT I Nr;" v Vn. 1 ' " i T r /- -»7 i - <- -> G r>. r HA'n' r'js ', ; r fi*jt.y G^N. NO.? P = ". r r r' 1 r; f n. ho. t- :if -.sin = BRANCH PhA^FG- / j u r " * #1 a< < '.? 17 * ' 1 r " t Ir j r r" - t, ~cj \l) ') O r r,r «', 1 <1 f 7 ".T 'O '' n.?s"7r "C r. r 77^ - '. 1 - it - <- 1 pystfi i:i i r "? r

89 80 CONTINGENCY N. U GENERATOR CHANGE GEN. NO. 2 DP- G E N. NO. ft - q ti m ~ ^ " r R P A N C H P H A G F E - rf " ->p -? r ^ r * 1 k \ c IP ' '. 1 UMF " ' - M M 11'-' 1 P H A S K A NGT. F T N"C'I A' I ONLY j C O S T = S'TE 2 P r S U E ti MG r ' ' 1 (, ' >" ' ' 1 ID v p 7 r, 1 G r - M f t T T r f - r t ^ r, r- ~? U " ~ A t. - '1- - t 'f i ; GFN. AND TOAP CGFT- 1 1Cr ''U G p N. NO. 2 ni'3 r. 22CF ^ 1 GFN. NO, ( > T P = ". t; 7U*- - " G!,DH O in. 1 pi, 11U" ^ 1 L 0 A n N n. 7 V '., ' T' ppsnch p HA G F S - -. y Q k I - ' _ '. JS S ' 1 '.? 5 '7 - ' i U 7 v- r > r ' i c i r ' ' " 4 C f. U ( ^, 1 u ^ - <1 c,t_- - - ". 1 F SF ' r r F ' f r " - '. 19F 1 PHA5F ANGLE I N F 0 1[ A ] TTY CONSTr<ftTN7 JOT s A ^ I '' r! r P- - T \ E rc' p r C O I. ' T I N G N r ', ' 1 '. 1 ' r * GEN".-'A? O1-' CH'N-Vf'^I.Y Gr N. no.? :> - :. 1 ' " ' U G EN. NO. P OP- ". 1 "' V * U S,JW=. r ; O BRANCH n ii j\ t p^ = '. u f, '. 1 r, v _ i '. ;i t f r - _ - 1 1f)»r 'f ".tiuff ' ', U4 U11 ' ( U 7nr r r r. li 7 c. f. i ' 1 T ^ t ~ 'l r itt r ' - p f r 1 1 ' ( * SYST^N IS SFCMP p r r

90 81 GFN. SEN. SI11 = PTJ I»JC M NO. NO. o. PI: C Y NO, *- AT Of CHANOF 2 D P - 6 n p = n '1, o o A F "!P y y ^ ^ ^ r> 2 15E " r u q p rr COST = O K I. Y r. 1 Onp P. 10 F r,y o U - n. RS7r - p2 *. 2 f>5 p p ' \pq F f. 'I p -j f, t? r i f ^ r q r. M f' i; : n r f r» r -1->c, tr ^ - * s y: a r u r.^ ^? ; v t r t? :ONTINGKNCY No. 7 GFNPPATOp f'han.;? ONI.Y GFN. NO. 2 PTi = GFN. NO. n I"> - o Jj«n = n ^ fl ^.*\v p -v PPANCH P H A S FF - ', y '. 2 2 o " ".7 V I '' ' '. 1 ^ T fr - ',r ^ (j U ;.- 1. U? L v '. ~ P ' ' 1. n. ' ' 1 p r ' 'm -, ). 7 H * 7 7 U ' P P A S F. T N O1 F 1 N p O' 1M. 7 T Y C O V S ^ I A7 M^ N o 7 7 h 7 t - c 7 v n _ ' - T*: s "rn> ; 7. n. a n n ;ost = i F. N. NO. F N. n a ii O \ O M*1 - N o. N O. N O. I O AO 1. 1 c- P P a 1 F - ' _ T at r' m r y pr> - OP - OT - n r - '. U2f*y \ 7 U r. U ^ U r. U17 7 " 1 r 2 1 ) *, U0 (J } -> r ' * - r f?q?r fr ^ *7^ r ' ^ * 1. 7~7^-. *FFF 17' 7 J7 7 T _ - 7 *- '.«7;>p '" ', 1 '! Ur ' " '. p o r t 01 AO A N r-1. F TN^t'iM! T 71 Y PN;: U t o A7 T F F! F n F ' (' r t r- -

91 82 C ONT I NO F NT Y NO. a C HP *i ; p Cl17"*. NO. 2 [)! = (SFN. NO. ft DP = f tin =? HP AN ' ;1 P f^ s ^ s = - 14 ^ ~ ' T)r,p f r u i n s r' 13UF rr S71 r _ r i PH ASF ANGLK IN fo n A 'IT v C 0^'T- ONLY ' r ^ c r ;nir '. r- ->J " - * 0 r i -j 'i i- * f r.uf:k ' 0. 1 s 7^ r-^ r-n\ t : * r. R Uf r ". r?!! ^ 1 r t f 1 f. r r ^ ">1 r r ' " a j (i i r n " * I \ N O " - ft A T I S F I TIP f. t I li S r On c * n^v. A N 0 L n r SFAPJH COST - ', 1 0 r r U S r N. NO. 2 pi = r. >7 v- - i p.17n, *:o. f- 11* - \ m i ' i i. o s. p 'O '. i p l - *. n r - ' n LOAD Nf. ^ P1, - '. M r ' 7 S1J* - ". ')7 7r - r 7 PL A NTH ph f, S t M rt ^ * r. v ir : 7. : : r-^ # 1 u T r " - ', l a cii' ' ' ', n i r.7 1 7P r <- '.til; " r '? i -? ' ^ ". 1 3 ft r "'r ^. 1 ft 7 T ^. u r u rr ' - ~. 4 ft n * ' - r 1 PH AS17 ANSI7" T N v OH A1 I Y C'ONS -^A r N" N ft"1 S C T ^ O ' o - - ] '' nr r

92 83 C on clu sion s From th e com puter r e s u l t s, we se e t h a t th e e q u a l i t y c o n s t r a i n t s w ere s a t i s f i e d in a l l i n s t a n c e s by r e s c h e d u lin g g e n e r a t i o n. The sy ste m co u ld be re s c h e d u le d f o r s e c u r e o p e r a t io n f o r t h r e e o f th e e i g h t c o n ti n g e n c i e s. The o t h e r f i v e c o n ti n g e n c i e s c o u ld n o t be re s c h e d u le d to s a t i s f y th e p h a se a n g le c o n s t r a i n t. In o r d e r t o have th e sy stem s e c u re f o r a l l c o n tin g e n c ie s i n th e c o n tin g e n c y s e t, th e l i m i t s o f some o r a l l o f th e c o n s t r a i n t s m ust be r e l a x e d. F o r exam ple, th e sy ste m w ould be o v e r a l l s e c u re f o r a l l c o n ti n g e n c i e s i f th e maximum a llo w e d p h ase a n g le w ere 0.72 r a d i a n s. The p a t t e r n r e c o g n i t i o n / s e a r c h a p p ro a c h u s in g a c o s t f u n c t i o n t h a t i n c o r p o r a te d th e s e c u r i t y c o n s t r a i n t s was q u i t e s u c c e s s f u l and d id n o t r e q u i r e an e x te n s iv e change o f v a r i a b l e s a s in th e fo rm e r l i n e a r program m ing a p p ro a c h w hich te n d e d to mask th e b a s i c v a r i a b l e s o f th e p ro b lem.

93 CHAPTER V CONCLUSIONS An o n - l in e l e a r n i n g c o n t r o l l e r was s u c c e s s f u l l y a p p lie d to th e p ro b lem o f th e o p tim a l m egaw att lo a d - fre q u e n c y c o n tr o l o f a 2 - a r e a i n te r c o n n e c t e d pow er sy ste m. The t i e - l i n e p a ra m e te r T. Q was v a r i e d and th e l e a r n i n g 1. *w sy stem e x h i b i t e d an o n - l i n e a d a p t i v i t y and was a b le to g e n e r a l i z e from i n i t i a l t e a c h e r d a ta to y i e l d an im provem ent in sy ste m r e s p o n s e. An i n t e r p o l a t i o n te c h n iq u e e n a b le d th e l e a r n i n g c o n t r o l l e r to b e t t e r a p p ro x im a te th e c o n tin u o u s o p tim a l c o n tr o l th a n c o u ld be a t t a i n a b l e by u s in g a q u a n tiz e d s e t of c o n t r o l s from w hich to choose a s u sed in my M a s t e r 's T h e s is [lac]. U n t i l one had a " f e e l " f o r th e sy stem re s p o n s e to th e l e a r n i n g c o n t r o l l e r, th e s e l e c t i o n o f th e ra n g e s o f th e l e a r n i n g p a ra m e te rs ( th o s e w hich a f f e c t th e r a t e o f l e a r n in g and s t a b i l i t y o f th e l e a r n i n g p r o c e s s ) to y i e l d th e " b e s t" r e s u l t s f o r th e p a r t i c u l a r sy stem was found to be a tim e -c o n su m in g p r o c e s s due to th e in h e r e n t i n t e r a c t i o n o f th e p a ra m e te r s. T h is c o in c id e s w ith a p p l i c a t i o n s of l e a r n i n g c o n t r o l l e r s to o t h e r sy ste m s found in th e l i t e r a t u r e, how ever, th e ran g e o f th e l e a r n i n g p a ra m e te rs seemed to be more c o n fin e d f o r t h i s power sy ste m exam ple 8*4-

94 85 th a n in o t h e r p r o c e s s e s r e p o r t e d. An e x a m in a tio n o f th e o p tim a l fe e d b a c k g a in s [Red] shows t h a t th e re s p o n s e i s v e ry s e n s i t i v e to s m a ll v a r i a t i o n s i n th e g a i n s. T h is i s p ro b a b ly why th e l e a r n i n g p a ra m e te r s w ere somewhat l i m i t e d in ra n g e. One e i t h e r g o t f a s t l e a r n i n g, o r i n s t a b i l i t y, o r v e ry slow l e a r n i n g. The s e c u r i t y e v a l u a t i o n p ro b lem was fo rm u la te d as a p a t t e r n r e c o g n i t i o n p ro b lem and s e a r c h m ethods w ere used to o p tim a lly r e s c h e d u le g e n e r a t i o n and demand to s a t i s f y s e c u r i t y c o n s t r a i n t s f o r no rm al o p e r a t io n. The p a t t e r n r e c o g n i t i o n and s e a r c h m ethod was c o n s id e r a b ly s im p le r (d id n o t r e q u i r e e x te n s iv e changes in v a r i a b l e s ) th a n th e l i n e a r program m ing (s im p le x ) te c h n iq u e c u r r e n t l y u se d. A lth o u g h l e a r n i n g c o n t r o l l e r s h av e been a p p li e d s u c c e s s f u l l y to s e v e r a l r e a l - w o r ld p r o c e s s e s [D~], th e work o f t h i s d i s s e r t a t i o n i n d i c a t e s o n ly t h a t a p a t t e r n r e c o g n i t i o n and l e a r n i n g sy ste m a p p ro a c h i s a f e a s i b l e a l t e r n a t i v e o n - l i n e c o n t r o l to be used a s an a id to c u r r e n t o f f - l i n e t e c h n iq u e s. The p ro b lem o f m aking th e l e a r n i n g sy ste m more " c o n t e x t - f r e e " by d e v is in g a h i e a r c h i a l l e a r n i n g sy stem to l e a r n th e l e a r n i n g p a ra m e te rs o f th e i n n e r sy stem i s an a r e a o f w o rth w h ile f u t u r e s tu d y. One w ould choobe th e l e a r n i n g p a ra m e te rs o f th e h i g h e r l e v e l l e a r n i n g a lg o r ith m w hich w ould be l e s s in f lu e n c e d by th e p r o c e s s b e in g c o n t r o l l e d and sh o u ld r e s u l t in l e s s s e n s i t i v e l e a r n i n g p a ra m e te r s.

95 86 The p u rp o se of t h i s d i s s e r t a t i o n was a c c o m p lish e d. P a t t e r n r e c o g n itio n and l e a r n i n g te c h n iq u e s w ere s u c c e s s f u l l y a p p lie d to power sy stem problem s t h a t had n o t y e t been so lv e d by th e s e te c h n iq u e s. The r e s u l t s o f th e a p p l i c a t i o n was e n c o u ra g in g, and in d ic a te d t h a t such h e u r i s t i c te c h n iq u e s a re f e a s i b l e and v ia b le a l t e r n a t i v e s tn r 1i s s i c a l a p p ro a c h e s.

96 BIBLIOGRAPHY A AF B Bea BM CA Cohn D Dy E lg Andrews, H. C. I n t r o d u c t io n to M a th em atica l T echniques in P a t te r n RegognitToru W iley- I n t e r s c i e n c e, New York, A th a n s, M ichael and P a lb, P. L. O ptim al C o n t r o l. M cg raw -H ill, New York, I B a tc h e lo r, B. G. P r a c t i c a l Approach tp P a t te r n C l a s s i f i c a t i o n. Plenum P r e s s, London 197*+. B e a t t i e, L. A. "E x p e rie n c e s w ith Power System S e c u r ity A n a ly s is v i a P a t t e r n R e c o g n itio n T e c h n iq u e s." IEEE Power E n g in e e rin g S o c ie ty Summer M e e tin g. San F r a n c is c o, J u ly 20-25, 1975* P a p er A 75 U49-9. Bush, R. R. and M o s te lle r, F. S t o c h a s ti c Models f o r L e a rn in g. John W iley and S ons, New York, Chang, A. and A d ib i, M. M. "Power System Dynamic E q u iv a le n ts," IEEE T r a n s. Power A p p aratu s and S y s te m s V ol. PAS- 8 9, no. 8, Nov-Dee pp ^. Cohn, N athan. "A utom atic C o n tro l o f Power S y ste m s," 1966 IEEE I n t e r n a t i o n a l C onvention R e co rd. P a r t 12, March , New York, 1966, pp. 2^ D av ies, W. D. T. " S ta te of th e A rt and Survey o f L e a rn in g C o n tro l A p p l i c a ti o n s," P ap er No. T P 1-1, C on feren ce D e c isio n and C o n tro l and th e 1 1 th Symposium on A d a p tiv e P r o c e s s e s, New O rle a n s, LA Dec I 3- I , pp D yliacco, T. E. C o n tro l Power System s v ia th e M u l t i- l e v e l c o n c e n t. PhD T h e s is, Case W estern R eserve U n iv e r s ity, C le v e la n d, O hio, June E lg e rd, 0. I. "A utom atic G e n e ra tio n C o n tr o l," P le n a ry S e s s io n A d d re ss, IFAC Symposium, M elbourne A u s t r a i l i a, Feb ,

97 88 EF FE F F2 Ga G1 HLG I JAL Jo E lg e rd, 0. I. and F osha, C. E. J r. "Optimum M egaw att-f requency C o n tro l o f M u ltia re a E l e c t r i c E nergy S y ste m s," IEEE T ra n s. Power A p p a ratu s and S y s te m s, V ol. PAS-8 9, no. 4, A p ril 1970, pp F osha, C. E. J r. and E lg e r d, 0. I. "The M egaw att-f requency C o n tro l Problem* A New A pproach v i a O ptim al C o n tro l T h eo ry." IEEE T ran s. Power A p p a ratu s and S y s te m s, VolT PAS- 9, no.april 1970, ppt Fu, K. S. "L e a rn in g System T h eo ry," System T h e o ry. e d ite d by L. A. Zadeh and E. P o la k, M cg raw -H ill, New York, pp. ^ 25- ^ 6 3 * Fu, K. S. "R ep o rt o f S p e c ia l S e s s io n s h e ld a t th e F i r s t I n t e r n a t i o n a l J o i n t C onference on P a t t e r n R e c o g n itio n." O ct. 30-Nov. 1, 1973, W ashington, D. C. G arden, M urray. "L e a rn in g C o n tro l o f V alve A c tu a to rs in D ir e c t D i g i t a l C o n tro l S y ste m s." P ro c. JACC U niv. o f P e n n s y lv a n ia, pp G lo r io s o, R. M. E n g in e e rin g C y b e r n e tic s. P r e n t i c e - H a l l, N. J., Hammer, A., Lynn, J, W., and G ra u p ie, D. " I n v e s t i g a t i o n of a L e a rn in g C o n tro l System w ith I n t e r p o l a t i o n. " IEEE T ra n s, on S y s te m s. Man. and C y b e r n e tic s. V ol. SM-2, no. 3, J u ly 1972, ppt I n b a r, G. F., E d i t o r, S ig n a l A n a ly s is and P a t t e r n R e c o g n itio n j.n B iom edical E n g in e e r in g. John W iley and S ons, New York, 1975* J o l i s s a i n t, C. H., A r v a n ti d i s, N. V., and L u en b e rg er, D. G. "D ecom position o f R eal and R e a c tiv e Power Flows 1 A method s u i t a b l e f o r o n - lin e a p p l i c a t i o n s. " IEEE T ra n s, on Power A p p a ratu s and S y s te m s. v o l. PAS-91 * pp , Jo n e s, L. E., I I I "On th e c h o ic e o f BUbgoals f o r L e a rn in g C o n tro l S y ste m s." IEEE T ran s, on A u to. C o n tr o l. AC-13, no. 6, Dec , pp. 3-62o7

98 89 KH K a lte n b a c h, J. C. and H adju, L. P. "An o p tim a l C o r r e c tiv e R e sc h e d u lin g f o r Power System S e c u r i t y, " IEEE T ra n s, on Power A p p aratu s and System s. V ol. PAS-90, no. 2, M arch -A p ril 1971* pp KPG K a lte n b a c h, J. L., and P esch o n, John, and G eh rig, E. H., "A M a th em atica l O p tim iz a tio n T echnique f o r th e E xpan sio n o f E l e c t r i c Power T ra n sm issio n S y ste m s," IEEE T r a n s. _on_power A p p aratu s and S y s te m s. Vol PAS-8 9, No. 1, J a n. 1970, pp Kn K n ig h t, U. G. Power S ystem s E n g in e e rin g and M a th e m a tic s. Permagon P r e s s, O xford, E ngland, LAC L ac am a, Ronald J. S t u d i e s, I n L e a rn in g C o n tro l S y s te m s. M. S. T h e s is, L o u is ia n a S t a t e U n iv e r s ity, Baton Rouge, May LS Lee, S te p h en T. Y., and Schweppe, F. C. " D ista n c e M easures and C oherency R e c o g n itio n f o r t r a n s i e n t S t a b i l i t y E q u iv a le n ts." IEEE T ra n s, on Power A p p a ra tu s and S y stem s, VoTi PAS-92, S e p t-o c t pp. i & O - i W. LM L eondes, C. T. and J. M. M endel. " A r t i f i c i a l I n t e l l i g e n c e C o n tr o l." D ouglas M iss le and Space S y s te m s. p a p e r no , 1 96 ^ MF M endel, J. M. and Fu, K. S., e d i t o r s. A d ap tiv e L e a rn in g and P a t t e r n R e c o g n itio n S ystem s. T heory and A p p l i c a t i o n s. Academic P r e s s, New York, Men M endel, J. M. "S urvey o f L e a rn in g C o n tro l System s For S p a c e -V e h ic le A p p l i c a ti o n s." D ouglas P a p er No. 3777, A ugust 1966, PouglaB M issle and Space System s D i v i s i o n. H u n tin g to n Beach, C a l i f. Men2 M endel, J. M. "Survey o f L e a rn in g C o n tro l System s f o r S p a c e -V e h ic le A p p lic a tio n s. " D ouglas P ap er No. 3777* A ugust 1966, D ouglas M issle and Space System s D i v i s i o n. H u n tin g to n Beach, C a l i f. N N ils s o n, N. J. L e a rn in g M a ch in e s. M cg raw -H ill, New York, PI Pang, C. K. S tu d ie s on S e c u r i t y E v a lu a tio n bv P a t t e r n R e c o g n itio n and U n it Committment by Dynamic Program m ing in Puwer S y s te m s. PhD D i s s e r t a t i o n, Purdue U n iv e rs ity, Dec

99 Pang, C. K., K oivo, A. J., and B l-'a biad, A. H. " A p p lic a tio n o f P a t t e r n R e c o g n itio n to S te ad y S t a t e S e c u r i t y E v a lu a tio n i n a Power S y stem." Tinra Trftne. on SyfltemB. Han, and C ybflnietiafl, V ol. SMC-3, No. 6, Nov PP Pang, C. K., P ra b h a k a ra, P. S., and E l-a b ia d, A. H., and K oivo, A. J. " S e c u r ity E v a lu a tio n in Power System s U sing P a t t e r n R e c o g n itio n." IEEE T ranb. on Power A p p a ratu s and S y s te m s. V ol. 93* M ay-june 197^. P P * **'* P a t r i c k, E. A. F undam entals o f P a t t e r n R e c o g n itio n. P r e n t i c e - H a l l, E l e c t r i c a l E n g in e e rin g S e r i e s, N. J P o w ell, M. P. D. "An E f f i c i e n t Method f o r F in d in g Minimum o f F u n c tio n s o f S e v e r a l V a r ia b le s w ith o u t C a lc u la tin g D e r iv a t iv e s." Com puter J o u r n a l, V ol. 7, no. 2, J u ly 196*+, pp Reddoch, T. W., J u l i c h, P. M., Tan, 0. T., and T a c k e r, E. C. "M odels and C ost F u n c tio n a l f o r Load F requency C o n tro l in In te r c o n n e c te d Power S y ste m s," P ro c e e d in g s o f th e IEEE De c i s i o n and C o n tro l C o n fe re n c e. Dec Reddoch, T. W., M odels and C ost F u n c tio n a l f o r O ptim al A utom atic G e n e ra tio n C o n t r o l l e r s. PhD D i s s e r t a t i o n, L o u is ia n a S t a t e U n iv e r s ity, B aton Rouge, May 1973* R oss, C. W. " E rr o r A d a p tiv e C o n tro l Com puter f o r In te r c o n n e c te d Power S y ste m s." TREE Tr a n s. on Power A p p aratu s and S y s te m s. V ol. PAS-8 5, no. 7, J u l y, 190 6, pp. 7*+2-7i+9. R oss, C. W. and G reen, T. A. "Dynamic P erform ance E v a lu a tio n o f a com puter C o n tr o lle d E l e c t r i c Power S y s te m," IEEE T ra n s, on Power A p p a ratu s V ol. PAS-91, No. 3. M ay-june 1972, S a y e rs, B. McA. " S e lf-o rg a n iz in g S y ste m s." An E x p o s itio n o f A d a n tiv e C o n tro l. J. H. W e s tc o tt, E d it o r, Permagon P re ss * New Y ork, S e b e sty e n, George S. " P a t te r n R e c o g n itio n by an A d ap tiv e P ro c e s s o f Sam ple S e t C o n s tr u c tio n." IRE T r a n s. I n f o. T h e o ry, Vol I T -?, No. 8, S e p t. 1962, pp