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UBC Theses and Dissertations

The development of a reactive power management technique for a planning environment Garrett, Bretton Wayne 1978

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THE DEVELOPMENT OF A REACTIVE POWER MANAGEMENT TECHNIQUE FOR A PLANNING ENVIRONMENT by BRETTON WAYNE GARRETT B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA Jufcy, 1978 (cp Bretton Wayne G a r r e t t , 1978 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I further agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying o r ,pub l i ca t i on of this thes is for f inanc ia l gain sha l l not be allowed without my written permission. Department of E l e c t r i c a l Engineering The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date March 12, 1973 A b s t r a c t A computer-aided a l g o r i t h m i s developed f o r the management of r e a c t i v e power flow i n an e l e c t r i c power system. The technique i s designed to a s s i s t t r a n s m i s s i o n planning engineers i n e s t a b l i s h i n g s a t i s f a c t o r y base-case power flow s o l u t i o n s . The o b j e c t i v e i n the a l g o r i t h m i s to reduce r e a l power l o s s i n the system through c o n t r o l of r e a c t i v e power flow, and so i s d i f f e r e n t than conventional "VAr a l l o c a t i o n " algorithms. The minimisation i s performed by a s p e c i a l l y adapted gradient search w i t h a sub-optimal s t e p - s i z e , which can be simply incorporated i n t o a standard Newton-Raphson power flow program. A s p e c i a l feature of t h i s t h e s i s i s the p r e s e n t a t i o n of a set of contour p l o t s of the o b j e c t i v e f u n c t i o n versus various p a i r s of c o n t r o l v a r i a b l e s . An a n a l y s i s of these p l o t s i s presented, and i s used to demonstrate the v a l i d i t y of the steepest descent minimi s a t i o n technique f o r t h i s problem. Comments are given on t e s t s conducted w i t h t h i s technique on a t y p i c a l B r i t i s h Columbia Hydro and Power A u t h o r i t y power flow s i m u l a t i o n c o n s i s t i n g of 245 busses and 327 branches, w i t h 47 c o n t r o l l a b l e generators and 44 c o n t r o l l a b l e v a r i a b l e - t a p transformers. The al g o r i t h m i s claimed to be e f f e c t i v e and e f f i c i e n t f o r s t u d i e s of t h i s s i z e . TABLE OF CONTENTS INTRODUCTION 1 Chapter I . PRINCIPLES AND TECHNIQUES OF REACTIVE POWER MANAGEMENT 5"' Manual Techniques Optimal (Automatic) Techniques I I . THE MANAGEMENT OF REACTIVE POWER USING LOSS REDUCTION 22 The Development of an Objective Function I I I . THE INVESTIGATION OF THE CONSTRAINED OBJECTIVE FUNCTION 33 Contours of Constant Loss The E f f e c t of the Penalty Terms Reactive Power L i m i t s on Generators IV. THE CHOICE OF A SUITABLE MINIMISATION TECHNIQUE 45 The Treatment of C o n s t r a i n t s The Method of M i n i m i s a t i o n Summary V. THE PERFORMANCE OF THE TECHNIQUE 5 7 - i v -VI. CONCLUSIONS 67 V I I . DIRECTIONS FOR FURTHER WORK 68 REFERENCES 70 APPENDIX 74 - V -LIST OF TABLES I . Progress of Convergence f o r 245 Bus Problem ... 60 - v i -LIST OF ILLUSTRATIONS 1. Three-bus example system 90 2. Contours of constant l o s s f o r system of f i g u r e 1 91 3. Contours of constant l o s s w i t h reduced generation 91 4. Three-bus example system w i t h added branch .... 92 5. Contours of constant l o s s f o r system of f i g u r e 4 93 6. Contours of constant l o s s w i t h reduced load ... 93 7. Contours of constant l o s s w i t h added transformer 94 8. Contours of constant l o s s w i t h transformer and branch of f i g u r e 4 94 9. Contours of constant l o s s w i t h reduced power f a c t o r 95 10. Contours of constant l o s s w i t h reduced power f a c t o r and load 95 11. Contours of constant l o s s w i t h bus 3 shunt .... 9 ^ 12. Contours of constant l o s s w i t h transformer and bus 3 shunt ^6 13. Contours of constant l o s s w i t h bus 3 shunt and branch of f i g u r e 4 97 - v i i -14. Contours of f i g u r e 2 w i t h added voltage p e n a l t y 97 15. Contours of f i g u r e 3 w i t h added voltage p e n a l t y 98 16. Contours of f i g u r e 11 w i t h added voltage p e n a l t y 98 17. Contours of f i g u r e 16 w i t h added r e a c t i v e power p e n a l t y 99 18. Contours of f i g u r e 14 w i t h added p e n a l t y f o r generator r e a c t i v e power 99 19. Contours of f i g u r e 14 w i t h o p t i m i s a t i o n path .100 20. Contours of f i g u r e 7 w i t h voltage p e n a l t y and o p t i m i s a t i o n path 100 21. Contours of f i g u r e 17 w i t h o p t i m i s a t i o n path 101 - v i i i -NOTATION Notes: 1) S c a l a r q u a n t i t i e s are w r i t t e n as simple or s u b s c r i p t e d v a r i a b l e s (e.g. V). Vector q u a n t i t i e s are w r i t t e n as simple v a r i a b l e s w i t h an overhead bar (e.g. V). M a t r i x q u a n t i t i e s are w r i t t e n as simple v a r i a b l e s w i t h i n square brackets (e.g. [YJ ) . 2) The n u l l matrix i s w r i t t e n [o] , whi l e the id e n t i t y -matrix i s w r i t t e n [l] . The zero vector i s w r i t t e n 0*. 3) A "A" preceding some v a r i a b l e (e.g. AV) i n d i c a t e s a v a r i a t i o n i n tha t v a r i a b l e . The r e a l and r e a c t i v e power mismatches are the exceptions to t h i s r u l e . They are w r i t t e n AP and AQ. That these are mismatches w i l l always be pointed out i n the t e x t . 4) A s u b s c r i p t ( i , j , or k) a p p l i e d to a s c a l a r v a r i a b l e (e.g. Vi ) i n d i c a t e s t h a t the v a r i a b l e p e r t a i n s t o the bus i n d i c a t e d by the s u b s c r i p t . S i m i l a r l y , a v a r i a b l e w i t h two s u b s c r i p t s i n d i c a t e s t h a t the v a r i a b l e p e r t a i n s t o the p a i r of busses i n d i c a t e d . A p a r t i a l d e r i v a t i v e of s u b s c r i p t e d v a r i a b l e s i n d i c a t e s t h a t the d e r i v a t i v e i s an element of the d e r i v a t i v e matrix (e.g. — — i s the aVj ( i , j ) element of the matrix jjjy^ J) - A s p e c i a l s u b s c r i p t , s, i s used t o i n d i c a t e the s l a c k bus. - i x -The u n q u a l i f i e d term "power" r e f e r s t o the complex power S. Absolute value i s i n d i c a t e d w i t h two v e r t i c a l bars (e.g. j.-lj = 1) , w h i l e the Euc l i d e a n norm i s i n d i c a t e d w i t h two p a i r s of v e r t i c a l bars (e.g. |Jl + j ljj = Yl) . Summations are given over elements of a s e t . For example, the sum of the squares of the r e a c t i v e power mismatches 2 fo r the power system i s ^t(AQi)• The s u p e r s c r i p t "des" i n d i c a t e s the d e s i r e d value of the v a r i a b l e . n t S i m i l a r l y , "sched" i n d i c a t e s the scheduled • els s value of the v a r i a b l e . E.g. Vj_ i n d i c a t e s the d e s i r e d value of the voltage magnitude at bus i . The s u p e r s c r i p t "max" i n d i c a t e s the maximum d e s i r e d value of the v a r i a b l e . S i m i l a r l y , "min" i n d i c a t e s the minimum d e s i r e d value. Where a vector or vector-valued expression i s shown t o be greater than 0, t h i s i s intended t o mean t h a t each element of the vector (or expression) i s greater than 0. An a s t e r i s k s u p e r s c r i p t i n d i c a t e s the complex conjugate, e.g. S i s the complex conjugate of the power. A "T" s u p e r s c r i p t i n d i c a t e s the transpose of the vec t o r or matrix. E.g. V i s the transpose of the voltage v e c t o r . A s u b s c r i p t "o" i n d i c a t e s the i n i t i a l value of the v a r i a b l e . -X-e.g. x Q i s the i n i t i a l value of vector x. 14) A "V" preceding a v a r i a b l e i n d i c a t e s the gradient of that v a r i a b l e , e.g. V F U i n d i c a t e s the gradient of F w i t h respect to u. A "rj2." s i m i i a r i y i n d i c a t e s the Hessian of the v a r i a b l e . 15) A s u p e r s c r i p t such as "k" or "k+1" i s used to i n d i c a t e the value of the v a r i a b l e at step k or k+1 i n the i t e r a t i v e process. - x i -Symbols: S - complex power. Sometimes used f o r the s e n s i t i v i t y m a t rix |jj^-j, but always i n d i c a t e d as such. P - r e a l (active) power = Re{sj. P o s i t i v e from bus i n t o system (or ground). Q - imaginary (r e a c t i v e ) power = Irn^S^. P o s i t i v e from bus i n t o system (or ground). V , - voltage magnitude. 8 - voltage angle r e l a t i v e t o an a r b i t r a r y reference voltage ( u s u a l l y the s l a c k bus v o l t a g e ) . Z - impedance (complex). R - r e s i s t a n c e = Re{zj. X - reactance = Im[z]. Y - admittance (complex) = Z~^~. G - conductance = Re £YJ. B - susceptance = Im^Y^. - tra n s m i s s i o n r e a l power l o s s , f - o b j e c t i v e f u n c t i o n to be minimised, g - the vector of e q u a l i t y c o n s t r a i n t s (equal to 0). h - the vector of i n e q u a l i t y c o n s t r a i n t s (greater than 0) . u - vector of c o n t r o l v a r i a b l e s : generator v o l t a g e s , transformer taps, and a l l o c a t e d r e a c t i v e power. - X l l -vector of independent v a r i a b l e s : load bus vo l t a g e s , voltage angles. weighting f a c t o r s f o r voltage p e n a l t y term, weighting f a c t o r s f o r r e a c t i v e a l l o c a t i o n term, tap s e t t i n g f o r v a r i a b l e tap transformers of the set NT. set of a l l busses i n the system. se t of a l l load and other busses i n the system f o r which the bus voltage i s not f i x e d . set of a l l generators i n the system e l i g i b l e f o r voltage adjustment dur i n g o p t i m i s a t i o n . s e t of a l l busses i n the system e l i g i b l e as l o c a t i o n s f o r shunt r e a c t i v e banks during o p t i m i s a t i o n . set of a l l busses at which the voltage i s to be he l d to w i t h i n some tolerance of nominal. s e t of a l l v a r i a b l e transformer taps e l i g i b l e f o r adjustment d u r i n g o p t i m i s a t i o n . s e t of a l l busses w i t h a connecting branch t o bus i . subset of NV f o r which voltages are higher than the d e s i r e d maximum. subset of NV f o r which voltages are lower than the de s i r e d minimum. ACKNOWLEDGEMENT The author would l i k e to thank the System Planning Department of the B r i t i s h Columbia Hydro and Power A u t h o r i t y f o r permission t o undertake t h i s p r o j e c t while employed by the Department, and f o r the system data used f o r the two t e s t cases. The author would p a r t i c u l a r l y l i k e t o thank Dr. B.A. Dixon of B.C. Hydro f o r h i s unceasing encouragement and he l p , without which t h i s p r o j e c t would never have begun. The author would a l s o l i k e t o thank Drs. H.W. Dommel and G.F. Schrack of the Department of E l e c t r i c a l Engineering of the U n i v e r s i t y of B r i t i s h Columbia, f o r t h e i r valued e f f o r t s as supervisors of t h i s p r o j e c t . -1-INTRODUCTION With power systems becoming l a r g e r and more t i g h t l y meshed, the development of good power flow s i m u l a t i o n s f o r each year and loa d i n g c o n d i t i o n of a c a p r i c i o u s ten-year system pla n i s becoming one of the most tedious procedures i n v o l v e d i n planning t r a n s m i s s i o n networks. The procedure i n v o l v e d i n producing a good power flow s o l u t i o n i s s u f f i c i e n t l y w e l l defined t h a t the m a j o r i t y of the adjustments can be done by an automated technique. Someone f a m i l i a r w i t h the power system (e.g. a good t e c h n o l o g i s t ) can set up the c o n s t r a i n t s f o r the process and evaluate the r e s u l t s . I f the r e s u l t s are not e x a c t l y r i g h t the f i r s t time, as i s l i k e l y (depending on the a b i l i t y of the u s e r ) , the technique can be used i t e r a t i v e l y . Provided the person "at the c o n t r o l s " understands what c o n s t i t u t e s a good s o l u t i o n , i t i s p o s s i b l e to achieve a good s i m u l a t i o n i n much l e s s time, and w i t h much l e s s f r u s t r a t i o n , than i s p o s s i b l e w i t h standard manual power flow s i m u l a t i o n techniques. There are b a s i c a l l y only three steps i n producing a power flow s i m u l a t i o n . F i r s t , the tr a n s m i s s i o n and generation must be modelled, and the system loads must be adjusted t o the load f o r e c a s t f o r the year and l o a d i n g c o n d i t i o n (e.g. 1984 heavy -2-winter peak) under c o n s i d e r a t i o n . Second, a reasonable genera-t i o n schedule, i n c l u d i n g import/export schedules, must be e s t a b l i s h e d t o support the t o t a l load. This schedule must take i n t o account the e f f e c t of upstream hydro p l a n t s on downstream p l a n t s (e.g. B.C. Hydro and Power A u t h o r i t y ' s S i t e One p l a n t i s dependent on the upstream G.M. Shrum p l a n t ) , the a v a i l a b i l i t y of water f o r hydro p l a n t s , the d e s i r a b i l i t y of operating thermal p l a n t s , the merit order of a v a i l a b l e hydro and thermal p l a n t s , e t c . T h i r d , the generator v o l t a g e s , v a r i a b l e transform-er taps, switchable c a p a c i t o r / r e a c t o r banks and the l i k e a l l must be adjusted so t h a t the voltages around the system are w i t h i n safe (and s t a b l e ) operating l i m i t s . The voltages are then f u r t h e r adjusted using a v a i l a b l e and planned r e a c t i v e power sources, so as t o o b t a i n a reasonable voltage p r o f i l e . This f i n a l trimming of voltages i s c a r r i e d out i n i n c r e a s i n g d e t a i l according to the nearness of the study date to the a c t u a l date, w i t h the most a t t e n t i o n being given t o the f i r s t year of the plan., This t h i r d and l a s t step, the adjustment of generators, * U s u a l l y t h i s i s a s u b j e c t i v e e v a l u a t i o n , which i n v o l v e s many i n t e r - r e l a t i n g f a c t o r s , such as whether the s i m u l a t i o n i s f o r a normal or emergency (outage) c o n d i t i o n , and the l o c a t i o n , s i z e , and nature of the a f f e c t e d l o a d . -3-v a r i a b l e transformer taps, switched r e a c t i v e banks, e t c . , and p a r t i c u l a r l y the d e t a i l e d i n v e s t i g a t i o n i n v o l v e d i n e a r l y budget years, g e n e r a l l y r e q u i r e s the most work i n any power flow s i m u l a t i o n , and i t i s thus f o r the r e s o l u t i o n of t h i s step t h a t many computer-aided techniques have been proposed. This t h e s i s summarizes s e v e r a l of the var i o u s techniques proposed f o r r e a c t i v e power management , and then presents the development and a n a l y s i s of an o b j e c t i v e f u n c t i o n d i f f e r e n t from t h a t adopted by most other authors. Instead of minimising the d e v i a t i o n of system voltages from "standard" values as i s u s u a l l y done, the technique presented here reduces the tran s m i -s s i o n l osses i n the system, which, i n the op i n i o n of the author, provides the s o l u t i o n which good "voltage d e v i a t i o n " techniques only approximate. This amounts t o s o l v i n g the r e a c t i v e h a l f of the general optimal power flow problem addressed by Dommel [ l j * Reactive power management i s defined here to be the c o n t r o l of generator v o l t a g e s , v a r i a b l e transformer tap s e t t i n g s , and e x i s t i n g switchable shunt c a p a c i t o r and r e a c t o r banks, . . and the a l l o c a t i o n of new c a p a c i t o r and r e a c t o r banks, i n a manner which best achieves the d e s i r e d goal of voltage c o n t r o l , l o s s r e d u c t i o n , or both. This i s more general than r e a c t i v e power a l l o c a t i o n ("VAr a l l o c a t i o n " ) which deals on l y w i t h the a l l o c a t i o n of new and e x i s t i n g b l o c k s of shunt compensation. -4-and Sasson [2] . F i n a l l y , the a n a l y s i s of the o b j e c t i v e f u n c t i o n i s used to s e l e c t and t e s t a computationally e f f i c i e n t s o l u t i o n technique which w i l l r e q u i r e a minimum of user i n t e r a c t i o n to be e f f e c t i v e . -5-CHAPTER I P r i n c i p l e s and Techniques of Reactive Power Management Although r e a c t i v e power does not u s e f u l l y c o n t r i b u t e t o the flow of energy i n the power system, i t has a s i g n i f i c a n t e f f e c t on power system performance and e f f i c i e n c y . By reducing unne-cessary r e a c t i v e power flows, i t i s p o s s i b l e t o increase therm-a l l y r e s t r i c t e d a c t i v e power c a p a c i t i e s of l i n e s , transformers, and generators. System s t a b i l i t y i s improved by reducing the wide v a r i a t i o n i n bus voltages which c h a r a c t e r i s t i c a l l y accom-pany high r e a c t i v e power flows. The lower c u r r e n t s improve voltage r e g u l a t i o n on d i s t r i b u t i o n c i r c u i t s , and reduce energy losses throughout the system. There are s e v e r a l ways i n which unnecessary r e a c t i v e power flows can be reduced. The most common way i s to supply the necessary r e a c t i v e power generation at the load i t s e l f . This i s u s u a l l y done using shunt c a p a c i t o r s or r e a c t o r s . Although, i d e a l l y , shunt r e a c t i v e devices should be provided at a l l busses where the power f a c t o r i s l e s s than u n i t y , the cos t of these devices p r o h i b i t s t h i s p r a c t i c e . Instead, i t i s usual to i n s t a l l shunt devices i n such a way tha t the b e n e f i t s are shared over s e v e r a l adjacent busses. A -6-t r a d e o f f i s thus made between the e f f e c t i v e n e s s of the c o r r e c -t i o n , and the cost of the equipment. The a l l o c a t i o n of new and e x i s t i n g shunt devices i n a manner which o f f e r s the gre a t e s t b e n e f i t f o r the lowest c o s t i s the problem addressed by most "VAr a l l o c a t i o n " techniques. Some techniques perform true r e a c t i v e power management, o f f e r i n g a d d i t i o n a l ways of c o n t r o l l i n g r e a c t i v e power flow. W i t h i n s t a t o r current, f i e l d c u r r e n t , and s t a b i l i t y r e s t r i c t i o n s , generator voltages may be c o n t r o l l e d , a l t e r i n g both the r e a c t i v e power production (or absorption) at the generator i t s e l f , and the r e a c t i v e power flow through adjacent, s t r o n g l y connected busses. Another method of c o n t r o l l i n g the r e a c t i v e power flow i n a system i s transformer tap adjustment. In overhead t r a n s m i s s i o n networks, the l i n e inductance i s t y p i c a l l y much greater than l i n e resistance. Under these c o n d i t i o n s , the a c t i v e power flow along a tr a n s m i s s i o n l i n e i s n e a r l y independent of the d i f f e r e n c e between the voltage magnitudes at the ends of the l i n e , w hile the r e a c t i v e power flow i s n e a r l y p r o p o r t i o n a l to t h i s voltage d i f f e r e n c e . By using transformer taps to a l t e r one or more bus vol t a g e s , i t i s p o s s i b l e t o c o n t r o l the flow of r e a c t i v e power through these busses independently of the flow of r e a l power. (Note t h a t only the r e a c t i v e power f l o w i n g through a bus may be -7-c o n t r o l l e d . Any r e a c t i v e power absorbed by a load must always be s u p p l i e d , i r r e s p e c t i v e of any adjustments i n transformer taps.) Manual Techniques Perhaps the most obvious method f o r a l l o c a t i n g r e a c t i v e power sources i s by i n s p e c t i o n of the power flows on va r i o u s l i n e s . Adjustments t o generator v o l t a g e s , e x i s t i n g (switchable) shunt r e a c t i v e banks, and v a r i a b l e transformer tap s e t t i n g s can be determined from c a r e f u l study of bus voltages and c i r c u i t r e a c t i v e power flows. A f t e r these adjustments have been determ-ined and checked w i t h a new power flow s i m u l a t i o n , remaining regions of high or low v o l t a g e , and t r a n s m i s s i o n l i n e s w i t h high r e a c t i v e power flows are i d e n t i f i e d from the newly c a l c u l a t e d r e s u l t s . New shunt r e a c t i v e devices can then be l o c a t e d at busses c e n t r a l t o the problem areas. This procedure i s repeated f o r each l o a d i n g c o n d i t i o n t o determine the t o t a l system requirements f o r the year being s t u d i e d . This technique, which i s o f t e n r e f e r r e d t o as the t r i a l and  e r r o r method, can a l s o be used to p l a n voltage support f o r outage cases. Lines can be removed from the study, as necessary, to i n v e s t i g a t e each system contingency. The power flows are c a l c u l a t e d as f o r the normal c o n d i t i o n base case, and r e a c t i v e -8-compensation l o c a t e d so as to c o r r e c t f o r adverse v o l t a g e s , and to minimise the power flows along h e a v i l y loaded t r a n s m i s s i o n l i n e s . The shunt compensation requirements f o r a l l cases would then be a s s i m i l a t e d i n t o a s i n g l e system p l a n . While t h i s method of a l l o c a t i n g r e a c t i v e power sources gives exact power flow s o l u t i o n s , i t does have s e v e r a l drawbacks. F i r s t l y , at l e a s t two power flows=r-one . i n i t i a l t o f i n d the uncorrected voltages and power flows, and at l e a s t one other to t e s t the a l l o c a t i o n scheme—are r e q u i r e d f o r each l o a d i n g and contingency c o n d i t i o n . Secondly, i t i s d i f f i c u l t to determine the c o r r e c t s i z e f o r each r e a c t i v e i n s t a l l a t i o n , p a r t i c u l a r l y when i t i s necessary t o keep voltages at s e v e r a l adjacent busses w i t h i n some tolerance of nominal. This problem i s f u r t h e r complicated by the need to combine r e a c t i v e requirements f o r a l l contingencies i n t o a s i n g l e p l a n f o r the system, i n that compensation added f o r one contingency w i l l a f f e c t the amount of compensation needed f o r other con t i n g e n c i e s . In s p i t e of the amount of work involved i n t h i s type of r e a c t i v e power a l l o c a t i o n study, the t r i a l and e r r o r technique i s the one used by most Canadian u t i l i t i e s [3] f o r s t u d i e s of the f i r s t one or two years of the system p l a n . Another manual technique which i s sometimes used (and i s -9-p r e s e n t l y i n use a t B.C. Hydro) i s the s e n s i t i v i t y technique £ 4 ] . In t h i s method, the p a r t i a l d e r i v a t i v e s of voltage change w i t h respect t o r e a c t i v e power ( - ^ 7 ^ ) are c a l c u l a t e d from the network data (these d e r i v a t i v e s can e a s i l y be obtained from the Jacobian matrix of a Newton-Raphson power flow program). I t i s then p o s s i b l e to f i n d the bus voltages which accompany a change i n r e a c t i v e power: AV = [Sj] AQ (1) where AV = expected voltage changes AQ = changes i n r e a c t i v e power out of busses d V 4Qj ( s e n s i t i v i t y matrix d e r i v e d from Jacobian m a t r i x ) . From t h i s equation, i t i s p o s s i b l e t o c a l c u l a t e the changes i n bus r e a c t i v e shunts which w i l l a djust the voltage at bus i by AV^. The most e f f e c t i v e l o c a t i o n f o r a shunt r e a c t i v e bank i s then at the bus w i t h the highest s e n s i t i v i t y c o e f f i c i e n t (element of [sj]) f o r the voltage at bus i ( i . e . bus k, where l s i k l = m a x ^ l S i r j | ' f o r a 1 1 3 ^ NQJ) • Both the l o c a t i o n and the s i z i n g of the bank are thus obtained d i r e c t l y . I t i s i n t e r e s t i n g t o note t h a t f o r d i f f e r e n c e s (9^ - 0j) ~ 0, -10-voltage magnitudes V ^ a 1 p.u., and low r e s i s t a n c e branches ( R i j < < : x i j ) ' t h e n which i s j u s t the s h o r t - c i r c u i t r a t i o of the branch between busses i and j . This, along w i t h the approximation has been e x p l o i t e d by M a l i s z e w s k i , Garver, and Wood £5]. Dommel ^6] has shown th a t the s e n s i t i v i t y technique can modified t o determine the amount of r e a c t i v e compensation Qj_ re q u i r e d at a s i n g l e bus i t o minimise the o b j e c t i v e f u n c t i o n f = S ( V i - v d e s ) 2 where V i = voltage at bus i des V"i = d e s i r e d voltage at bus 1. I n s e r t i n g the f i r s t - o r d e r s e n s i t i v i t y r e l a t i o n of Vj_ about V( gives -11-f = ^ ( v o i + s ± i AQ, - v f e s ) 2 where s i j ^Qj i s t h e expression f o r &VL = ^AQA from (1) At the minimum d f - 0. or dAQj from which AQ, = 1 J ' 3 I s * The s e n s i t i v i t y technique has the advantage of being a true r e a c t i v e power management technique, and can be used to c a l c u l a t e r e q u i r e d adjustments i n transformer taps and generator voltages i n the same way as was done above f o r shunt compensation. The major disadvantage of the s e n s i t i v i t y technique i s that i t assumes th a t the r e l a t i o n s h i p between AV and AQ i s a l i n e a r one, while i n p r a c t i c e i t i s very complex. The c a l c u l a t e d values, t h e r e f o r e , are v a l i d only f o r small changes i n bus -12-v o l t a g e . * A f t e r an a l l o c a t i o n scheme i s s e l e c t e d , i t must be t e s t e d w i t h a power flow t o f i n d the exact bus v o l t a g e s . I f the voltages are s t i l l u n s a t i s f a c t o r y , the procedure can be repeated. Note the s i m i l a r i t y between t h i s technique and the t r i a l and e r r o r approach: they both begin w i t h the s o l u t i o n of an i n i t i a l power flow, and end w i t h the s o l u t i o n of a con f i r m i n g power flow. The d i f f e r e n c e i s s o l e l y i n the way the r e a c t i v e i n s t a l l a t i o n s are l o c a t e d and s i z e d . In the t r i a l and e r r o r technique, the planning engineer s e l e c t s the l o c a t i o n on the b a s i s of the r e a c t i v e power flows and voltage l e v e l s . He must then use judgement t o s i z e the i n s t a l l a t i o n — a t a s k complicated by the d i f f i c u l t y of p r e d i c t i n g the e f f e c t of the new i n s t a l l a -t i o n on voltages at adjacent busses. The s e n s i t i v i t y technique provides t h i s i n f o r m a t i o n d i r e c t l y ( a l b e i t approximately), and much more q u i c k l y . The s e n s i t i v i t y technique i s f a s t , f a i r l y simple, handles a l l c o n t i n g e n c i e s , a l l l o a d i n g c o n d i t i o n s , and a l l types of * The accuracy of the p r e d i c t i o n s always improves as the mag-nitude of voltage c o r r e c t i o n decreases. For example, us i n g t h i s technique on a t y p i c a l B.C. Hydro system, t o change a bus voltage from 0.998 p.u. t o 1.025 p.u., the p r e d i c t e d change i n Q at th a t bus b r i n g s the voltage to 1.026—an e r r o r of 0.12%. For a s t a r t i n g voltage (at the same bus) of 1.007 p.u., the new p r e d i c t e d change i n Q b r i n g s the voltage to 1.0253 p.u.—an e r r o r of only 0.03%. -13-r e a c t i v e power management (shunt compensation, transformer taps, and generator v o l t a g e s ) . Unfortunately, i t i s o n l y approximate, and i t shares w i t h a l l manual techniques, and many of the "optimal" automatic techniques, the d i f f i c u l t y of combining a l l o c a t i o n s f o r a l l l o a d i n g c o n d i t i o n s and contingencies i n t o a s i n g l e , l e a s t - c o s t system p l a n . Optimal (Automatic) Techniques The major d i f f e r e n c e between manual and optimal r e a c t i v e power a l l o c a t i o n techniques i s t h a t optimal techniques both l o c a t e and s i z e shunt r e a c t i v e banks a u t o m a t i c a l l y . The term "optimal" i m p l i e s — i n some cases c o r r e c t l y — t h a t the r e s u l t a n t a l l o c a t i o n scheme i s the best p o s s i b l e , subject t o c o n s t r a i n t s such as bus voltage l i m i t s , maximum and minimum acceptable i n s t a l l a t i o n s i z e s , e t c . Optimal r e a c t i v e power a l l o c a t i o n techniques can be g e n e r a l l y d i v i d e d i n t o two types. The s i m p l e s t , and the one on which the m a j o r i t y of the l i t e r a t u r e has been w r i t t e n , i s the group of l i n e a r i z e d techniques: l i n e a r programming, i n t e g e r programming, and 0-1 programming. These techniques u s u a l l y deal o n l y w i t h the a l l o c a t i o n of shunt r e a c t i v e banks, and always assume a l i n e a r " o b j e c t i v e f u n c t i o n " (which i n t h i s case means t h a t AV -14-and AQ are assumed p r o p o r t i o n a l ) , subject to l i n e a r c o n s t r a i n t s . (There i s a v a r i a t i o n of l i n e a r programming known as q u a d r a t i c  programming, which allows the use of a quadratic o b j e c t i v e f u n c t i o n subject to l i n e a r c o n s t r a i n t s . ) The second type of optimal technique, and the most complex, i s the group of nonlinear techniques o f t e n termed nonli n e a r  programming. These techniques d e a l w i t h a l l types of r e a c t i v e power management, can handle a wide range of o b j e c t i v e f u n c t i o n s , and can have both l i n e a r and n o n l i n e a r c o n s t r a i n t s . This great f l e x i b i l i t y i s not without p e n a l t y , however, as the computation time r e q u i r e d f o r s o l u t i o n increases r a p i d l y w i t h problem complexity, and the s o l u t i o n technique used must o f t e n be t a i l o r e d to the problem i n order to f i n d any s o l u t i o n at a l l . Nevertheless, i f the e x t r a f l e x i b i l i t y i s r e q u i r e d , i t i s n e a r l y always p o s s i b l e t o develop a workable al g o r i t h m . Since the o b j e c t i v e f u n c t i o n i s u n r e s t r i c t e d , these techniques produce as exact an answer as the p r e c i s i o n of the computer, the data, and the convergence behaviour of the a l g o r i t h m w i l l permit. L i n e a r Optimal Techniques The b a s i c l i n e a r optimal r e a c t i v e power a l l o c a t i o n technique i s l i n e a r programming. In the l i n e a r programming approach, the -15-voltage changes i n the system are assumed to be p r o p o r t i o n a l to the changes i n r e a c t i v e power; AV = [s] AQ (2) where AV = expected changes i n bus voltages AQ = changes i n r e a c t i v e power out of busses [s^] = constant matrix, which may or may not be the [Sj] of equation (1) . The c o n s t r a i n t s on AV, AV ± > V m i n - V ± f o r a l l i € NV A v ± ^ v m a x - V ± are combined w i t h (2) to ob t a i n the set of i n e q u a l i t i e s .SgijAQj * v f n - V i j (3) J 6 N Q m a x \ f o r a l l i £ NV ^>S±A AQ-i < V i - V i ) (4) J6NQ J J 7 The c o n s t r a i n t s on AQ are: -16-AQi 2t 0, or AQj ^ Q m a x - Qj -AQj — 0 f o r a l l j £ NQ (5) (6a) (6b) where equation (6a) i s used f o r i n d u c t i v e a l l o c a t i o n s , and equation (6b) i s used f o r c a p a c i t i v e a l l o c a t i o n s . The o b j e c t i v e f u n c t i o n to be minimised i s the sum of the absolute values of a l l the r e a c t i v e a d d i t i o n s : The s e r i e s of equations (3) - (7) form a standard l i n e a r programming problem, and can be solved by any of the a v a i l a b l e l i n e a r programming algorithms. The r e s u l t of the o p t i m i s a t i o n i s a set of r e a c t i v e a l l o c a t i o n s AQ f o r the busses chosen by the planning engineer. In some methods, AQ i s permitted to assume any v a l u e , w h i l e i n others i t i s c o n s t r a i n e d t o be one of a s e t of standard s i z e s . In the case where any value i s p e r m i s s i b l e , the s i z e s must l a t e r be rounded to the nearest standard s i z e by the p l a n n i n g engineer. A power flow can then be run w i t h the standardized a l l o c a t i o n s to o b t a i n the exact bus v o l t a g e s . Although d i f f e r i n g i n some p o i n t s — e s p e c i a l l y i n the f(Q) = £ | A Q j | • (7) -17-for m u l a t i o n of equation ( 2 ) — a l l l i n e a r programming a l l o c a t i o n techniques have t h i s general form [5,7J. They are u s u a l l y solved w i t h the Simplex l i n e a r programming algorithm, f o r which standard code i s a v a i l a b l e . These techniques can be made to handle m u l t i p l e c o n t i n -gencies a u t o m a t i c a l l y . M a l i s z e w s k i et a l . [5] accomplish t h i s by s o l v i n g a l l contingencies together as one massive l i n e a r programming problem. I f necessary, some of the c a p a c i t y needed f o r one contingency can be removed during other contingencies to prevent the voltage from exceeding t h a t experienced under normal c o n d i t i o n s . The i n s t a l l e d c a p a c i t y i s permitted to take any value, l a t e r being rounded t o the nearest standard value by the user. The l i n e a r programming techniques based on (2) through (7) use l i n e a r i z e d s e n s i t i v i t y i n f o r m a t i o n . A d i f f e r e n t technique, by K o h l i and K o h l i ^ 8 ] , uses an i n t e g e r programming technique to a l l o c a t e c a p a c i t o r s i n u n i t s i z e s . This technique b u i l d s a " t r e e " of a l l p o s s i b l e c a p a c i t o r c o n f i g u r a t i o n s , subject t o the r e s t r i c t i o n s determined by the pla n n i n g engineer. These c o n f i g u r a t i o n s are then s y s t e m a t i c a l l y t e s t e d to see i f they s a t i s f y bus voltage c o n s t r a i n t s . The search i s ordered so t h a t the f i r s t f e a s i b l e s o l u t i o n w i l l be optimal ( l e a s t number of c a p a c i t o r s ) . The authors suggest t h a t the extent of the -18-(otherwise s u b s t a n t i a l ) search can be reduced by assuming t h a t the voltage on a bus w i l l be a f f e c t e d the most by c a p a c i t o r s at that bus. Unfortunately t h i s i s not always t r u e . (The voltage increase which accompanies c a p a c i t o r a d d i t i o n s i s due t o a reduc t i o n of curren t i n the tr a n s m i s s i o n feeding the bus i n question. The c u r r e n t r e d u c t i o n — a n d hence voltage i n c r e a s e — w i l l be the greatest when the c a p a c i t y i s added d i r e c t l y at the p o i n t of the r e a c t i v e l o a d , due to the saving i n r e a c t i v e power losses betwee the bus i n question and the load p o i n t . The exception to t h i s i when there i s a constant voltage bus i n the v i c i n i t y , i n which case p r e d i c t i o n i s complex, and i s best done d i r e c t l y from the s e n s i t i v i t i e s . ) As a r e s u l t , the search remains s u b s t a n t i a l , and so i s of l i t t l e , i f any, p r a c t i c a l value f o r use w i t h other than s m a l l subsystems of usual power networks. Convergence of a l l of these techniques may be hampered by automatic transformer tap s w i t c h i n g or generator Q l i m i t s , as these upset the assumed network l i n e a r i t y . Nonlinear Optimal Techniques Nonlinear optimal r e a c t i v e power management techniques may t r e a t the r e a c t i v e power problem separately,,, or as a p a r t of the - I n -complete optimal power flow problem, which would a l s o optimise r e a l power flows. s e p a r a t e l y . They minimise the t o t a l c a p a c i t y a l l o c a t e d , subject to voltage and shunt c a p a c i t y c o n s t r a i n t s . This technique solves the non l i n e a r problem w i t h a s e r i e s of l i n e a r approximations designed to make each gradient step terminate on a c o n s t r a i n t boundary. A f t e r each gradient step, a power flow i s solved to f i n d the exact bus v o l t a g e s . By assuming t h a t the optimal s o l u t i o n l i e s at the i n t e r s e c t i o n of c o n s t r a i n t boundaries, t h i s technique a p p l i e s l i n e a r approximation  programming [lo] to f i n d the s o l u t i o n . This technique i s very s i m i l a r t o the l i n e a r programming approach of [5] . Sachdeva and B i l l i n t o n | l l ) s olve the r e a c t i v e power man-agement problem as a p o r t i o n of a complete optimal power flow. This technique performs true r e a c t i v e power management i n c l u d i n g transformer taps and generator v o l t a g e s . The shunt compensation i s a l l o c a t e d i n u n i t s i z e s . Although the technique does handle m u l t i p l e c o n t i n g e n c i e s , t h i s increases the data storage requirements c o n s i d e r a b l y . A modified method which uses l e s s storage i s presented i n \l2~]. This modified technique solves the optimal r e a l power flow f i r s t , and then the optimal r e a c t i v e power flow (with r e a c t i v e shunt t r e a t the r e a c t i v e power problem -20-a l l o c a t i o n ) . This c y c l e i s repeated i f the optimal r e a c t i v e step a l t e r s the voltage angles by an amount s u f f i c i e n t t o change the r e a l power flow s i g n i f i c a n t l y from the optimum. Both of these methods, the f u l l and the decoupled, use a Fletcher-Reeves or F l e t c h e r - P o w e l l technique f o r the m i n i m i s a t i o n , w i t h a l l c o n s t r a i n t s t r e a t e d as p e n a l t y f u n c t i o n s ( s i m i l a r l y to [2 ] ) -Two other optimal power flow techniques are a l s o a p p l i c a b l e to r e a c t i v e power management. Sasson e t a l . [13] use a technique s i m i l a r to t h a t of [ l l ] , w i t h a l l c o n s t r a i n t s handled as p e n a l t y f u n c t i o n s . The c o e f f i c i e n t s of the p e n a l t y terms are a l t e r e d during the s o l u t i o n t o speed convergence to a f e a s i b l e s o l u t i o n . The major d i f f e r e n c e between the techniques of [l3] and [ll] i s i n the m i n i m i s a t i o n , f o r which Sasson e t a l . use the Hessian, which i s computed from the Newton-Raphson Jacobian m a t r i x . Dommel and Tinney [ l ] use a gradient search (and a l s o a Hessian approximation) f o r m i n i m i s a t i o n , t r e a t i n g the power flow equations ( e q u a l i t y c o n s t r a i n t s ) w i t h Lagrange m u l t i p l i e r s . The i n e q u a l i t y c o n s t r a i n t s are t r e a t e d as absolute l i m i t s f o r independent v a r i a b l e s (e.g. generator v o l t a g e s ) , and as p e n a l t y f u n c t i o n s f o r dependent v a r i a b l e s (e.g. load bus v o l t a g e s ) . The gradient technique, w h i l e sometimes slower to converge than F l e t c h e r - P o w e l l or Hessian techniques, r e q u i r e s c o n s i d e r a b l y -21-l e s s storage and computation time per step. A l l of the techniques [ l , 2,11-13] solve the optimal power flow (and the r e a c t i v e power management extension) without assuming o b j e c t i v e f u n c t i o n l i n e a r i t y . -22-CHAPTER I I The Management of Reactive Power Using Loss Reduction Most of the many papers w r i t t e n on r e a c t i v e power a l l o c a t i o n propose the use of l i n e a r programming and r e l a t e d techniques to determine the l e a s t amount of shunt capacitance which must be added at v a r i o u s busses t o ensure t h a t system voltages are above u s e r - s p e c i f i e d minimums. Most of these l i n e a r programming techniques are adaptable to shunt inductor s w i t c h i n g a l s o , a l l o w i n g the scheduling of a l l shunt r e a c t i v e devices i n a roughly- l e a s t cost way. The computer program thus does i n an o r d e r l y f a s h i o n e x a c t l y what a planning engineer would do i n an "educated" manual way: add c a p a c i t o r s or inductors as necessary to adjust a l l voltages t o w i t h i n set l i m i t s . I f t h i s i s a l l t h a t i s d e s i r e d , these techniques work very w e l l : they are f a s t , easy to use, and inexpensive t o run. The a l l o c a t i o n of new shunt i n s t a l l a t i o n s cannot reasonably begin, however, u n t i l a l l generator v o l t a g e s , v a r i a b l e transform-er taps, s t a t i c and synchronous compensators, and e x i s t i n g switched shunt r e a c t i v e banks have been adjusted to o b t a i n the * These techniques minimise the t o t a l a d d i t i o n a l c a p a c i t y , n e g l e c t i n g the f a c t t h a t the cost of a new i n s t a l l a t i o n i s l i k e l y much more than the cost of e n l a r g i n g an e x i s t i n g one. -23-best p o s s i b l e base case. Only then i s i t reasonable to attempt i d e n t i f y i n g s i z e s and l o c a t i o n s f o r any new i n s t a l l a t i o n s . The task of o b t a i n i n g the i n i t i a l "optimum" base case i s co n s i d e r a b l y more d i f f i c u l t — a t l e a s t f o r major transm i s s i o n networks--than l o c a t i n g new r e a c t i v e banks. A transmission-system i s planned f o r continuous expansion.. With a load growth of 7% or l e s s per year, i t should never be necessary t o add e x t r a r e a c t i v e banks at more than a few busses at a t i m e — e v e n i n large systems. This i s e s p e c i a l l y true i n view of the f a c t t h a t i t i s cheaper to add one or two large banks (or expand e x i s t i n g banks) than t o add s e v e r a l smaller ones, due mainly t o the cost of the switchgear, buswork, concrete pads, e t c . that are r e q u i r e d to transform a symbol on a diagram i n t o a r e a l i t y on the system. I t i s even reasonable to do t h i s manually, using V-Q s e n s i t i v i t i e s when necessary. The major d i f f i c u l t y , not only f o r r e a c t i v e power a l l o c a t i o n , but whenever power flow stud i e s are made, i s t o obta i n the best p o s s i b l e base case. To many people, p a r t i c u l a r l y p r a c t i s i n g power engineers, a "good" base case i s one w i t h a uniform, or n e a r l y so, voltage p r o f i l e . This e x p l a i n s the frequent appearance i n the l i t e r a t u r e of "VAr a l l o c a t i o n " techniques designed to c o r r e c t bus voltages to w i t h i n s p e c i f i e d bounds. The need f o r s p e c i f y i n g these bounds, however, and i n p a r t i c u l a r the d i f f i c u l t y and importance of -24-determining the " c o r r e c t " bounds to use, l i m i t s the e f f e c t i v e -ness of t h i s approach. A poor choice of voltage l i m i t s f o r any of the l i n e a r programming techniques discussed e a r l i e r can prevent convergence by p l a c i n g incompatible c o n s t r a i n t s on the s o l u t i o n . Some busses, p a r t i c u l a r l y those near constant voltage busses, w i l l be i n s e n s i t i v e t o changes i n shunt r e a c t i v e power at busses to which they are connected through r e l a t i v e l y large branch impedances. I f the voltage c o n s t r a i n t s on the bus and i t s neighbour do not a l l o w f o r a r e a l i s t i c r e l a t i o n s h i p between the two v o l t a g e s , the problem w i l l be e i t h e r unsolvable, or the s o l u t i o n w i l l r e q u i r e an unreasonably l a r g e amount of shunt compensation. The avoidance of t h i s type of d i f f i c u l t y i s g e n e r a l l y l e f t as the r e s p o n s i b i l i t y of the user. Although i t i s not, perhaps, unreasonable to assume th a t an experienced engineer can set s a t i s f a c t o r y voltage c o n s t r a i n t s , one of the most common a r -guments given i n favour of r e a c t i v e power a l l o c a t i o n techniques i n general i s tha t they r e q u i r e l e s s experience w i t h the power system to use e f f e c t i v e l y than do manual techniques. The d i f f -i c u l t y of s e t t i n g proper voltage l i m i t s weakens t h i s argument co n s i d e r a b l y f o r v o l t a g e - c o r r e c t i o n techniques. Although there i s l i t t l e question t h a t v o l t a g e - c o r r e c t i o n techniques can a s s i s t w i t h the a l l o c a t i o n process, and may even -25-o c c a s i o n a l l y provide a l l o c a t i o n schemes r e q u i r i n g l i t t l e or no f u r t h e r m o d i f i c a t i o n , the requirement f o r u s e r - s e l e c t e d voltage l i m i t s can lead to an unnecessary amount of p r e l i m i n a r y work, and the r e s u l t i n g a l l o c a t i o n scheme i s always s e n s i t i v e to the l i m i t s s e l e c t e d . The more experience one gains w i t h power flow s t u d i e s , the more apparent i t becomes th a t uniform (or other selected) v o l -tages are not of primary importance at tr a n s m i s s i o n , and to a l e s s e r extent subtransmission system l e v e l s . Of more importance are the magnitudes, and the r e l a t i v e magnitudes, of the r e a l and r e a c t i v e power flows. Whereas r e a l power flows are u s u a l l y determined by a v a i l -a b i l i t y of (extremely expensive) generation sources, energy reserves, and the curren t system load, r e a c t i v e power flows are more f l e x i b l e . As. there: i s no energy r e q u i r e d f o r the generation of r e a c t i v e power, i t i s much e a s i e r and l e s s expensive to gen-erate than i s r e a l power, and so can be produced nearer the load, reducing t r a n s m i s s i o n l o s s e s , equipment loadings, and voltage drops. By reducing unnecessary r e a c t i v e power flows, i t i s p o s s i b l e t o approximate a. uniform voltage c o n d i t i o n . Indeed, i t i s by c o n t r o l l i n g r e a c t i v e power flows t h a t v o l t a g e - c o r r e c t i o n type "VAr a l l o c a t i o n " techniques attempt to s a t i s f y voltage c o n s t r a i n t s . -26-Reactive power flow i s , i n t h i s way, more fundamental to a "good" base case than i s a s e l e c t e d voltage p r o f i l e . I t i s always p o s s i b l e t o minimise r e a c t i v e power flows; i t i s not always p o s s i b l e to achieve a d e s i r e d voltage p r o f i l e (given normal operating c o n s t r a i n t s ) . I t i s f o r these reasons t h a t minimum r e a c t i v e power flow, and not voltage p r o f i l e , w i l l be used i n t h i s t h e s i s as the primary c r i t e r i o n f o r the s e l e c t i o n of a "good" base case. The Development of an Objective Function From the foregoing d i s c u s s i o n , i t should be c l e a r t h a t a s u i t a b l e o b j e c t i v e f u n c t i o n f o r a m i n i m i s a t i o n process w i l l be r e l a t e d d i r e c t l y to the r e a c t i v e power flow i n the t r a n s m i s s i o n system. There are s e v e r a l such functions which would be s u i t - , a b le: -(a) the sum of the squares of the currents i n each branch (b) the sum of the squares of the r e a c t i v e power flows i n each branch (c) the r e a c t i v e power l o s s i n the system (sum of ||l|| 2 | x j f o r a l l branches, which i s e f f e c t i v e l y a weighted sum of squares of currents) -27-(d) the r e a l power l o s s i n the system (sum of R f o r a l l branches, s i m i l a r l y t o (c)) (e) the s l a c k bus r e a l power (constant term + r e a l power l o s s i f r e a l power i n j e c t i o n s are not a l t e r e d ) or any s i m i l a r f u n c t i o n . Function (a.) i s general, s i n c e i t i s the cu r r e n t which produces both the r e a l and the r e a c t i v e power l o s s e s . U n f o r t -unately, i t would tend to e q u a l i s e branch c u r r e n t s i r r e s p e c t i v e of the equipment represented by the branch. S i n g l e - c i r c u i t , low c a p a c i t y branches would thus be loaded at cu r r e n t l e v e l s com-parable t o m u l t i - c i r c u i t , high c a p a c i t y b r a n c h e s — o b v i o u s l y an untenable prospect: Function (b) p a r t i a l l y avoids the problem, s i n c e i t places no r e s t r i c t i o n s on the r e a l power flow. I t does tend to balance r e a c t i v e power flows between branches i n the same way as (a) balances c u r r e n t s , however, which, although not r e s u l t i n g i n q u i t e such an unreasonable s i t u a t i o n as (a), i s s t i l l an un d e s i r -able c h a r a c t e r i s t i c f o r reasons analogous t o those given above f o r (a). Function (c) i s a v a r i a t i o n of (a) which tends t o avoid the problem of e q u a l i s i n g c u r r e n t flows by weighting the squares of the currents i n each branch w i t h a f a c t o r equal t o the branch reactance. E f f e c t i v e l y , t h i s f u n c t i o n tends to e q u a l i s e the -28-product (|l|{|x|*. A branch of twice the reactance of another branch w i l l , t h e r e f o r e , be scheduled to c a r r y about h a l f of the c u r r e n t i n the low reactance branch (assuming t h a t the branches are e f f e c t i v e l y i n p a r a l l e l , and t h a t the t o t a l c u r r e n t i s con-? s t a n t ) . This i s a near i d e a l sharing of c u r r e n t , and has the advantage of minimising the t o t a l r e a c t i v e power generation which must be provided t o meet a .(fixed) r e a c t i v e power load. Function (d) i s an i n t e r e s t i n g analog to ( c ) . I t produces a r e s u l t s i m i l a r to t h a t of (c) i n most cases, and i d e n t i c a l i f the r a t i o X/R i s constant f o r every branch. The advantage of t h i s f u n c t i o n over (c) l i e s i n the c o s t d i f f e r e n c e , economically, environmentally, and s o c i a l l y , between r e a l and r e a c t i v e power. Real power comes from large dams and r e s e r v o i r s , conventional thermal s t a t i o n s burning i r r e p l a c e a b l e and expensive s u p p l i e s of c o a l or petroleum, or nuclear thermal s t a t i o n s which produce unmanageable, or n e a r l y so, f i s s i o n by-products. Reactive power, i n c o n t r a s t , i s generated n a t u r a l l y by t r a n s m i s s i o n l i n e s and c a b l e s , and can be generated d e l i b e r a t e l y i n shunt c a p a c i t o r banks. Function (d) shares w i t h f u n c t i o n s (a) t o (c) a d i f f i c u l t y * See appendix s e c t i o n A6 f o r a proof of t h i s . -29-of c a l c u l a t i o n , i n that currents must be c a l c u l a t e d f o r each branch, and then summed according t o the f u n c t i o n employed. Function (e) i s more elegant, as the s l a c k bus power may be c a l c u l a t e d w i t h t r i v i a l e f f o r t from the s o l u t i o n v o l t a g e s . For constant r e a l power i n j e c t i o n s at busses other than the s l a c k bus, as i s the case here, t h i s f u n c t i o n produces r e s u l t s i d e n -t i c a l to those of (d). I t i s t h i s o b j e c t i v e f u n c t i o n which w i l l be used i n t h i s t h e s i s . In order t o prevent the s o l u t i o n a l g o r i t h m from d r i v i n g the bus voltages e x c e s s i v e l y high, i t i s necessary e i t h e r to augment the o b j e c t i v e f u n c t i o n w i t h a term designed t o increase or " p e n a l i s e " the o b j e c t i v e f u n c t i o n as the bus voltages deviate from nominal, or t o f o r m a l l y c o n s t r a i n the o b j e c t i v e f u n c t i o n w i t h a voltage c o n s t r a i n t d u r i n g the mi n i m i s a t i o n . The former technique permits a voltage t o deviate f a r from nominal i f t o do so s i g n i f i c a n t l y reduces e i t h e r other voltage d e v i a t i o n s , or the primary o b j e c t i v e f u n c t i o n (system r e a l power l o s s ) . The l a t t e r technique w i l l not permit the voltages t o v i o l a t e t h e i r r e s p e c t i v e c o n s t r a i n t s even i f the c o n s t r a i n t s are preventing s o l u t i o n of the problem. The augmented o b j e c t i v e f u n c t i o n ("penalty f a c t o r " ) technique i s the r e f o r e p r e f e r a b l e f o r t h i s , use, as i t at l e a s t ensures the existence of a s o l u t i o n . I f the alg o r i t h m i s to be u s e f u l f o r the a l l o c a t i o n of new -30-shunt r e a c t i v e power sources, the o b j e c t i v e f u n c t i o n w i l l r e q u i r e an a d d i t i o n a l term accounting f o r the cost of the a d d i t i o n a l shunt c a p a c i t y which i s t o be s u p p l i e d . This may e a s i l y be accomplished by adding a term s i m i l a r t o the vo l t a g e (penalty) term, which w i l l add to the o b j e c t i v e f u n c t i o n an amount equal to the weighted sum of squares of the compensation added. The f i n a l form of the o b j e c t i v e f u n c t i o n i s , t h e r e f o r e , f(u,x) = P_(u,x) + 2lw- (v, - V ? i n ) 2 + <cr / T T max. 2 , 2 2_w i(v i -V-; ) + 2zk Bk (8) where term w i ( V i - V i i n ) 2 i s low voltage p e n a l t y f o r bus i term Wj (Vj - V j 1 3 * ) 2 i s high voltage p e n a l t y f o r bus j 2 . term z^B^ i s shunt c a p a c i t y p e n a l t y f o r bus k P s = s l a c k bus r e a l power Wj_ = voltage weighting f a c t o r f o r bus i = shunt c a p a c i t y weighting f a c t o r f o r bus k B^ . = r e a c t i v e shunt added at bus k The c o n s t r a i n t s on f(u,x) are: g = = 0 f o r a l l busses (9) -31-which are the power flow equations r e q u i r i n g the power mismatches to be zero at a l l busses (AP and AQ are the vectors of r e a l and r e a c t i v e power mismatches, r e s p e c t i v e l y ) , (10a) f o r a l l busses € NG (10b) which are the minimum and maximum voltage l i m i t s f o r a l l c o n t r o l l e d voltage (generator) busses, (Ha) f o r a l l busses € NT ( l i b ) which are the minimum and maximum tap l i m i t s f o r a l l c o n t r o l l e d transformers, (12a) f o r a l l busses €. NQ (12b) which are the minimum and maximum l i m i t s of r e a c t i v e compensation t o be added t o e l i g i b l e busses (of the s e t NQ), and f i n a l l y V - V™1*1 > 0 "max ~" -V - V > 0 — . mxn — T - T > 0 "max — — T - T > 0 — mxn — B - B > 0 max — — B - B > 0 -32-min Q - Q > 0 (13a) Q max - Q > 0 f o r a l l generator busses (13b) which are the minimum and maximum r e a c t i v e power l i m i t s f o r the generators. The i n e q u a l i t i e s (10) - (13) c o l l e c t i v e l y c o n s t i t u t e the i n e q u a l i t y c o n s t r a i n t s e t h > 0. Note th a t the l i m i t s on the c o n t r o l v a r i a b l e s , equations (10) through (12) are l i n e a r , which w i l l permit the use of a simpler c o n s t r a i n e d o p t i m i s a t i o n technique. -33-CHAPTER I I I The I n v e s t i g a t i o n of the Constrained Objective Function The choice of a numerical minimi s a t i o n technique r e q u i r e s knowledge of the nature of the con s t r a i n e d o b j e c t i v e f u n c t i o n . One of the si m p l e s t ways of g e t t i n g t h i s i n f o r m a t i o n i s through the use of contour p l o t s , i n which contours of constant objec-t i v e f u n c t i o n value are p l o t t e d versus the various c o n t r o l v a r -i a b l e s . Such contour p l o t s can be produced by using a power flow program to evaluate the o b j e c t i v e f u n c t i o n f o r various values of two c o n t r o l v a r i a b l e s . The contour p l o t s i n t h i s t h e s i s were produced by a. power flow program which a u t o m a t i c a l l y v a r i e d the c o n t r o l v a r i a b l e s on the two axes through each of e i g h t val u e s , g i v i n g a t o t a l of 64 power flows (or f u n c t i o n e v a l u a t i o n s ) . The values of the o b j e c t i v e f u n c t i o n were then i n t e r p o l a t e d between these p o i n t s to o b t a i n the contours f o r p l o t t i n g . The o b j e c t i v e f u n c t i o n chosen (equation 8) i s composed of •k three p a r t s : the s l a c k bus r e a l power , the voltage p e n a l t i e s , * The r e a l power l o s s i s equal to the s l a c k bus r e a l power p l u s a constant. The contours of r e a l power l o s s thus have id e n -t i c a l form t o the contours of s l a c k bus r e a l power. The contours of r e a l power l o s s w i l l be used h e r e i n . -34-and the a l l o c a t e d r e a c t i v e power p e n a l t i e s . Although i t i s not d i f f i c u l t t o p l o t the contours f o r t h i s o b j e c t i v e f u n c t i o n , i t w i l l be d i f f i c u l t t o analyse such a complex set of contours d i r e c t l y . I t i s simpler t o analyse the l o s s contours f i r s t , and then analyse the e f f e c t t h a t the p e n a l t i e s w i l l have on these contours. Contours of Constant Loss The system f o r which contours have been obtained i s shown i n f i g u r e 1. This three-bus system i s a s i m p l i f i e d r epresenta-t i o n of the p o r t i o n of B.C. Hydro's 230 kV system from Bridge Ri v e r (bus 2) through Cheekeye (bus 3) to Vancouver (bus 1). The contours of constant l o s s versus the voltages at busses 1 and 2 are shown i n f i g u r e 2. These contours appear to be s t r o n g l y p a r a b o l i c , w i t h the a x i s p a r a l l e l to and s l i g h t l y d i s p l a c e d from the l i n e V-j_ = V 2 . The r e d u c t i o n of l o s s w i t h i n c r e a s i n g voltage a t busses 1 and 2 i s due t o the r e s u l t a n t increase i n voltage at bus 3, which reduces the c u r r e n t necessary t o provide the load P and Q. This reduced c u r r e n t flow then r e s u l t s i n reduced l o s s i n the two branches. On e i t h e r s i d e of the a x i s the l o s s i n c r e a s e s . This i s not -35-due to a v a r i a t i o n i n the voltage at bus 3, which i s e s s e n t i a l l y -constant along a locus at r i g h t angles to the a x i s ( p a r a l l e l to the d i r e c t r i x ) . For example, along the l i n e AB defined by A = (1.20,0.85) and B = (0.85,1.20), the per u n i t bus 3 voltages are: 1.00, 1.00, 1.00, 1.00, 1.00, 1.01, 1.01, 1.01 f o r increments of 0.05 pu i n and V 2. (The f a c t t h a t the voltages increase s l i g h t l y as V 2 increases and V-^  decreases i n d i c a t e s t h a t the l i n e AB defined above a c t u a l l y i n t e r s e c t s the a x i s at an angle o s l i g h t l y l e s s than 90 , which means that the a x i s i s not q u i t e p a r a l l e l t o the l i n e V-^  = V 2 . ) The reason f o r the increase of l o s s t o e i t h e r side of the a x i s becomes apparent upon c l o s e examination of the power flows corresponding t o each p o i n t on the p l o t : p o i n t s o f f the a x i s correspond to a t r a n s f e r of r e a c t i v e power (and hence current) from one generator to the other. The a x i s of the contours i s the locus of s o l u t i o n voltages f o r which t h i s interchange of r e a c t i v e power i s zero. The f o l l o w i n g t a b l e shows t h i s e f f e c t f o r three p o i n t s on the p l o t of f i g u r e 2:-vl • V 2 °- from 2 to 3 Q from 1 t o 3 system l o s s (pu) (pu) (pu) (pu) (pu) 0.95 1.10 1.00 1.05 1.05 1.00 2.364 0.746 -0.718 1.014 0.485 2.128 0.239 0.216 0.250 -36-The a x i s i n t h i s case runs approximately along the l i n e CD defined by C = (0.85,0.93) and D = (1.15,1.20). The r e a c t i v e power i s seen t o flow from generator 2 to generator 1 f o r voltage (0.95,1.10), and from 1 t o 2 f o r voltage (1.05,1.00). The l o c a -t i o n of the a x i s i s a f f e c t e d by such parameters as the r e l a t i v e impedances of the branches, the r e a l power flows along each of the branches, and the R/X r a t i o s of the branches. By way of example, f o r the case of f i g u r e 2, the r e a l power flows from bus 2 to bus 1. For the case of f i g u r e 3, however, both gen-e r a t o r s supply power to bus 3 e q u a l l y , r e s u l t i n g i n a smaller displacement of the a x i s from the l i n e = V than i s the case f o r f i g u r e 2. Figure 3, which i s f o r a case i d e n t i c a l t o that of f i g u r e 2 except f o r the generation, a l s o shows more elongated contours due to the reduced power, and so c u r r e n t , f l o w i n g on the branch from bus 2 to bus 3. The l o s s (£) thus decreases l e s s r a p i d l y w i t h i n c r e a s i n g voltage than f o r the more h e a v i l y leaded case (the d e r i v a t i v e -j-j" = 2IR and so i s p r o p o r t i o n a l . to the c u r r e n t f l o w ) , w h i l e the l o s s due t o the interchange of r e a c t i v e power between generators i s a f f e c t e d r e l a t i v e l y l e s s by the r e d u c t i o n i n load c u r r e n t . Hence the greater e l o n g a t i o n of the contours. The r e l a t i o n s h i p between the shape of the contours and the l i n e l o a d i n g i s f u r t h e r apparent from i n s p e c t i o n of f i g u r e 5. This i s the contour p l o t f o r the s i t u a t i o n of f i g u r e 4, w i t h a (hypothetical) branch from bus 1 to bus 2 p a r a l l e l i n g the previous path. Any r e a c t i v e power f l o w i n g from generator 2 to generator 1 the r e f o r e has an a l t e r n a t e path around the load bus 3. This reduces s i g n i f i c a n t l y the cu r r e n t on the other l i n e s . The contours have been elongated to the p o i n t where they are v i r t u a l l y p a r a l l e l l i n e s . •Ki This extreme e l o n g a t i o n of the contours i s due, as w i t h the case of f i g u r e 3, t o the f a c t t h a t the cur r e n t s a s s o c i a t e d w i t h the t r a n s f e r of r e a l power are now very s m a l l (due to equal load-sharing by the two generators), and the v a r i a t i o n i n l o s s w i t h c u r r e n t i s s i g n i f i c a n t l y l e s s than would be the case f o r a higher c u r r e n t flow. The v a r i a t i o n i n l o s s due t o the exchange of r e a c t i v e power between the two generators, however, i s much greater than f o r the case of f i g u r e 3. This i s due t o the f a c t that w i t h a reduced impedance between the two generator busses, a given d i f f e r e n c e i n voltage magnitude produces higher c u r r e n t than p r e v i o u s l y . Since the cu r r e n t increases more r a p i d l y than before, the square of the cu r r e n t increases much more r a p i d l y , w i t h a correspondingly r a p i d increase i n l o s s . Figure 6 shows the contours f o r the same case: as f i g u r e 2, but w i t h a load at bus 3 of only 40% of the previous value. -38-Here again the dramatic e l o n g a t i o n of the contours i s very-evident. This i s again due t o the s u b s t a n t i a l decrease i n l o a d i n g of the t r a n s m i s s i o n l i n e between bus 2 and the load at bus 3. The contours of f i g u r e 7 were obtained by p l a c i n g a t r a n s -former of zero impedance i n the t r a n s m i s s i o n l i n e between busses 2 and 3 (with the tap at bus 3). In t h i s way the tap of the transformer could c o n t r o l the voltage at load bus 3 without d i r e c t l y a f f e c t i n g the c u r r e n t flow along the l i n e between busses 2 and 3. The contours of f i g u r e 7 t h e r e f o r e , are essen-t i a l l y the constant l o s s contours f o r the t r a n s m i s s i o n l i n e between bus 3 and bus 1. These contours are very s i m i l a r t o the l i g h t load contours f o r the whole system. Figure 8 shows contours f o r a case i d e n t i c a l to f i g u r e 4, but w i t h a zero-impedance transformer as f o r f i g u r e 7. The contours are again elongated, but t h i s time they appear to be e l l i p t i c a l , w i t h a minimum w i t h i n the p l o t range. The voltage at bus 2 f o r t h i s case, as f o r the case of f i g u r e 7, i s one p e r - u n i t . This e x p l a i n s the minimum l o c a t e d about a tap of one and a voltage on bus 1 of one p e r - u n i t . The c l o s e d contours i n t h i s example r e s u l t from the f a c t t h a t the tap now c o n t r o l s a c i r c u l a t i n g r e a c t i v e power flow throughout the system. The minimum l o s s c o n d i t i o n occurs f o r a c i r c u l a t i n g Q flow of zero, -39-which occurs f o r a tap s e t t i n g of 1 and f o r no t r a n s f e r of r e a c t i v e power between g e n e r a t o r s — i . e . when the voltage at bus 1 approximately equals the voltage of bus 2 at one p e r - u n i t . One of the o b j e c t i v e s of the a l g o r i t h m being developed i s to a l l o c a t e r e a c t i v e power i n such a way as t o minimise the tra n s m i s s i o n losses i n the system, and so i t i s reasonable to i n v e s t i g a t e the e f f e c t of the power f a c t o r of the load a t bus 3 on the tr a n s m i s s i o n l o s s e s . For the cases of f i g u r e s 9 and 10 the power f a c t o r of the load was reduced from the 95% of the previous cases t o 80%. As can be seen from comparison w i t h the previous cases ( f i g u r e s 2 and 6, r e s p e c t i v e l y ) , a l t e r i n g the power f a c t o r has a n e g l i g i b l e e f f e c t on the shape of the con-tours . This i s f u r t h e r demonstrated by the contours of f i g u r e s 11 to 13. In each case, and p a r t i c u l a r l y i n the case of f i g u r e 13 (branch between bus 1 and bus 2), the contours are very e l o n -gated f o r the v a r i a t i o n i n shunt at bus 3. This i s true because, i n t h i s p a r t i c u l a r case, the voltage magnitude at bus 3 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 changes i n the shunt at bus 3. Since the r e a c t i v e power flow, and so the system l o s s , i s determined by the voltage of bus 3 r e l a t i v e t o the voltages of busses 1 and 2, the system l o s s i s a l s o r e l a t i v e l y i n s e n s i t i v e to changes i n the shunt at bus 3. (For V"2 = 1.0 pu and = 1.0 pu, -40-V 3 v a r i e s from 0.9791 pu to 0.9893 pu over the range of shunts 0.0 to 0.7 pu f o r the case of f i g u r e 11.) There are four c o n c l u s i o n s , then, t h a t can be drawn from these r e s u l t s . F i r s t l y , i n the absence of extreme c o n d i t i o n s , the l o s s w i l l depend more on generator voltages and on t r a n s -former taps than-on load power f a c t o r s . Secondly, the l o s s decreases w i t h a simultaneous increase i n generator v o l t a g e s . T h i r d l y , the l o s s decreases as the generator voltage magnitudes approach a uniform v a l u e . * And l a s t l y , the degree of e l o n g a t i o n of the contours increases very r a p i d l y as the l o a d i n g decreases. (Conversely, the contours become more c i r c u l a r as the l o a d i n g i n c r e a s e s . This l a s t p o i n t suggests t h a t the o p t i m i s a t i o n process w i l l be more d i f f i c u l t under l i g h t - l o a d c o n d i t i o n s because of the e l o n g a t i o n of the contours, i n d i c a t i n g poor s c a l i n g of the v a r i a b l e s . As the magnitude of the l o s s i s l e s s under l i g h t - l o a d , however, the o p t i m i s a t i o n can be terminated before .- completion w i t h l i t t l e p e n a l t y ; t h i s d i f f i c u l t y i s not a s e r i o u s one.) * This p o i n t i s important i n t h a t i t spans the gap between the "equal v o l t a g e " c r i t e r i o n used by most u t i l i t y engineers and the "minimum l o s s " c r i t e r i o n presented here. G e n e r a l l y the two c r i t e r i a produce very s i m i l a r r e s u l t s , as they must i f the"minimum l o s s " c r i t e r i o n i s t o c a r r y c r e d i b i l i t y . -41-The E f f e c t of the Penalty Terms The e f f e c t of the p e n a l t y terms depends to a l a r g e extent on the p e n a l t y f a c t o r s used. By a d j u s t i n g the p e n a l t y f a c t o r s , the balance i n the o b j e c t i v e f u n c t i o n between the l o s s and the two p e n a l t y terms can be s h i f t e d to o b t a i n d i f f e r e n t perform-ance c h a r a c t e r i s t i c s . Because the dependence of the o h j e c t i v e f u n c t i o n on l o s s i s l i n e a r , while the dependence on the p e n a l -t i e s i s q u a d r a t i c , i t i s n o t . p o s s i b l e t o equate, f o r example, a l o s s of 1 MW w i t h a voltage d e v i a t i o n of 0.01 pu or a r e a c t i v e shunt a l l o c a t i o n of 1 MVAr. Nevertheless, i t i s p o s s i b l e to determine the general e f f e c t of the p e n a l t y terms on the contours by using values of 7.5 and 1.0 f o r the voltage and r e a c t i v e power p e n a l t i e s , r e s p e c t i v e l y . (These values were found to give acceptable performance i n the f i n a l program.) Figure 14 i l l u s t r a t e s the e f f e c t of the voltage p e n a l t y on the contours of f i g u r e 2. The contours have been c l o s e d a t the high voltage end of the p l o t range, and have become g e n e r a l l y rounded. The most extreme e f f e c t of the voltage p e n a l t y i s observed i n f i g u r e 15, which corresponds t o the l i g h t - l o a d case of f i g u r e 6. Here the elongated contours have been rounded t o n e a r - c i r c u l a r . In both cases, the minimum i s c l e a r l y bounded, as opposed to the o r i g i n a l cases. The rounding of the contours occurs i n a l l cases except -42-those i n which one parameter i s the shunt on bus 3. Figure 16 shows the contours of f i g u r e 11 augmented w i t h the voltage pen-a l t y term. This l a c k of rounding i s due mainly to the r e l a t i v e i n s e n s i t i v i t y of the bus 3 voltage t o the bus 3 s h u n t — a c h a r a c t e r i s t i c which was. pointed out e a r l i e r . This voltage i n s e n s i t i v i t y r e s u l t s i n only a very small change i n voltage p e n a l t y w i t h shunt, so that the moderating e f f e c t of the voltage perialtyt;on::the r.coh.tour . shape i s minimal. The e f f e c t of the r e a c t i v e power p e n a l t y can be seen i n f i g u r e 17, which corresponds to the same case as f i g u r e 16. The contours have now become n e a r l y c i r c u l a r . This i s again due t o the r e l a t i v e i n s e n s i t i v i t y of the voltage of bus 3 to the shunt, which r e s u l t s i n the r e a c t i v e power pen a l t y term dominating the o b j e c t i v e f u n c t i o n i n the absence of r e a c t i v e power t r a n s f e r between generators. Both the r e a c t i v e power pe n a l t y term and the l o s s due to an interchange of r e a c t i v e power are quadratic terms, and so w i l l produce c i r c u l a r contours w i t h a s u i t a b l e choice of r e a c t i v e p e n a l t y f a c t o r . I t may appear from the foregoing d i s c u s s i o n t h a t there i s l i t t l e p o i n t i n a l t e r i n g bus shunts. For the example used here, t h i s i s q u i t e t r u e . In cases where the s e n s i t i v i t y of a load bus voltage to a bus shunt i s great, however, there w i l l be a strong v a r i a t i o n of l o s s w i t h shunt, and the a s s o c i a t e d contours -43-w i l l be d i s t i n c t l y more rounded than f o r t h i s example system. As i t i s only f o r cases e x h i b i t i n g t h i s strong s e n s i t i v i t y t h a t shunt c o n t r o l would be used, there i s no cause t o doubt the e f f e c t i v e n e s s or usefulness of shunt c o n t r o l from the preceding r e s u l t s . Reactive Power L i m i t s on Generators The set of equations (8) t o (12) does not q u i t e describe the r e a c t i v e power management problem completely. A l l generators have l i m i t s on the r e a c t i v e power they may absorb or produce, and so are co n s t r a i n e d by equations (13), analogous t o the equa-t i o n s (12) f o r a l l o c a t i o n busses. Because of the analogous s i t u a t i o n w i t h the r e a c t i v e a l l o c a t i o n busses, i t may seem d e s i r a b l e t o t r e a t generator r e a c t i v e power l i m i t s i n the same manner—i.e. as p e n a l t y terms. This would, i n f a c t , be a bad choice, as the contours of f i g u r e 18 i n d i c a t e . Generator busses 1 and 2 i n t h i s case were p e n a l i s e d (quite l i g h t l y , i n f a c t ) t o 0.95 power f a c t o r . The e l o n g a t i o n of the c o n t o u r s — e v e n w i t h the voltage term i n c l u d e d — i s c l e a r l y the worst yet encountered. The e x p l a n a t i o n i s t h a t the r e a l and r e a c t i v e l o s s e s i n a power system are very c l o s e l y r e l a t e d — i n f a c t , they are p r o p o r t i o n a l f o r each branch i n the -44-system. Although the sum of the r e a c t i v e l o s s e s i n the system w i l l not be q u i t e p r o p o r t i o n a l to the sum of the r e a l l o s s e s (unless the X/R r a t i o i s constant f o r every branch), i t i s nevertheless true that the minimum r e a c t i v e power l o s s s i t u a t i o n w i l l correspond g e n e r a l l y t o the minimum r e a l power l o s s s i t u a t i o n . Since the minimum r e a c t i v e power l o s s s i t u a t i o n i s the minimum r e a c t i v e power generation c o n d i t i o n , the generators w i l l be c a l l e d upon t o generate the most r e a c t i v e power when the r e a l power l o s s i s hi g h , and v i c e v e r s a . P e n a l i s i n g the generator Q v i o l a t i o n s as a square term i s thus very much l i k e u sing an o b j e c t i v e f u n c t i o n of f =oC + k<£ a where k = constant which e x h i b i t s very elongated contours. I t i s b e t t e r , then, t o handle the generator Q l i m i t s i n another way, thereby a v o i d i n g the mi n i m i s a t i o n problems which attend very poor s c a l i n g . An e f f e c t i v e method of handling the generator l i m i t s w i l l be presented i n the f o l l o w i n g chapter. CHAPTER IV The Choice of a S u i t a b l e M i n i m i s a t i o n Technique The problem described by equations (8) through (13) re q u i r e s the m i n i m i s a t i o n of a nonlinear o b j e c t i v e f u n c t i o n subject t o l i n e a r and non l i n e a r c o n s t r a i n t s . There are a great many poss-i b l e approaches t o the s o l u t i o n of t h i s type of problem. These many approaches d i f f e r mainly i n the degree t o which they u t i l i s e i n f o r m a t i o n about the o b j e c t i v e f u n c t i o n (such as f i r s t , second, and higher order d e r i v a t i v e s ) , and the manner i n which they t r e a t the c o n s t r a i n t s . Constrained o p t i m i s a t i o n i s gen-e r a l l y much more complex than unconstrained o p t i m i s a t i o n , and i t i s th e r e f o r e important t h a t the c o n s t r a i n t s be t r e a t e d i n the way l e a s t l i k e l y t o upset the mi n i m i s a t i o n procedure. The Treatment of C o n s t r a i n t s Both 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 can be handled i n e i t h e r of two ways. F i r s t l y , they can be handled d i r e c t l y , wherein the e q u a l i t y c o n s t r a i n t s are solved along w i t h the other c o n d i t i o n s f o r a minimum as a set of simultaneous equations, and the a c t i v e i n e q u a l i t y c o n s t r a i n t s are observed at each step as -46-a d d i t i o n a l equations on the c o n t r o l v a r i a b l e s . In each case, the c o n s t r a i n t s are met t o w i t h i n the p r e c i s i o n of the c a l c u l a t i o n s at every step, so that a l l intermediate s o l u t i o n s are f e a s i b l e . This approach i s e s s e n t i a l l y i d e n t i c a l whether i t i s implemented as Lagrange m u l t i p l i e r s , g r a d i e n t . p r o j e c t i o n , or gradient reduc- t i o n . The second approach i s t o use the pe n a l t y f u n c t i o n technique, i n which the steps are permitted t o enter and leave the f e a s i b l e r e g i o n at w i l l . For steps t e r m i n a t i n g outside the f e a s i b l e r e g i o n , a "penalty term" i s added t o the o b j e c t i v e f u n c t i o n . This p e n a l t y term u s u a l l y increases q u a d r a t i c a l l y as the step leaves the f e a s i b l e r e g i o n . The steps are thus "encouraged", and not, as w i t h the previous approach, "forced" t o remain w i t h i n the boundaries of the f e a s i b l e r e g i o n . As the minimisa-t i o n progresses, the p e n a l t y term i s o f t e n m u l t i p l i e d by an e v e r - i n c r e a s i n g f a c t o r , which tends t o keep the intermediate s o l u t i o n s s u c c e s s i v e l y l e s s i n f e a s i b l e u n t i l , a t the s o l u t i o n p o i n t , the s o l u t i o n i s f e a s i b l e t o w i t h i n some t o l e r a n c e . Note th a t both techniques are e s s e n t i a l l y the same—as of course they must b e — i n t h a t the second approach uses the min i m i s a t i o n process t o solve the same equations as are solved a l g e b r a i c a l l y by the f i r s t technique. Because the two techniques solve the c o n s t r a i n t equations d i f f e r e n t l y , i t i s reasonable t o -47-expect one technique to be s u p e r i o r to the other f o r c e r t a i n forms of c o n s t r a i n t s . I f the equations of c o n s t r a i n t can be s o l v e d a n a l y t i c a l l y without an e x t r a o r d i n a r y amount of computa-t i o n , the f i r s t method i s c l e a r l y s u p e r i o r , e s p e c i a l l y as the intermediate s o l u t i o n s f o r t h i s method are a l l f e a s i b l e , and so usable--although 1' sub-optimal.. An important disadvantage of the p e n a l t y f u n c t i o n approach i s t h a t the p e n a l t y terms d i s t o r t the contours, of t e n making the r e s u l t i n g augmented o b j e c t i v e f u n c t i o n much more d i f f i c u l t t o minimise (the generator Q l i m i t s of the l a s t chapter are an example). For cases of c o n s t r a i n t equations which may not be e a s i l y solved a n a l y t i c a l l y , however, the p e n a l t y f u n c t i o n approach i s p r e f e r a b l e . The c o n s t r a i n t s on the c o n t r o l v a r i a b l e s — t r a n s f o r m e r taps, generator v o l t a g e s , and shunt r e a c t i v e source a l l o c a t i o n s — c a n be e a s i l y t r e a t e d as absolute l i m i t s on the allowable v a r i a t i o n i n the c o n t r o l v a r i a b l e s , and so handled d i r e c t l y . The other c o n s t r a i n t s , however, must be t r e a t e d by one of the two methods discussed above. While the e q u a l i t y c o n s t r a i n t s could l i k e l y be t r e a t e d w i t h e i t h e r of the two techniques, the i n e q u a l i t y c o n s t r a i n t s on the generator r e a c t i v e power l i m i t s should not be handled as p e n a l t y terms. As was demonstrated i n the l a s t chapter, these p e n a l t i e s have a severe i n f l u e n c e on the o b j e c t i v e f u n c t i o n contours, even w i t h a s m a l l m u l t i p l y i n g f a c t o r . Both -48-e q u a l i t y and generator r e a c t i v e power i n e q u a l i t y c o n s t r a i n t s w i l l thus be t r e a t e d here using the a n a l y t i c method of c o n s t r a i n t handling. I t i s not p o s s i b l e to solve e x p l i c i t l y the nonlinear equa-t i o n ( 9 ) f o r the dependent v a r i a b l e s as a f u n c t i o n of the indepen-dent v a r i a b l e s . I f the f u n c t i o n g (equation 9 ) i s expanded about the cur r e n t intermediate s o l u t i o n p o i n t i n a f i r s t - o r d e r Taylor expansion, however, i t i s p o s s i b l e to solve the new l i n e a r r e l a -t i o n e x p l i c i t l y f o r the independent v a r i a b l e s . This r e l a t i o n s h i p between the dependent and independent v a r i a b l e s may then be sub-s t i t u t e d i n t o the expression f o r the gradient of the o b j e c t i v e f u n c t i o n to o b t a i n a reduced expression i n which the gradient i s a f u n c t i o n of the independent v a r i a b l e s only. The equations are c a l c u l a t e d i n appendix s e c t i o n A 2 . Because of the l i n e a r i z a t i o n of the c o n s t r a i n t equation, each intermediate step w i l l not: n e c e s s a r i l y end w i t h i n the f e a s -i b l e r e g i o n , and i t i s thus necessary to adjust the s o l u t i o n v e c t o r at each step to c o r r e c t t h i s . The most e f f i c i e n t way t o do t h i s f o r the e q u a l i t y c o n s t r a i n t s ( 9 ) i s to solve the set of I t i s worth no t i n g here th a t i f the o b j e c t i v e f u n c t i o n were l i n e a r i z e d along w i t h the c o n s t r a i n t s , and i f the minimum was assumed to l i e along a c o n s t r a i n t boundary, the problem could be solved u s i n g a standard l i n e a r programming program. This i s the method used i n [ 9 ] . -49-simultaneous n o n l i n e a r equations using a. conventional Newton-Raphson power flow program (the reader w i l l r e c a l l t h a t c o n s t r a i n t s (9) are j u s t the s e t of power flow equations). There are a. number of advantages to t h i s approach. F i r s t , the power flow program provides the s l a c k bus power and load bus voltages which are necessary f o r the e v a l u a t i o n of the o b j e c t i v e f u n c t i o n at each step. Second, as was pointed out by Dommel and Tinney i n \l\ , the gradient may be e a s i l y formulated at each step from terms of the Jacobian matrix (derived i n appendix s e c t i o n A l ) produced i n the power flow program. As an a d d i t i o n a l advan-tage, t h i s approach permits the bus-type s w i t c h i n g p o r t i o n of the power flow program to ensure the s a t i s f a c t i o n of the generator r e a c t i v e power c o n s t r a i n t s of equation (13). Bus-type s w i t c h i n g i s the most common way of ensuring the operation of generator busses w i t h i n t h e i r r e a c t i v e power gen-e r a t i o n r e s t r i c t i o n s . I t acts by s w i t c h i n g generator (constant r P, constant V) busses t o load (constant P, constant Q) busses whenever they are no longer able to ho l d the scheduled voltage without exceeding a Q l i m i t ( i . e . when the c o n s t r a i n t s become a c t i v e ) . When the scheduled voltage may again be h e l d without exceeding a Q l i m i t , the switched bus i s permitted t o r e v e r t back t o constant v o l t a g e . Switching busses from type P,V t o P,Q amounts t o changing -50-the a c t i v e i n e q u a l i t y c o n s t r a i n t s i n t o e q u a l i t y c o n s t r a i n t s , reducing the dimension of the s o l u t i o n space by one f o r each bus switched. This moves one bus voltage from the set of independent v a r i a b l e s t o the set of dependent v a r i a b l e s , thereby reducing the d i m e n s i o n a l i t y of the gradient by one (which i s the same as p r o j e c t i n g the f u l l g r a dient onto the appropriate c o n s t r a i n t boundary). The bus-type s w i t c h i n g technique used i n power flow programs i s ther e f o r e i d e n t i c a l i n i t s e f f e c t t o gradient reduc-t i o n or gradient p r o j e c t i o n i n a constra i n e d o p t i m i s a t i o n . Summarizing, i t i s not p o s s i b l e to use p e n a l t y f u n c t i o n methods on the generator i n e q u a l i t y c o n s t r a i n t s due t o the adverse e f f e c t s these p e n a l t i e s have on the o b j e c t i v e f u n c t i o n contours. I t i s p o s s i b l e , however, t o use a gradient r e d u c t i o n technique, by using the l i n e a r i z e d equations f o r the e q u a l i t y and a c t i v e i n e q u a l i t y c o n s t r a i n t s t o reduce the d i m e n s i o n a l i t y of the gr a d i e n t , and then c o r r e c t i n g f o r the e f f e c t s of the l i n e a r i z a t i o n at the end of each step. I f t h i s c o r r e c t i o n i s done using a conventional Newton-Raphson power flow program, the o b j e c t i v e f u n c t i o n and i t s gradient may be evaluated w i t h l i t t l e e x t r a e f f o r t , and the generator r e a c t i v e power ( i n e q u a l i t y ) c o n s t r a i n t s are a u t o m a t i c a l l y s a t i s f i e d by being converted t o e q u a l i t y c o n s t r a i n t s , when they become a c t i v e , by the bus-type s w i t c h i n g a l g o r i t h m . -51-The Method of M i n i m i s a t i o n I t has been presumed above that the gradient would be an e s s e n t i a l p a r t of any s e l e c t e d m i n i m i s a t i o n technique. While the m i n i m i s a t i o n can, of course, be performed without knowledge of the gra d i e n t , b e t t e r performance can u s u a l l y be r e a l i z e d by t a k i n g advantage of t h i s and any other i n f o r m a t i o n about the o b j e c t i v e f u n c t i o n . With the method of c o n s t r a i n t handling described above, the c a l c u l a t i o n of the gradient r e q u i r e s a r e l a t i v e l y minor amount of computation (most of which c o n s i s t s of one repeat s o l u t i o n w i t h the f a c t o r i z e d Jacobian matrix from the power flow s t e p ) . I t i s a l s o p o s s i b l e , as has been po i n t e d out by Sasson (13J to c a l c u l a t e the matrix of second p a r t i a l d e r i v a t i v e s — c a l l e d the Hessian m a t r i x — u s i n g the terms of the Jacobian matrix as f o r the g r a d i e n t . , Using the Hessian matrix the m i n i m i s a t i o n problem may be solved using a g e n e r a l i z e d v e r s i o n of the Newton-Raphson method of s o l v i n g c n o n l i n e a r equations. . For o b j e c t i v e f u n c t i o n s w i t h elongated contours, the g e n e r a l i z e d Newton-Raphson method e x h i b i t s more r a p i d and r e l i a b l e convergence than steepest descent or modified steepest descent methods, * See appendix s e c t i o n A3. -52-which make use of the gradient only. The disadvantage of Hessian-based techniques i n t h i s a p p l i c a t i o n i s the large amount of storage and computation r e q u i r e d t o produce the Hessian m a t r i x . Further, Himmelblau [l4J p o i n t s out t h a t , while Hessian-based techniques e x h i b i t q u a d r a t i c convergence i n the v i c i n i t y of the minimum, steepest descent methods may be s u p e r i o r f a r away from the minimum. For t h i s a p p l i c a t i o n , i t i s not necessary t o know the optimum e x a c t l y , but only to w i t h i n , perhaps, a few percent, so t h a t the major p o r t i o n of the o p t i m i s a t i o n e f f o r t w i l l occur away from the minimum. Bearing i n mind the observations of the l a s t chapter, where the o b j e c t i v e f u n c t i o n contours were found t o be only moderately e l l i p t i c a l ( f o r a reasonable s e l e c t i o n of p e n a l t y f a c t o r s ) , w i t h no i r r e g u l a r i t i e s i n shape to cause convergence f a i l u r e , the steepest descent method appears to be a s l i g h t l y b e t t e r choice f o r t h i s a p p l i c a t i o n than Hessian-based methods. I t was the steepest descent method, coupled w i t h a Lagrangian treatment of e q u a l i t y c o n s t r a i n t s , which was chosen by Dommel and Tinney i n [ l j . In order t o gain the maximum improvement at each step, steepest descent searches g e n e r a l l y use step lengths c a l c u l a t e d to terminate each step at the f u n c t i o n minimum i n each successive search d i r e c t i o n . These searches are termed "optimal s t e p - s i z e " -53-searches. The optimal step length can be approximated from the value of the o b j e c t i v e f u n c t i o n , and perhaps i t s d e r i v a t i v e s , at one or more p o i n t s i n the cu r r e n t d i r e c t i o n of search. The number of values needed i s dependent upon the d e s i r e d accuracy of the approximation, which determines the order of the polynomial used f o r i n t e r p o l a t i o n (or e x t r a p o l a t i o n ) i n the curren t search d i r e c t i o n . I f only the f u n c t i o n value i s known a t each p o i n t , v a r i o u s types of d i r e c t searches (see, f o r example, Himmelblau \l5^) may be used. The f i r s t d e r i v a t i v e of the o b j e c t i v e f u n c t i o n i n the d i r e c t i o n of steepest descent i s the negative of the gradient of the o b j e c t i v e f u n c t i o n ( t h i s i s the d i r e c t i o n a l d e r i v a t i v e of the o b j e c t i v e f u n c t i o n i n the d i r e c t i o n of sea r c h ) . As the gradient i s evaluated at each p o i n t t o determine the next d i r e c -t i o n of search, both the value of the o b j e c t i v e f u n c t i o n and i t s f i r s t d i r e c t i o n a l d e r i v a t i v e are a v a i l a b l e immediately, without f u r t h e r work. To i n t e r p o l a t e w i t h a second-order polynomial (the lowest order polynomial which can reasonably describe the o b j e c t i v e f u n c t i o n i n the d i r e c t i o n of search), one other piece of i n f o r m a t i o n i s r e q u i r e d . While t h i s missing p i e c e of in f o r m a t i o n could be obtained from a f u r t h e r f u n c t i o n e v a l u a t i o n i n the d i r e c t i o n of search, -54-i t i s computationally more e f f i c i e n t t o approximate the second d i r e c t i o n a l d e r i v a t i v e of the o b j e c t i v e f u n c t i o n . The reason the second d i r e c t i o n a l d e r i v a t i v e must be approximated i s t h a t i t s exact c a l c u l a t i o n would r e q u i r e the Hessian (second-order gradient) matrix. I f t h i s matrix were a v a i l a b l e , which would r e q u i r e considerable e f f o r t , i t could be used d i r e c t l y f o r a Hessian-based m i n i m i s a t i o n . Two observations permit the Hessian t o be e a s i l y approximated. F i r s t , Smirnov [l6] has p o i n t e d out t h a t , as can be i l l u s t r a t e d g r a p h i c a l l y , f o r each two-dimensional p r o j e c t i o n of the ( e l l i p -t i c a l ) contour space, a. steepest descent search converges to the optimum along the major a x i s of the e l l i p s e . In the m u l t i -dimensional case, the search w i l l converge along the major a x i s of the h y p e r - e l l i p s o i d , which i s i n the d i r e c t i o n of the e i g e n -vector corresponding t o the minimum eigenvalue of the Hessian matr i x . The second d e r i v a t i v e i n subsequent d i r e c t i o n s of search may, t h e r e f o r e , be approximated as a constant equal t o the minimum eigenvalue. Second, as has already been po i n t e d out, the contours discussed i n the previous chapter are n e a r l y c i r c u l a r . This i n d i c a t e s t h a t the Hessian matrix has diagonal terms which are a l l of the same order of magnitude, and o f f - d i a g o n a l terms which are r e l a t i v e l y s m a l l . This f u r t h e r improves the usefulness of -55-of Smirnov's observation f o r t h i s type of problem. The second d i r e c t i o n a l d e r i v a t i v e may, t h e r e f o r e , be c a l -c u l a t e d on the b a s i s of the previous step, and then used i n a second-order Taylor expansion t o p r e d i c t the optimum step length f o r the cu r r e n t step (see appendix s e c t i o n A4). Although t h i s method f o r c a l c u l a t i n g the s t e p - s i z e -is approximate, the approx-imation improves during the m i n i m i s a t i o n , and s u b s t a n t i a l l y l e s s computer time can be r e q u i r e d than f o r the c a l c u l a t i o n of the Hessian m a t r i x . The best scheme f o r the s o l u t i o n of equations (8) through (13) i s , t h e r e f o r e , 1) Newton-Raphson power flow s o l u t i o n , s a t i s f y i n g e q u a l i t y c o n s t r a i n t s , and e v a l u a t i n g the o b j e c t i v e f u n c t i o n and i t s g r a d i e n t . (2) steepest-descent search, using a sub-optimal s t e p -s i z e c a l c u l a t e d from the preceding step assuming c i r c u l a r o b j e c t i v e f u n c t i o n contours ( i . e . assuming Summary th a t the Hessian i s of the form k = c o n s t a n t ) . -56-(3) e q u a l i t y c o n s t r a i n t s (9) are handled using gradient r e d u c t i o n . (4) i n e q u a l i t y c o n s t r a i n t s (10) - (12) are handled as absolute l i m i t s on c o n t r o l v a r i a b l e v a r i a t i o n s (analogous t o gradient r e d u c t i o n f o r a c t i v e c o n s t r a i n t s ) . (5) i n e q u a l i t y c o n s t r a i n t s (13) are a u t o m a t i c a l l y handled using gradient r e d u c t i o n by a bus-type s w i t c h i n g feature i n the power flow program used at stage (1). This approach i s v i r t u a l l y i d e n t i c a l t o the general approach of Dommel and Tinney, and i s i n c o n t r a s t t o the more computa-t i o n a l l y complex scheme of Sasson e t a l . I t i s worth no t i n g t h a t , although Dommel and Tinney t r e a t e d the e q u a l i t y c o n s t r a i n t s as Lagrange terms, the equations f o r the o p t i m i s a t i o n are mathematically i d e n t i c a l t o those developed f o r the gradient r e d u c t i o n approach used here. I f the e q u a l i t y c o n s t r a i n t s are to be considered using Lagrange m u l t i p l i e r theory, then the bus-type s w i t c h i n g scheme i n the power flow r o u t i n e causes the a c t i v e i n e q u a l i t y c o n s t r a i n t s of (13) t o be t r e a t e d as Kuhn-Tucker terms. The approach of [Y] i s thus i d e n t i c a l t o that o u t l i n e d here. -57-CHAPTER V The Performance of the Technique Because the technique o u t l i n e d i n the l a s t chapter i s e s s e n t i a l l y an enhancement (to account f o r generator r e a c t i v e power c o n s t r a i n t s and shunt c a p a c i t o r and r e a c t o r a l l o c a t i o n s ) of the r e a c t i v e power o p t i m i s a t i o n technique described by Dommel and Tinney, i t was implemented by modifying an a v a i l a b l e program based on the technique described i n £l} . As there was an i n t e r p o l a t i o n scheme in!the o r i g i n a l p r o-gram, i t was r e t a i n e d on the assumption th a t i t would improve the estimate of optimal s t e p - s i z e near the minimum, thereby speeding convergence. As i t i s i n the v i c i n i t y of the minimum tha t steepest descent e x h i b i t s the worst performance, the i n t e r p o l a t i o n process was a c t i v a t e d only near the end of the o p t i m i s a t i o n process. This i n t e r p o l a t i o n r o u t i n e c a l c u l a t e s the approximate second d e r i v a t i v e assuming th a t the f i r s t d i r e c t i o n a l d e r i v a -t i v e depends only on the c o n t r o l v a r i a b l e s ( u j j , using the equations d e r i v e d i n appendix s e c t i o n A5. This was the only m o d i f i c a t i o n to the scheme developed i n the l a s t chapter. -58-The progress of the o p t i m i s a t i o n technique when performing an unconstrained m i n i m i s a t i o n on the three-bus system of f i g u r e 1 i s p l o t t e d i n f i g u r e 19 on the o b j e c t i v e f u n c t i o n contours of f i g u r e 14 (f o r the c o n t r o l of two generator v o l t a g e s ) . In f i g u r e 20, the progress i s p l o t t e d on the contours f o r the c o n t r o l of one transformer tap and one generator voltage (as f o r f i g u r e 7), and i n f i g u r e 21, i t i s p l o t t e d on the contours of f i g u r e 17 f o r the c o n t r o l of one bus voltage and one reac-t i v e shunt (at bus 3). As can be seen from these p l o t s , the o p t i m i s a t i o n progresses w e l l i n the f i r s t few steps, and reaches the minimum (to w i t h i n p r a c t i c a l tolerances) a f t e r 3 steps. To t e s t the procedure on a r e a l i s t i c , f u l l y c o n s t r a i n e d problem, a 1976 winter heavy load r e p r e s e n t a t i o n of the B.C. Hydro system was used, c o n s i s t i n g of 245 busses and 327 branches, w i t h 47 c o n t r o l l a b l e generators, and 44 c o n t r o l l a b l e (on-load tap-changing) transformers. A f t e r 16 i t e r a t i o n s of the min i m i s a t i o n , which r e q u i r e d a t o t a l of 72 power flow i t e r a t i o n s , the s o l u t i o n was acceptably c l o s e t o the optimum. C a r e f u l observation of the progress of convergence revealed t h a t o c c a s i o n a l l y an i t e r a t i o n would apparently diverge, r e s u l t i n g i n an increase i n the value of the o b j e c t i v e f u n c t i o n and/or the d e r i v a t i v e , both of which -59-must reduce f o r the process t o be convergent. This apparent divergence would occur for one or two steps, w i t h the next s e v e r a l steps converging normally. This process may occur s e v e r a l times d u r i n g a m i n i m i s a t i o n (see Table I ) . The most probable cause f o r t h i s p e c u l i a r behaviour i s t h a t the s t e p - s i z e chosen at each i t e r a t i o n i s only an approximation to the optimal step. This approximation i s based on the assumptions t h a t : a) the o b j e c t i v e f u n c t i o n i s of order 2 or l e s s i n the d i r e c t i o n of search. b) the second d e r i v a t i v e of the o b j e c t i v e f u n c t i o n i s constant f o r a l l d i r e c t i o n s of search ( i . e . the Hessian matrix i s di a g o n a l , w i t h a l l diagonal terms e q u a l ) . c) the e q u a l i t y and a c t i v e generator r e a c t i v e power i n e q u a l i t y c o n s t r a i n t s are approximately l i n e a r over the r e g i o n of the step. d) no i n a c t i v e i n e q u a l i t y c o n s t r a i n t s w i l l become a c t i v e d u r i n g the step, and no a c t i v e i n e q u a l i t y c o n s t r a i n t s w i l l become i n a c t i v e . I f any of these assumptions are i n v a l i d f o r a given s t e p — a s , i n general, a t l e a s t one w i l l be-- the c a l c u l a t e d s t e p - s i z e w i l l be sub-optimal. Depending on the degree t o which the assump-t i o n s are i n v a l i d , the c a l c u l a t e d s t e p - s i z e may become sub-TABLE I Progress of Convergence f o r 245 Bus Problem Step No.- F i r s t D i r e c t i o n a l Slack Bus Power D e r i v a t i v e p l us p e n a l t i e s 1 7.37 15.88 2 2.68 15.58 3 2.23 15.52 4 3.27 15.50 5 2.61 15.48 6 1.03 15.46 7 0.853 15.45 8 0.674 15.44 9 0.797 15.42 10 9.37 15.59 11 9.05 15.64 12 1.14 15.44 13 0.911 15.43 14 1.56 15.42 15 1.73 15.43 16 0.382 15.41 17 0.271 15.41 18 0.238 15.41 19 0.364 15.41 20 0.175 15.41 21 0.122 15.41 22 0.098 15.41 -61-optimal t o the p o i n t of being divergent. While--for . a steepest descent s e a r c h — a s t e p - s i z e of l e s s than the optimal amount w i l l g e n e r a l l y only h o l d the conver-gence r a t e down somewhat, too l a r g e a s t e p - s i z e can cause the new value f o r the o b j e c t i v e f u n c t i o n and/or gradient to be greater than the previous value, g i v i n g a divergent step. For a s t e p - s i z e l a r g e r than the optimum, whether or not the step i t s e l f w i l l be divergent depends on by how much the step length i s too l a r g e , and on how r a p i d l y the o b j e c t i v e f u n c t i o n and gradient change i n the new d i r e c t i o n of search. Of the above four assumptions, there are two which are the most l i k e l y to d i s t u r b the approximation t o the optimum st e p -s i z e . For problems w i t h a l a r g e number of i n e q u a l i t y con-s t r a i n t s , the f o u r t h assumption w i l l l i k e l y be v i o l a t e d at most steps. The s w i t c h i n g of c o n s t r a i n t s from the i n a c t i v e set to the a c t i v e set produces a d i s c o n t i n u i t y i n the o p t i m i -s a t i o n process which could give r i s e t o sporadic divergence. This c o n s t r a i n t s w i t c h i n g was observed to be o c c u r r i n g at most steps i n the m i n i m i s a t i o n . The second assumption i s a l s o known to be sometimes u n r e l i a b l e , as i t i m p l i e s t h a t the o b j e c t i v e f u n c t i o n contours must be c i r c u l a r , whereas they are known to be always e l l i p t i -c a l t o some degree. From t h i s observation, we can p r e d i c t -62-t h a t the c a l c u l a t e d s t e p - s i z e w i l l be too small i n some d i r e c t , t i o n s , and too large i n others. Since the estimate f o r the optimal s t e p - s i z e i s based upon the r a t e of change of the gradient i n the d i r e c t i o n of search f o r the previous step (and evaluated over the span of th a t s t e p ) , a greater-than-optimal s t e p - s i z e w i l l most l i k e l y r e s u l t when the new d i r e c t i o n of search i s more perpendicular to the major a x i s of the contours than was the l a s t . The e f f e c t i s more pronounced, and more r e a d i l y leads t o divergence, when the contours are s t r o n g l y e l l i p t i c a l . I t i s m i t i g a t e d somewhat by c o n s t r a i n t a c t i v a t i o n , i n th a t the successive search d i r e c t i o n s are thereby a l t e r e d from the usual near-orthogonal search d i r e c -t i o n s of an unconstrained ( n e a r l y - o p t i m a l s t e p - s i z e ) search. Of the two assumptions l i s t e d above as p o t e n t i a l causes of the observed sporadic divergence, i t i s the second which i s the probable major c o n t r i b u t o r . This would account f o r the f a c t t h a t the divergent steps occur r a r e l y ; the steps would u s u a l l y be convergent u n t i l the contours became too e l l i p t i c a l . One way t h i s c o uld occur, f o r example, i s when a generator on a long, l i g h t l y loaded feeder "came o f f " a minimum-Q l i m i t (that i s , the generator had been, but i s no longer, l i m i t e d to-i t s minimum a v a i l a b l e r e a c t i v e power output). In the plane of t h i s and some other generator v o l t a g e , i t i s reasonable t h a t the -63-second generator voltage may a f f e c t the o b j e c t i v e f u n c t i o n c o n s i d e r a b l y more than the f i r s t — l e a d i n g to e l l i p t i c a l contours i n the plane corresponding to these two generator v o l t a g e s . When the f i r s t generator reaches a mdnimum-Q l i m i t (which i s more l i k e l y , under the circumstances, than a high l i m i t ) , t h i s plane vanishes because the f i r s t generator voltage i s no longer a c o n t r o l v a r i a b l e . This would then ( e f f e c t i v e l y ) reduce the e l l i p t i c i t y of the multi-dimensional contours, s t a b i l i z i n g the adaptive s t e p - s i z e and thus the convergence. I t may appear from the above d i s c u s s i o n t h a t a b e t t e r approximation t o the optimal s t e p - s i z e , or perhaps even a more powerful unconstrained m i n i m i s a t i o n technique i s needed. These are not n e c e s s a r i l y s o l u t i o n s , however, i n tha t the improvement gained w i l l be much l e s s than would be expected due to the e f f e c t of the i n e q u a l i t y c o n s t r a i n t s , and may not be s u f f i c i e n t to warrant the e x t r a c a l c u l a t i o n necessary. The e f f e c t of the i n e q u a l i t y c o n s t r a i n t s i s r a t h e r hard t o p r e d i c t , other than t h a t they are l i k e l y t o d i s t u r b the steady convergence of the same unconstrained problem. This disturbance i s so powerful t h a t improvements i n the unconstrained o p t i m i s a t i o n technique used at each step do not n e c e s s a r i l y speed up the o v e r a l l s o l u t i o n . In p a r t i c u l a r , knowing the optimal s t e p - s i z e i s of l i t t l e advantage i f the search d i r e c t i o n i s d e f l e c t e d from -64-the negative gradient d i r e c t i o n by i n e q u a l i t y c o n s t r a i n t s . The same i s true f o r Hessian and r e l a t e d searches, i n th a t the search d i r e c t i o n c a l c u l a t e d may bear l i t t l e r e l a t i o n t o the f i n a l d e f l e c t e d search d i r e c t i o n . Another problem which e x h i b i t e d i t s e l f was the e f f e c t of the voltage p e n a l t y f a c t o r s on the convergence r a t e . Values f o r the voltage p e n a l t y f a c t o r s which are too lar g e can lead t o e r r a t i c convergence, probably due t o the r e s u l t i n g s e n s i t i v i t y of the o b j e c t i v e f u n c t i o n t o the voltage p e n a l t i e s . This i s not unduly s u r p r i s i n g , s i n c e i f the " c i r c u l a r i z i n g " e f f e c t of the voltage p e n a l t i e s demonstrated i n chapter I I I i s c a r r i e d t o extremes, the contours w i l l become e l l i p t i c a l again, t h i s time w i t h the minor and major axes interchanged. Machine p r e c i s i o n may a l s o be a problem w i t h l a r g e voltage p e n a l t y m u l t i p l i e r s due to the l a r g e r second d e r i v a t i v e terms and<-consequent s h o r t e r step lengths. The use of moderate voltage p e n a l t y f a c t o r s permitted the 500 kV busses at. remote generator s i t e s to r i s e w e l l over the d e s i r e d value of 1.05 per u n i t t o values as high as 1.10 pu. Larger values of voltage p e n a l t y f a c t o r s on these busses wor-sened convergence without decreasing the f i n a l voltages s i g n i f -i c a n t l y , i n d i c a t i n g that the high remote s i t e voltages were important t o the mi n i m i s a t i o n of the balance of the t o t a l -65-o b j e c t i v e f u n c t i o n — i ,e. the maintenance of reasonable voltages on other system busses and the re d u c t i o n of system l o s s . The best way to reduce these v o l t a g e s , where necessary, i s to c o n t r o l the maximum value of generator voltage on the asso-c i a t e d generator busses. Since the generator voltage i s t r e a t e d as an absolute l i m i t , t h i s w i l l prevent the high-voltage bus from exceeding safe voltage l e v e l s . This problem i s most l i k e l y t o a r i s e when transformer taps, r e a c t o r banks, e t c . which were not made c o n t r o l l a b l e are p o o r l y s e t . This was, i n f a c t , the problem w i t h the t e s t case, as too many r e a c t o r s had been used a t s t a t i o n s between the remote generation and the load center. I t i s important to r e a l i z e t h a t , w h i l e problems such as the high voltages noted above appear s e r i o u s , they can i n f a c t be important c l u e s t o d e f i c i e n c i e s i n the power system on which the m i n i m i s a t i o n was oper a t i n g . The o p t i m i s a t i o n process acts on the c o n t r o l v a r i a b l e s i n any way necessary t o achieve i t s o b j e c t i v e . Provided always t h a t the o b j e c t i v e i s a reasonable one, unconventional s o l u t i o n s may i n d i c a t e poor adjustment of other parameters not c o n t r o l l a b l e by the program, o r — a s i s perhaps too o f t e n the c a s e — m e r e l y i n v e t e r a t e t h i n k i n g on the p a r t of the person e v a l u a t i n g the s o l u t i o n . Other than these i s o l a t e d high v o l t a g e s , the r e s u l t i n g system s t a t e was much b e t t e r than the author had been able to -66-achieve when using t h i s case p r e v i o u s l y f o r system s t u d i e s w i t h a conventional manual power flow, one major improvement being the system voltage p r o f i l e . -67-CHAPTER VI  Conclusions The optimal r e a c t i v e power flow problem can be solved using a steepest descent search w i t h gradient r e d u c t i o n (or e q u i v a l e n t l y , Lagrange) c o n s t r a i n t terms, u s i n g an o b j e c t i v e f u n c t i o n composed of the s l a c k bus r e a l power, and voltage and r e a c t i v e power p e n a l t y terms f o r load bus voltages and a l l o c a t i o n (load) busses, r e s p e c t i v e l y . This o b j e c t i v e f u n c t i o n i s g e n e r a l l y w e l l - s c a l e d , and a sub-optimal s t e p - s i z e search performs e f f e c t i v e l y on the f u l l y c o n s t r a i n e d problem, provided t h a t the voltage p e n a l t y f a c t o r s are not exc e s s i v e . The r e s u l t i n g set of generator and transformer s e t t i n g s , and s i z i n g s f o r shunt r e a c t i v e compensation banks produce a g e n e r a l l y good power flow case w i t h l e s s engineering e f f o r t than would be r e q u i r e d u s i n g conventional manual adjustments. -68-CHAPTER VII D i r e c t i o n s f o r Further Work The most important remaining work i s the e v a l u a t i o n of the technique i n a production environment. The two problems encountered—the sporadic divergence and the d i f f i c u l t y i n h o l d i n g down remote bus voltages w i t h p e n a l t y terms a l o n e — a r e not thought t o be s e r i o u s . The only way of con f i r m i n g t h i s b e l i e f however, i s by o b t a i n i n g production experience. In a d d i t i o n t o the aforementioned production t e s t i n g , the technique may need t o be extended t o cover the many automatic features a v a i l a b l e i n modern power flow programs (e.g. blocks of shunt r e a c t i v e c a p a c i t y a c t i v a t e d by voltage magnitude, generators w i t h r e a c t i v e power adjusted to ho l d remote bus voltages w i t h i n l i m i t s , e t c . ) . In many cases, these features of power flow programs w i l l be made unnecessary by an optimal r e a c t i v e power management fe a t u r e . Nevertheless, allowance must be made f o r p o s s i b l e c o n f l i c t s between the o p t i m i s a t i o n process and the ( g e n e r a l l y r a t h e r crude) automatic c o n t r o l provided by such f e a t u r e s , and the features should be removed, i n h i b i t e d d uring o p t i m i s a t i o n , or f o r m a l l y incorporated i n t o the o p t i m i s a -t i o n process. -69-One f u r t h e r problem which should be i n v e s t i g a t e d i s m u l t i p l e contingency o p t i m i s a t i o n . This i s somewhat d i f f e r e n t than o p t i m i s a t i o n f o r a normal operating c o n d i t i o n , i n that the outage c o n d i t i o n s must meet c e r t a i n minimum operating l i m i t s ( g e n e r a l l y on load bus voltage) w i t h only on-load transformer taps, gen-e r a t o r v o l t a g e s , and switchable banks of shunt compensation being adjusted from normal operating s e t t i n g s . This means tha t o f f - l o a d transformer taps must be set so t h a t i t i s p o s s i b l e t o achieve the minimum operating l i m i t s using only the adju s t a b l e parameters. Although the o p t i m i s a t i o n may be performed sep-a r a t e l y on each contingency c o n d i t i o n , a method i s r e q u i r e d f o r e f f i c i e n t l y combining the separate contingency optima i n t o a s i n g l e r e s u l t . -70-REFERENCES H.W. Dommel and W.F. Tinney. "Optimal Power Flow S o l u t i o n s . " IEEE Trans., v o l . PAS-87, Oct. 1968, pp. 1866-76. A l b e r t M. Sasson. "Combined Use of the Powell and F l e t c h e r - P o w e l l Nonlinear Programming Methods f o r Optimal Load Flows." IEEE Trans. , vol. ;^PAS-88, No. 10, Oct. 1969, pp. 1530-1537. A Survey of Canadian U t i l i t y P r a c t i c e s i n Planning  & A p p l i c a t i o n of S t a t i o n Shunt Cap a c i t o r Banks. By K. Nishikawara, Chairman. Toronto, O n t a r i o : Canadian E l e c t r i c a l A s s o c i a t i o n Power System Planning & Operation S e c t i o n , 1976. W.F. Tinney and H.W. Dommel. "Steady State S e n s i t i v i t y A n a l y s i s . " Report No. 3.1/10, 4th Power Systems Computation Conference, Grenoble (France), Sept. 11-16, 1972. -71-Raymond M. M a l i s z e w s k i , Len L. Garver, and A l l e n J . Wood. "Linear Programming as an A i d i n Planning K i l o v a r Requirements." IEEE Trans., v o l . PAS-87, No. 12, Dec. 1968, pp. 1963-67. H.W. Dommel. "Input-Output S e n s i t i v i t i e s . " Unpublished notes. A. Kishore and E.F. H i l l . " S t a t i c O p t i m i z a t i o n of Reactive Power Sources by Use of S e n s i t i v i t y Parameters." IEEE Trans., v o l . PAS-90, No. 3, May/June 1971, pp. 1166-73. N.P. K o h l i and J.C. K o h l i . "Optimal Capacitor A l l o c a t i o n by 0-1 Programming." Paper A 75 476-2, IEEE Summer Power Meeting, San F r a n c i s c o , C a l i f . , J u l y 20-25, 1975. A. Kuppurajulu and K. Raman Nayar. "Minimisation of Reactive-Power I n s t a l l a t i o n i n a Power System." Proc. IEE, v o l . 119, No. 5, May 1972, pp. 557-563. -72-10 David M. Himmelblau. A p p l i e d Nonlinear Programming. McGraw-Hill, New York, 1972, chapter 6. 11 S.S. Sachdeva and R. B i l l i n t o n . "Optimum Network VAr Planning by Nonlinear Programming." IEEE Trans., v o l . PAS-92, July/Aug. 1973, pp. 1217-25. 12 . "Optimum Network VAr Planning Using Real and Reactive Power Decomposition Non-linear A n a l y s i s . " Proc. 8th Power Industry Computer A p p l i c a t i o n s  Conference, Minneapolis, Minn., 1973, pp. 339-347. 13 A.M. Sasson, F. V i l o r i a , and F. Aboytes. "Optimal Load Flow S o l u t i o n Using the Hessian M a t r i x . " IEEE Trans., v o l . PAS-92, Jan./Feb. 1973, pp.31-41. 14 Himmelblau. A p p l i e d Nonlinear Programming, pp. 87-88. 15 I b i d . Chapter 2. 16 K.A.Smirnov. "Optimization of the Performance of a Power System by the Decreasing Gradient Method." I z v e s t i y a Akademii Nauk SSSR, s e r . Energetika i -73-Transport (News of the Academy of Sciences USSR, Power Engineering and Transport S e r i e s ) , Moscow, No. 2, 1966, pp. 19-28, i n Russian. T r a n s l a t e d f o r B o n n e v i l l e Power A d m i n i s t r a t i o n by the J o i n t P u b l i c a t i o n s Research S e r v i c e s , May 1967. -74-APPENDIX D e r i v a t i o n of Relevant Equations A l The Terms of the Jacobian M a t r i x The power e n t e r i n g the system at bus i i s given by S* = (P. - j Q.) = V2± (G.. + j B..) + V i (cos © i - j s i n 6 i ) * 2 (G.. + j B..) V. (cos 9. + j s i n 9.) ;ewcr J J J J J Breaking t h i s equation i n t o r e a l and imaginary components P i = V V^ j [cos Q± ( G ^ cos - B-j-j s i n 9^) + s i n e ± (G ±j s i n 9j + cos 9j)] + V?G i ; L Qi = - V i ^ V ^ r c o s 0. {G-- s i n 0. + B H, cos 9.) -s i n 9 ± ( G ± j cos 9j - B i j s i n 9j)] -The terms of the Jacobian matrix are: H l l - a e , : N " " V i ^ J l 1 " ^ e ; L l l " V l ^ -75-where AP and AQ are the r e a l and r e a c t i v e power mismatches ( p o s i t i v e i n t o bus), r e s p e c t i v e l y Define so «ij = G i j c o s 9 j " B i j s i n e j / ^ i j = G i j s^-n e j + B i i c o s 9 i ^4- = -3- • = o(. Now, since ^ p i ~ pgen^ ~ p l o a d ^ ~ p i ^Qi = Q g e n i - Qioad ± - Qi vl'.r where then p g e n i ' P l o a d i ' Qgen.^' a n d Qloadi a r e constant, H i i = " V ^ V j V i j cos 9. s i n 9.) Q. + v V . -76-N i j = - v * f t N i i = - v. a P i J i j = - ^ = V J . . =-. l l — 1 L . . = — —j - V . 4 & = L. . = 11 - V l®i' = • = - V i Vj (oCij cos 9 i + ^ i j s i n 6 i ) -= ~VH $ V . (*.. cos 9- s i n 6 -2 V 2G.. = - P. - V 2G.. l l l l l l l ± Vj (<Xij cos 9 i + s i n 0 ±) V. 2.V. (of. . cos 0. +/3\ . s i n G.) 1J£A/CIJ ij x f xj i P. + V 2G.. l l l l j a v T v i v J c o s e i s i n e i ) pencil /"ID i -2 V ? B ± i = -Q± + V ? B i ± The Terms of the Gradient The o b j e c t i v e f u n c t i o n i s f(x,u) = P s(x,u) + .^.w i(V i - v ? c h e d ) 2 + . sched.2 ^z. (B. - B. ) where -T x _T u l e v ] [t v c B ] V c = voltage on busses of the set NG B = r e a c t i v e shunt on busses of the set NQ -77-Let F(u) = f ( x , u ) . The gradient 7 F U may be found by observing t h a t the f i r s t - o r d e r v a r i a t i o n of F i s given by A F = -|j-Au + ~ A x = V F ^ Au (A2.1) Now, expanding the e q u a l i t y c o n s t r a i n t (9) i n a f i r s t - o r d e r Taylor expansion: g(x,u) = g ( x 0 , u Q ) + j j J ^ A x +[-|J]AU Since g(x,u) = g ( x Q , u 0 ) = 0, S u b s t i t u t i n g (A2.2) i n t o (A2.1) A " = V F J AU The gradient V F U i s thus given by = i^ _r42.r^ _rii£ („. 3 ) -78-This i s the same expression f o r the gradient as Dommel and Tinney o b t a i n f o r the gradient i n £lj using the Lagrange m u l t i p l i e r approach. The terms of t h i s expression are: 2AQ ZAP a e i A f av dV (A2.4) i£_T I ax L + 2w(V - VS C h e d i _ av/ (A2.5) au Mf at ^>AP -0.-aaa at iAQ avc aaa ae (A2.6) Equation (A2.3) can be c a l c u l a t e d e a s i l y by t a k i n g advantage of the f a c t t h a t the expression (which corresponds to the ve c t o r of Lagrange m u l t i p l i e r s i n the -79-Dommel and Tinney approach) can be obtained from one repeat s o l u t i o n using the transposed f a c t o r i z e d Jacobian matrix from the Newton-Raphson power flow. Expressed i n terms of the Jacobian matrix, t h i s e x p r e s s i o n becomes -tf - J The terms f o r (A2.4) and (A2.5) can be obtained d i r e c t l y from the Jacobian matrix. The terms f o r (A2.6) and (A2.7) are M P i it W i j cos 0 i H - ^ i j s i n e ±) = - ML i f i i s non-tap side bus, or 2 \jf Gij i f i i s tap s i d e bus. S i m i l a r l y , 'J Li i f i i s non-tap side bus, or + V i f i i s tap side bus -80-- N. N. . l l = L ID = 0 - Vi A3 The Terms of the Hessian M a t r i x F o l l o w i n g the same procedure as f o r the c a l c u l a t i o n of the terms of the grad i e n t , l e t F(u) = f ( x , u ) , so th a t AF = PF^Au + h A u T ] > F u u ] Au Using the same expression f o r Ax i n terms of 4u as i n s e c t i o n A2, -81-The e v a l u a t i o n of t h i s matrix r e q u i r e s a considerably greater amount of computation than does the e v a l u a t i o n of the gradient The a d d i t i o n a l matrices needed are: <72f "uu £-Ps 0 0 2Z V 2f xx a2Ps <J8a© aea*/ avai/ + 2( 7 2 f xu aeat ae avc a^Ps ^avat av^i/c 0 -82-f o r which the terms are: <r P S \2 P p. Ms i 5_£s_ - - J s - non-tap side bus, or gs/?Gsj' s = tap side bus. _ 2 % .0 _ A/5/' - N*i + -A/sJ ^Ps _ A/sJ 32PS _ 2 G 3 V/ ^ ss ^ _ 0 " N s j f% - - p„ + s Nsj <?V, _ 0 S-H --fPs _ *-y s = non-tap; side bus, or WsGsj s _ s i ^ e > j U S > v s G s s V B s ss -83-A4 The Approximation t o the Optimal S t e p - s i z e Let F(u) = f(x,u) as i n s e c t i o n A2. Now expand F (u) to second order: F(u) = F + VTF„ An + h A u T T^F ~1 Au O U I- UU J For steepest descent ^ = _ c£AJ>Tu where [A] represents the r o t a t i o n and s c a l i n g of flu as a r e s u l t of c o n s t r a i n t r e f l e c t i o n (diagonal m a t r i x ) . S u b s t i t u t i n g , F(u) = F - ^cCAiVF^ j_ C ' ^ ^ M ' C ^ U W ^ so t h a t -84-do 1 j l C A l ^ ^ f Assuming th a t the o b j e c t i v e f u n c t i o n i s of order 2 or l e s s i n the d i r e c t i o n of search, so that 7-^= G (constant), then d£ k + /_ i £ k The d e r i v a t i v e at step k.+ 1 i n the d i r e c t i o n of step k i s so t h a t G = -The optimal s t e p - s i z e w i l l ensure t h a t — = 0, and i s given a c by k+1 = VTfu LA] -85-and r - k + l _ - c LAl t7Fu ~ KlAl**' VFk+'[\ S u b s t i t u t i n g [AJVF* = '^^^1 and !ldu k|/ 2 = ( c k ) 2 At step k+1, i s s t i l l unknown, and so we assume D T 1 - [ i ] g i v i n g — k+1 4 u For the f i r s t step, there i s no previous i n f o r m a t i o n , so there i s no i n f o r m a t i o n f o r — — — i n a cr -86-F 2 = F 1 -H^FJU C 1 + hie1)2 0 -In order to solve t h i s f o r c without knowing , we assume that an optimal step w i l l reduce the o b j e c t i v e f u n c t i o n by an a r b i t r a r y amount. Experience i n d i c a t e s t h a t 2% i s reasonable, so 2 1 1 F - F = -0.02 F and -0.02 F 1 = - g r i l l e 1 + ^ ( c 1 ) 2 ^ - -a c** Since the step i s optimal, which i m p l i e s t h a t and -0.02 F 1 = - f j ^ u l / c 1 + h c ^ T F u l l = -h c^TPJK -87-Thus and A5 '.' I n t e r p o l a t e d S t e p - s i z e I f the s i g n of any p a r t i a l d e r i v a t i v e changes, the corresponding v a r i a b l e can be i n t e r p o l a t e d , as the minimum i n that d i r e c t i o n has been passed. Assuming = Rj_ (constant) 3u* then 1E1 6ui k+1 IE1 a«*i k+2 For an optimal step, = 0 and -88-k+1 k+1 • i f _ A6 Proof t h a t the Objective Function ^ ) < l ( ( 2 | x ( f o r a l l  System Branches E q u a l i z e s the Product /jljf )x| Consider the f o l l o w i n g s e c t i o n of a power system: 3 The t o t a l c u r r e n t I T = 1^ + i s assumed constant, while the two component currents Ij_ and I 2 may be a l t e r e d by a d j u s t -ment of the transformer tap t . The r e a c t i v e power l o s s i n these two branches i s given by = I ! x l + *2 X2 = X 1 X 1 + <I T " I l ) 2 x 2 -The value of I j f o r which ^ g i s a minimum can be determined by s e t t i n g the f i r s t d e r i v a t i v e of w i t h respect t o 1^ equal to zero. = 2 I 1 X 1 - 2 ( I T - I n ) X 9 = 0 T • l ' A 2 -89-which i m p l i e s that l x l - I 2 X 2 ' Q.E.D. = 5.14 pu 5 =-2.076 pu Q3 =-0.535 pu Y„ = 4.989-j 29.72 pu Y,3=-4.989 + j 29.84 pu YM= 6.063-j 28.4 pu Y„=-6.063 + j 28.67 pu ^3=11.05-j 58.12 pu on 100 MVA a 230 kV Figure 1. Three-bus example system. BUS I VOLTAGE Figure 2. Contours of constant l o s s f o r system of f i g u r e 1. Voltages are per u n i t based on 230 kV, and contour values are MW. BUS I VOLTAGE Figure 3. Contours of constant l o s s f o r system of f i g u r e 1, except t h a t P2 = 1 .0 pu, so that both generators provide approximately h a l f of the bus 3 r e a l power each. o o t £ = 1.0 pu Yfl = 6.985-j 41.37 pu ? £=-2.076 pu Y I2=-l.996 + j 11.94 pu Q3=-0.535 pu Y3 = -4.989 + j 29.84 pu Y2l= 8.059 - j 40.05 pu Y„=-6.063+j 28.67 pu Y33= 1105-j 58.12 pu on 100 MVA a 230 kV Figure 4.^ Three-bus example system of f i g u r e 1, wit h a d d i t i o n a l h y p o t h e t i c a l branch between busses 1 and. 2, and P.2""•= 1.0 pu. BUS I VOLTAGE Figure 5. Contours of constant l o s s f o r system of f i g u r e 4. 0 0.85 0.95 BUS 100 1.05 VOLTAGE Figure 6. Contours of constant l o s s f o r system of f i g u r e 1, but w i t h only 40% of the load at bus 3 (P 2 = 2.056 pu, S3 = -0.83 - j 0.214 pu). BUS I VOLTAGE Figure 7. Contours of constant l o s s versus voltage at bus 1, and tap s e t t i n g of hypothet-i c a l , zero impedance transformer i n s e r t e d at the bus 3 end of the branch between busses 1 and 3. The tap i s on the bus 3 s i d e . Otherwise the system i s i d e n t i c a l t o t h a t of f i g u r e 1. BUS I VOLTAGE Figure 8. Contours of constant l o s s versus voltage at bus 1, and tap s e t t i n g of hypothet-i c a l transformer i d e n t i c a l to that f o r f i g -ure 7. Otherwise the system i s i d e n t i c a l to that of f i g u r e 4. 42.0 38.0 CM 30.0 270 24.0 21.0 18.0 15.0 12.0 9.0 BUS I VOLTAGE 0.95 1.0 1.05 BUS I VOLTAGE Figure 9. Contours of constant l o s s versus ' vo l t a g e . System i s tha t of f i g u r e 1, except -that the power f a c t o r of the load at bus 3 i s only 80% (so tha t S 3 = -1.715 - j 1.286 pu) . Figure 10. Contours of constant l o s s versus voltage. System i s that of f i g u r e 1, except th a t the load at bus 3 has been reduced to only 40% (as f o r f i g u r e 6), and the power f a c t o r reduced to 80%. j 0.515 pu. Thus S 3 = -0.686 -UJ § o > CO 03 ro O I-CVJ " o cr. 25.5 lj_ 27.0 0_ • 28.5 I— 30.0 BUS 3 SHUNT Figure 11. Contours of constant l o s s versus voltage at bus 2, and the value of r e a c t i v e shunt a t bus 3. Other than the bus 3 shunt, the system i s th a t of f i g u r e 1. -28.0 -27.0 - 26.0 - 25.0 24.0 I 25.0 (J\ I 26.0 27.0 BUS 3 SHUNT Figure 12. Contours of constant l o s s versus the tap on the h y p o t h e t i c a l transformer (as f o r f i g u r e 7), and bus 3 r e a c t i v e shunt (as fo r f i g u r e 11). Otherwise the system i s that of f i g u r e 1. Figure 13. Contours of constant l o s s versus Figure 14. Contours of f i g u r e 2 augmented voltage at bus 2, and the value of r e a c t i v e w i t h a voltage p e n a l t y term f o r bus 3 v o l -shunt at bus 3. Other than the bus 3 shunt, tage. The p e n a l t y f a c t o r i s 7.5, and the the system i s t h a t of f i g u r e 4. maximum and minimum unpenalized voltages are 1.05 and 1.00 pu, r e s p e c t i v e l y . Figure 15. Contours of f i g u r e 3 augmented Figure 16. Contours of f i g u r e 11 augmented w i t h a voltage p e n a l t y term as f o r f i g u r e 14. w i t h a voltage p e n a l t y term as f o r f i g u r e 14. BUS 3 SHUNT Figure 17. Contours of f i g u r e 16 augmented wit h a p e n a l t y term f o r the shunt r e a c t i v e power i n j e c t e d at bus 3. Any amount of r e a c t i v e i n j e c t i o n i s p e n a l i z e d , w i t h the p e n a l t y f a c t o r being 1.0. Figure 18. Contours of f i g u r e 14 augmented w i t h a pen a l t y term f o r the r e a c t i v e power produced or absorbed by generators 1 and 2. The generators are allowed t o produce (or absorb) r e a c t i v e power to a power f a c t o r of 0.95, excess r e a c t i v e power being p e n a l i z e d w i t h a pen a l t y f a c t o r of 1.0. Figure 19. Contours of f i g u r e 14, on which Figure 20. Contours of f i g u r e 7 augmented has been p l o t t e d the progress of the prog- w i t h a voltage p e n a l t y term as f o r f i g u r e rammed o p t i m i s a t i o n method. 14, on which has been p l o t t e d the progress of the programmed o p t i m i s a t i o n method. Figure 21. Contours of f i g u r e 17, on which has been p l o t t e d the progress of the prog-rammed o p t i m i s a t i o n method. 

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