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New formulation of the load-flow problem Jalali-Kushki, Hossein 1973

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A NEW FORMULATION OF THE LOAD-FLOW PROBLEM by HOS'SEIN OALALI-KUSHKI Bo A.Sc., Arya-Mehr U n i v e r s i t y of T e c h n o l o g y , 197 1 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOE THE DEGREE OF MASTER CF APPLIED SCIENCE i n t h e Department of E l e c t r i c a l E n g i n e e r i n g Ke a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITI.SH COLUMBIA June 1973 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia Vancouver 8 , Canada Date ABSTRACT The new formulation of the load-flow problem presented in t h i s thesis yields a set of equations each of which has only one nonlinear term» The equations are derived from the corrections required to make the f i n a l values equal to the i n i t i a l estimated values. The resultant set of equations can be used when the i n i t i a l estimated values are adjusted to t h e i r f i n a l values. However, derivation of the equations f o r t h i s l a t t e r case r e s u l t s i n a set of equations with (n - 1 ) nonlinear terms i n each equation for an n-bus power system. Five algorithms based upon the new formulation are described. Numerical tests on several sample power systems show that some of the new algorithms possess better convergence and speed c h a r a c t e r i s t i c s than the commonly used Ward-Hale and Newton algorithms. TABLE OF CONTENTS ABSTRACT . . . . o i i TABLE OF CONTENTS i i i LIST OF SYMBOLS i v ACKNOWLEDGMENT . „ v i i 1. INTRODUCTION 1 2. PREVIOUS METHODS . 6 2.1 The Ward-Hale Method 6 The Algorithm „ 6 Acceleration : 10 Storage Requirements And Programming Techniques .... 11 2.2 Newton's Method 12 The Algorithm „, . 13 Storage Problem 15 Ordering Of The Equations 16 Programming Techniques 18 3. PROPOSED METHOD 20 3- 1 Introduction a * . . . . . . . * . . . 20 3.2 Derivation Of The Equations ........... < > . . . . . . . . a . . . 20 3.3 The Algorithms 25 Algorithm A 26 Algorithm B 28 Algorithm C 29 Algorithm D 30 Algorithm E 31 3.4 Programming And Storage 32 4. NUMERICAL RESULTS 33 4.1 Introduction 33 4.2 Results 36 Comparison of Convergence C h a r a c t e r i s t i c s 36 Comparison Of Computing Speed „ 42 Comparison of Storage Requirements 43 4.3 Summary 44 5. CONCLUSIONS 45 REFERENCES o 46 APPENDIX A 48 APPENDIX B 50 APPENDIX C 52 i v LIST OF SYMBOLS VOLTAGE V k vo l t a g e of bus k ' V magnitude of V k ek phase angle of A r e a l p a r t of Vfc f k imaginary part of u k magnitude of vol t a g e at (P-V) type bus k [ V ] v o l t a g e v e c t o r A v k vo l t a g e c o r r e c t i o n at bus k r e a l p a r t of AV ek imaginary part of AV K. CURRENT \ \ k [ I ] c u r r e n t e n t e r i n g bus k r e a l part of I imaginary part of I K.. c u r r e n t v e c t o r ADMITTANCE Y i k G i k Ik 1 V 1 element (i,k) of the nodal admittance matrix r e a l p a r t of Y (conductance) i k imaginary part of Y (suseptance) ik magnitude of Y ik ik phase angle of Y ik m nodal admittance matrix POWER A S * k C . s ] power e n t e r i n g bus k r e a l power e n t e r i n g bus k r e a c t i v e power e n t e r i n g bus k t o t a l change of power at bus k power v e c t o r JACOB I AN' D kj s e n s i t i v i t y of P, with r e s p e c t to e k j s e n s i t i v i t y of P with r e s p e c t to |V | k j (normali2ed) s e n s i t i v i t y of with r e s p e c t to e . s e n s i t i v i t y of 0 with r e s p e c t t c |V | k j (normalized) Jacobian sub-matrix GENERAL unprimed * m j n n present assumed or c a l c u l a t e d value f i n a l or scheduled value complex conjugate r e a l p a r t of a complex v a r i a b l e complex o p e r a t o r , -1 t o t a l number of buses i n the system number of. (P-Q) type buses i n the system number of (P-V) type buses i n the system s e t of a l l the (P-Q) type buses s e t of a l l the (P-V) type buses r e a l part of V! Y* ,- t * imaginary part of V. Y ACKNOWLEDGMENT I would l i k e Co thank my s u p e r v i s o r Dr. H.R.Cheann f o r h i s i n t e r e s t , a d v i c e and encouragement t h r o u g h o u t the r e s e a r c h and w r i t i n g of t h i s t h e s i s . ' I would a l s o l i k e t o thank Dr., Y.N.Yu f o r p r o o f - r e a d i n g the t h e s i s and o f f e r i n g f u r t h e r s u g g e s t i o n s and a l s o f o r f i n a n c i a l s u p p o r t d u r i n g the month of May. I am a l s o g r a t e f u l t o M i s s e s Norma Duggan and B e t t y Cockburn f o r t h e i r h e l p i n t y p i n g the t h e s i s . The f i n a n c i a l s u p p o r t o f t h e U n i v e r s i t y of B r i t i s h Columbia i n the form of UBC gr a d u a t e f e l l o w s h i p i s acknowledged. 1 1. INTRODUCTION. Load-flow s t u d i e s are r e q u i r e d i n the design of any power system. Such s t u d i e s are a l s o r e q u i r e d to o p t i m a l l y d i s p a t c h power i n the system and to f i n d the i n i t i a l c o n d i t i o n s of the system f o r t r a n s i e n t s t a b i l i t y s t u d i e s . Numerous load-flow s t u d i e s of v a r i o u s system c o n f i g u r a t i o n s have to be made i n p l a n n i n g new systems and extending present systems. The l o a d - f l o w problem i s to f i n d the v o l t a g e s of the v a r i o u s buses (nodes) of a power system such t h a t c e r t a i n c o n s t r a i n t s are s a t i s f i e d . These c o n s t r a i n t s may r e q u i r e t h a t a d e s i r e d p r e s c r i b e d or scheduled value of v o l t a g e magnitude e x i s t at some of the buses, or t h a t p r e s c r i b e d or scheduled values of a c t i v e , or r e a l , and r e a c t i v e powers e x i s t at some of the other buses. A d d i t i o n a l c o n s t r a i n t s i n the form of an a c c e p t a b l e flow of r e a c t i v e power i n the system or an a c c e p t a b l e 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 s may a l s o be imposed. To s a t i s f y these a d d i t i o n a l c o n s t r a i n t s repeated use of the b a s i c load-flow computer program i s necessary and d i f f e r e n t system c o n f i g u r a t i o n s may have to be s t u d i e d [ 1 ] . A l o a d - f l o w program c o n s i s t s of t h r e e p a r t s as f o l l o w s : 1) The input part which assembles a l l the i n p u t data and forms the necessary m a t r i c e s . 2) The procedure or the main p a r t of a l o a d - f l o w program which f i n d s the bus v o l t a g e s i n some i t e r a t i v e process. 3) The output p a r t which c a l c u l a t e s the i n d i v i d u a l l i n e power flows and p r i n t s out the r e s u l t s of the l o a d - f l o w s tudy. 2 In the i n p u t part, each bus of the power system i s d e f i n e d as one of the f o l l o w i n g types: a) A "(P-Q) type bus" or " l o a d bus" at which the r e a l and r e a c t i v e powers are scheduled. b) A " (P-V) type bus", "generator bus" or " v o l t a g e c o n t r o l l e d bus" a t which the r e a l power and the v o l t a g e magnitude are scheduled. c) A " s l a c k bus", "swing bus", " f l o a t i n g bus" or " r e f e r e n c e bus" at which the v o l t a g e i s scheduled both i n magnitude and phase angle. Real and r e a c t i v e powers are not s p e c i f i e d f o r these buses. These buses compensate f o r the unknown t r a n s m i s s i o n l o s s e s i n the system. Each system has a t l e a s t one s l a c k bus but some systems have more than one. In t h i s t h e s i s we w i l l assume t h a t there i s o n l y one s l a c k bus i n the system. However, a l l the c o n c l u s i o n s and r e s u l t s are v a l i d even i f t h e r e i s more than one s l a c k bus i n the system s i n c e these buses do not i n c r e a s e the number of unknowns and the number of equations used i n the computation. A d d i t i o n a l c o n s t r a i n t s , such as upper and lower bounds f o r the (P-Q) type bus v o l t a g e magnitudes and (P-V) type bus r e a c t i v e powers, may a l s o be given as p a r t of the bus i n p u t data. T r a n s m i s s i o n l i n e s are normally r e p r e s e n t e d by t'heir nominal n c i r c u i t s , as shown i n Appendix A. The procedure part of a l o a d - f l o w program i n c l u d e s some kind of i t e r a t i v e numerical process. Algorithms d i f f e r i n procedure but, i n g e n e r a l , a s e t of i n i t i a l v a l u e s f o r the bus v o l t a g e s i s estimated and c o r r e c t e d i n s u c c e s s i v e s t e p s u n t i l 3 the f i n a l s o l u t i o n i s achieved. A f a v o r a b l e i n i t i a l value i s necessary f o r a s u c c e s s f u l s o l u t i o n . Normally the f l a t - s t a r t i s used with a l l v o l t a g e magnitudes s e t to t h e i r scheduled v a l u e s o r , i f not scheduled, to the magnitude of the s l a c k bus and a l l a n g l e s s e t equal to z e r o . Sometimes the s o l u t i o n of a p r e v i o u s study may be used as the i n i t i a l v a l u e s f o r the study of a system c o n f i g u r a t i o n which i s s i m i l a r to the previous system. The i t e r a t i v e p a r t of an a l g o r i t h m uses some stop c r i t e r i o n to determine i f the s o l u t i o n i s complete. Any of the f o l l o w i n g stop c r i t e r i a or any combination of them may be used f o r t h i s purpose: 1) The g r e a t e s t v o l t a g e c o r r e c t i o n made i n an i t e r a t i o n c y c l e should be l e s s than a d e s i r e d t o l e r a n c e . 2) The magnitude of v o l t a g e s and the power flows should be w i t h i n a d e s i r e d t o l e r a n c e of t h e i r scheduled v a l u e s . 3) The sum of the r e s i d u a l s of r e a l and r e a c t i v e powers of a l l the buses, which i s c a l l e d the s l a c k bus power mismatch, should be l e s s than a d e s i r e d p r e s c r i b e d value. The t h i r d s t o p c r i t e r i o n i s used i n a l l the a l g o r i t h m s d e s c r i b e d i n t h i s t h e s i s . Although some of the e a r l i e r l oad-flow programs used mesh equa t i o n s , recent methods use nodal equations [1 ] because nodal equations are e a s i e r to form and modify and need l e s s computer st o r a g e . Laughton and Humphrey Davies [1] d i v i d e e x i s t i n g load-flow a l g o r i t h m s i n t o two c a t e g o r i e s , • the " i t e r a t i v e methods" and the " d i r e c t methods". The d i r e c t methods are those that s o l v e a set of l i n e a r e q uations a t each step of t h e i r i t e r a t i v e procedure. These methods r e q u i r e i n v e r s i o n of a matrix i n one way or another. The i t e r a t i v e methods make use of the nodal admittance matrix i n an i t e r a t i v e process and c o r r e c t the bus v o l t a g e s one by one. No matrix i n v e r s i o n i s necessary. The i t e r a t i v e methods do not have the huge computer st o r a g e requirements of the d i r e c t methods. In 1956 J.B.Ward and H.W.Hale [ 2 ] presented an a l g o r i t h m based upon the G a u s s - S e i d e l i t e r a t i v e procedure. . Because of i t s s i m p l i c i t y and a b i l i t y to converge f o r most problems t h i s a l g o r i t h m was commonly used u n t i l very r e c e n t l y . Use of d i f f e r e n t a c c e l e r a t i o n techniques with t h i s a l g o r i t h m were Implemented by o-thers [3,<t,5]. The a l g o r i t h m b e n e f i t s most from the use of l i n e a r a c c e l e r a t i o n f a c t o r s [ 3 ] and r e l a x a t i o n t echniques [ 4 ] . In 1959, a new f o r m u l a t i o n of the load-flow problem was presented by Van Ness [ 6 ] . T h i s f o r m u l a t i o n r e l a t e d s m a l l changes i n r e a l and r e a c t i v e powers to the s m a l l changes i n v o l t a g e magnitudes and phase angles by means of f i r s t o rder d e r i v a t i v e s . Van Ness t r i e d s e v e r a l i t e r a t i v e a l g o r i t h m s based upon the new f o r m u l a t i o n . These a l g o r i t h m s were e s s e n t i a l l y the Ward-Hale method used with p o l a r c o - o r d i n a t e s . L a t e r i n 1961, Van Ness and G r i f f i n [ 7 ] used the e l i m i n a t i o n method to s o l v e the s e t of l i n e a r equations which r e l a t e d the s m a l l changes i n power to the s m a l l changes i n v o l t a g e s . Nowadays t h i s method i s c a l l e d Newton's method. Although use of the e l i m i n a t i o n process reduced, to a c e r t a i n degree, the computer storage problem i n v o l v e d i n i n v e r t i n g the matrix of c o e f f i c i e n t s , c a l l e d the 5 J a c o b i a n matrix, the method was not yet capable of h a n d l i n g l a r g e power systems and, t h e r e f o r e , was not p r a c t i c a l l y u s e f u l . In 1967 Tinney and Walker [ 8 ] suggested s e v e r a l schemes f o r o r d e r i n g the nodes of a network such that when the e l i m i n a t i o n process i s used the number of non-zero terms accumulated i n the upper t r i a n g u l a r matrix w i l l tend to be minimum and Tinney and Hart [ 9 ] suggested an a l g o r i t h m t h a t used the e l i m i n a t i o n process along with sub-optimal o r d e r i n g of the buses. T h i s made Newton's method a p p l i c a b l e to l a r g e power systems and because of i t s f a s t e r and b e t t e r convergence, t h i s method soon r e p l a c e d the Ward-Hale method. Other c o n t r i b u t i o n s have been made to Newton's method to make i t more g e n e r a l and more e f f i c i e n t [10,11,12,13,14,15]. Use of the Hessian matrix i n s t e a d of the Jacobian matrix has a l s o been i n v e s t i g a t e d [16] but so f a r i t has not shown any s i g n i f i c a n t improvement over the Newton's method. In Chapter 2 of t h i s t h e s i s the Ward-Hale method and Newton's method are reviewed. In Chapter 3 a new s e t of l o a d - f l o w eguations i s d e r i v e d and s e v e r a l a l g o r i t h m s based upon t h i s f o r m u l a t i o n are d e s c r i b e d . These new a l g o r i t h m s are compared with the p r e v i o u s methods f o r a number of sample power systems and the r e s u l t s are presented i n Chapter 4. 6 2. PREVIOUS METHODS 2. 1_2he_Ward-Hale_Method This method of solving the load-flow problem was f i r s t suggested by J. B. Ward and H.W.Hale in 1956 [2], Further contributions to the o r i g i n a l paper [3,4,5] made the algorithm faster and more e f f i c i e n t . This method was widely used u n t i l very recently. The method makes use of the Gauss-Seidel i t e r a t i v e procedure which usually converges to a s a t i s f a c t o r y f i n a l solution a f t e r a cert a i n number of i t e r a t i o n s . However, the number of i t e r a t i o n s depends upon the s i z e , configuration and parameters of the power system. It also depends upon the closeness of the i n i t i a l estimate to the f i n a l s o l u t i o n . There are some cases where th i s method i s unable to converge to the solutions which are known to ex i s t . The algorithm i s simple and easy to program. I t i s also capable of handling very large power systems by using improved programming techniques and taking advantage of spa r s i t y of the admittance matrix. Xhe_Algorithm Nodal equations are used to express the r e l a t i o n s h i p between currents and voltages: 7 [I] = [Y] [VJ (2-1) where [ Y ] i s t h e n o d a l a d m i t t a n c e matrix,. T h i s m a t r i x i s formed e a s i l y i f t r a n s m i s s i o n l i n e s and t r a n s f o r m e r s a r e r e p r e s e n t e d by t h e i r II e q u i v a l e n t s . Appendix A shows how the n e q u i v a l e n t c i r c u i t s of t r a n s m i s s i o n l i n e s and t r a n s f o r m e r s a r e found and how [ Y ] i s formed. The i t e r a t i v e p rocedure s t a r t s once [Y ] i s formed. With primed symbols r e f e r r i n g t o the p r e s e n t v a l u e s a t each i t e r a t i o n s t e p and unprimed symbols t o t h e f i n a l or s c h e d u l e d v a l u e s , the p r o c e d u r e i s as f o l l o w s : S t e p 1) An i n i t i a l e s t i m a t e f o r the v o l t a g e s i s made. A f a v o r a b l e s t a r t i n g a p p r o x i m a t i o n i s n e c e s s a r y f o r a s u c c e s s f u l s o l u t i o n . I f the i n i t i a l e s t i m a t e i s top f a r from the f i n a l s o l u t i o n t h e method may d i v e r g e or the number of i t e r a t i o n s w i l l be p r o h i b i t i v e . N o r m a l l y , t h e f l a t - s t a r t i s s u f f i c i e n t . Step 2) U s i n g e q u a t i o n (2-1), the c u r r e n t e n t e r i n g any bus k can be c a l c u l a t e d : where n i s t h e number of buses. The power e n t e r i n g bus k can then be c a l c u l a t e d : O b v i o u s l y i f [ V 8 ] i s not e q u a l t o [ V ] , th e n S« w i l l not be k e q u a l to t h e s c h e d u l e d v a l u e f o r power, S . n (2-2) k St e p 3) The c o r r e c t i o n t o t h e v o l t a g e of bus k. AV k' i s 8 d e t e r m i n e d from t h e d i f f e r e n c e s between c a l c u l a t e d and s c h e d u l e d powers. I t i s assumed t h a t the v o l t a g e s o f a l l t h e o t h e r buses remain c o n s t a n t when t h i s c o r r e c t i o n i s c a l c u l a t e d . AV^ s h o u l d s a t i s f y the f o l l o w i n g c o n d i t i o n : S k - (V- + AYk) (I' + AI k) (2-4) Because V s j i s the o n l y v o l t a g e t h a t changes we can w r i t e : Al. = Y AV. k kk k (2-5) So (2-4) can be w r i t t e n as: • * s k - 4 v k i , . ; + V . „ K K * AV, ; + 4 V k v k ; ( 2 „ 6 ) Assuming t h a t AV"k i s s m a l l , the l a s t term on the r i g h t hand s i d e of (2-6) can be n e g l e c t e d : A S k " K A v k + vk \ , t < (2-7) E q u a t i n g r e a l and i m a g i n a r y p a r t s i n (2-7) we w i l l have: A P k m l m2 m3 m4 (2-8) where A Vy ~ *\ + J 3j. ar>3 ra ^ ra2 ' m 3 a n c^ m4 a r e f u n c t i ° n s °f t n e r e a l and i m a g i n a r y p a r t s of the known v a l u e s , I k, Y k k , Vk' as f o l l o w s : 9 L e t : Ykk " Gkk + J \k K - \ + J \ and v k = e k + j f k then: m l = \ + \ G k k + f i B i , k kk m 2 = \ - \ Bkk + fk Gkk ra3 - -bk * • fk Gkk " k kk m 4 = \ " fk \k " k kk For a (P-Q) type bus, (2-8) i s s u f f i c i e n t f o r c a l c u l a t i n g c o r r e c t i o n s a and a . On the other hand, at a (P-V) type k k bus, the v a l u e of Q i s not scheduled so t h a t AQ i s not k k s p e c i f i e d , but the magnitude of V^  i s to be held c o n s t a n t . T h i s c o n s t r a i n t i s formulated as: ( vk + A V (v'k + A V * = l \ l ' 2 ( 2 _ 9 ) \ A\ + vk* A\ + Ki2 = K\2~K\2 (2~10) N e g l e c t i n g the second order term |A V | 2 and using AV = a, + j 3 y i e l d s : 10 2e, thus i n the case of (P-V) type buses we have: (2-11) _ m l m2 " \ M|\| 2)_ 2e, k 2\ (2-12) Step 4) Step 3 i s repeated f o r a l l buses except the s l a c k bus. C o r r e c t i o n s are a p p l i e d to the v o l t a g e s as soon as they are c a l c u l a t e d . Step 5) V a r i o u s stop c r i t e r i a may be used a t each i t e r a t i o n c y c l e . In the o r i g i n a l Ward-Hale method, the stop c r i t e r i o n made use of s u c c e s s i v e c a l c u l a t e d v a l u e s of v o l t a g e s . I f the stop c r i t e r i o n i s s a t i s f i e d , then the process i s terminated. Otherwise, the process w i l l be repeated s t a r t i n g from step 2. Accele ra t i o n The a l g o r i t h m d e s c r i b e d above can be made more e f f i c i e n t by implementing c e r t a i n changes such as the use of a c c e l e r a t i o n f a c t o r s . The i d e a i s to m u l t i p l y the c o r r e c t i o n v o l t a g e values by f a c t o r s before adding them to the present values of v o l t a g e s . These f a c t o r s can be found e x p e r i m e n t a l l y so t h a t using them w i l l speed up the a l g o r i t h m . U s u a l l y the values that show the most s a t i s f a c t o r y r e s u l t s l i e i n the range of 1.5 to 1 . 8 . O r d i n a r i l y d i f f e r e n t f a c t o r s are a p p l i e d to the r e a l and 11 imaginary p a r t s of AV k« Another method of speeding up the a l g o r i t h m i s through the use of " r e l a x a t i o n t e c h n i q u e s " [ 4 ] . In t h i s method, the order i n which the unknowns are c a l c u l a t e d may be v a r i e d i n each i t e r a t i o n c y c l e . The r e s i d u a l s are c a l c u l a t e d and the next bus to be c o r r e c t e d i s the one having the l a r g e s t r e s i d u a l . Sometimes the c o r r e c t i o n s are based upon the v a l u e s g r e a t e r or s m a l l e r than the a c t u a l r e s i d u a l s . These are c a l l e d " o v e r - r e l a x a t i o n " and " u n d e r - r e l a x a t i o n " techniques, r e s p e c t i v e l y . Sforage^Requirements_And.Programming^Techniques Computer s t o r a g e i s needed mainly f o r the nodal admittance matrix [ Y ] . However, i f a l l the zero and non-zero elements of [ Y ] are s t o r e d , t h i s imposes a s e r i o u s r e s t r i c t i o n on the s i z e of the power system t h a t can be handled. For example, a 400-bus system cannot be s o l v e d on a computer that has 1000k bytes of memory. F o r t u n a t e l y , [ Y ] i s a very s p a r s e matrix. Taking advantage of i t s s p a r s i t y by s t o r i n g only non-zero elements, programs can be w r i t t e n to handle power systems of much l a r g e r s i z e on the same computer. Appendix B d e s c r i b e s the storage scheme used i n w r i t i n g t h i s program. T h i s i s a new scheme t h a t makes i t p o s s i b l e t o access any element of the matrix with l i t t l e scanning e f f o r t . Also i t s t o r e s the elements and the p o i n t e r s such t h a t , i n a t i m e - s h a r i n g computer system with v i r t u a l 12 memory, the number of READ/WRITE o p e r a t i o n s on a secondary s t o r a g e d e v i c e i s l e s s than that used i n c o n v e n t i o n a l schemes. 2o_ 2 Ne^tonjj;_Mej:hod Newton's method [7,9] f a l l s i n t o the categor y of s o - c a l l e d d i r e c t methods and has l a r g e l y superseded the Ward-Hale method i n l o a d - f l o w s t u d i e s of power systems. I t c o n s i s t s of the d i r e c t s o l u t i o n of a set of l i n e a r eguations i n an i t e r a t i v e process. These l i n e a r eguations r e l a t e the s m a l l changes i n r e a l and r e a c t i v e powers to the smal l changes i n v o l t a g e magnitudes and phase angles. When the changes are not s m a l l , these eguations w i l l be approximations to the a c t u a l e x p r e s s i o n s and using them i n an i t e r a t i v e p r ocess w i l l r e s u l t i n the f i n a l answers. I f changes are too l a r g e , then the eguations w i l l not be a b l e to converge to the f i n a l s o l u t i o n . The method might d i v e r g e or converge to a wrong answer. Thus, the c l o s e n e s s of i n i t i a l e s t i m a t e s to the f i n a l s o l u t i o n i s a very important f a c t o r i n determining the convergence of Newton's method. U s u a l l y the range of i n i t i a l e s t i m a t e s f o r which Newton's method converges t o the r i g h t s o l u t i o n i s s m a l l e r than t h a t f o r the Ward-Hale method. Storage i s the main problem i n t h i s method i f s p a r s i t y of the m a t r i c e s i s not taken i n t o account through sub-optimal o r d e r i n g of buses [8,9]. T h i s i s e x p l a i n e d l a t e r . Even then the s t o r a g e requirements are l a r g e r than i n the Ward-Hale method. I f sto r a g e i s not c r i t i c a l , t h i s method i s p r e f e r a b l e 13 to the Ward-Hale method because i t can solve some problems that the Ward-Hale method cannot. Also for problems that both methods can solve, Newton's method converges faster. lgj^ri thm Newton's method can be formulated using either polar or rectangular co-ordinates. Polar co-ordinates are, commonly used because i t has proven to be more advantageous [17]. Formulation of Newton's method using polar co-ordinates can be b r i e f l y explained as follows: Let ek + J f k . (2-13) and (2-14) then (2-15) Also l e t a . + j b . £ ( e . (2-16) From (2-15) the f i r s t order s e n s i t i v i t i e s can be found [ 6 ] 14 3P, 3 j V. J 9Q. k = J, 9Q, 3 6 J k j 9|V ' V ^ k j r kj (2-17) where f o r and for k=j K. . = L, . = a. f, - b . e kj kj j k i j k N = - J = a. e. + b . f, k J kj j k j k (2-18) \ k \ k N. kk - Q k - \ k l \ Qk " \ k l \ k kk k (2-19) Jkk Pk " Gkk l V k l Thus the Jacobian matrix, which relates small changes in voltages to small changes in power flows, has the following form: |AP] AQ IH N Ae A| Vl L Ivf (2-20) For any (P-Q) type bus, equation (2-20) can be written. But for 1 5 a (P-V) type bus onl y the f i r s t e quation of (2-20) can be w r i t t e n because Q i s not scheduled f o r such a bus and, t h e r e f o r e , AQ i s unknown. Also at these buses only A 8 i s unknown. T h i s makes the number of equations and unknowns e q u a l . In a system with n^ (P-Q) type and (P-V) type buses t h i s number w i l l be: m=2n1+n2 The c o r r e c t i o n s to the magnitudes and phase an g l e s of bus v o l t a g e s can then be found by s o l v i n g the s e t of l i n e a r e q u ations (2-20). Applying these c o r r e c t i o n s to the v o l t a g e s w i l l r e s u l t i n a new s e t of v o l t a g e s which are not n e c e s s a r i l y the f i n a l s o l u t i o n s i n c e the c o r r e c t i o n s may not be very s m a l l . These v o l t a g e s are used as i n i t i a l v a l u e s f o r the next i t e r a t i o n and c o r r e c t i o n s are c a l c u l a t e d i n the same way. T h i s process i s repeated as many times as necessary u n t i l the f i n a l s o l u t i o n i s achieved. §torage_Problem I f i n forming the Ja c o b i a n matrix each sub-matrix (2-21) i s c o n s i d e r e d as one element, then the J a c o b i a n matrix w i l l have the same format as the [ Y ] matrix except t h a t i t i s not n e c e s s a r i l y symmetric. Thus the Jacobian matrix i s a sparse matrix. However, i n order to s o l v e the s e t of l i n e a r equations 16 (2-20) , the i n v e r s e of the Jacobian matrix has t o be determined and the i n v e r s e i s not n e c e s s a r i l y sparse. In f a c t , i t i s u s u a l l y a f u l l matrix. For l a r g e power systems, the amount of s t o r a g e needed to s t o r e a f u l l matrix, such as the i n v e r s e of the Jacobian matrix, i s p r o h i b i t i v e . T h i s puts a s e r i o u s r e s t r i c t i o n on the a p p l i c a t i o n of t h i s method. The same problem occurs i n any of the s o - c a l l e d d i r e c t methods where a s e t of l i n e a r sparse eguations has to be s o l v e d . When the Gauss e l i m i n a t i o n method i s used, i t may be p o s s i b l e to order the eguations such t h a t the number of non-zero terms appearing i n the upper t r i a n g u l a r matrix t h a t r e s u l t s w i l l be minimal. However, no method has yet been found f o r f i n d i n g such an a b s o l u t e o p t i m a l o r d e r i n g of the eguations. Some sub-optimal o r d e r i n g schemes are d i s c u s s e d i n the next s e c t i o n [ 9 ] . any of these schemes w i l l make Newton's method p r a c t i c a l f o r l a r g e power systems. Ordering, Of ..The Equations Any one of the f o l l o w i n g schemes can be used to order the equations i n a sub-optimal manner. They are l i s t e d i n i n c r e a s i n g order of e f f i c a c y and e x e c u t i o n time: 1) Order the rows s t a r t i n g with the one having the l e a s t non-zero elements, p r i o r to the e l i m i n a t i o n p r o c e s s , and ending with the one having the most. 2) Order the rows such that a t each s t e p of the e l i m i n a t i o n process the row to be processed next i s the one having the l e a s t 17 non-zero terms. 3) Order the rows such t h a t at each s t e p of the e l i m i n a t i o n process the row to be processed next i s the one t h a t w i l l produce the l e a s t number of new non-zero terms i n the matrix a f t e r e l i m i n a t i n g t h i s row. Scheme 1 i s very simple but i t does not take i n t o account any changes t h a t occur d u r i n g the e l i m i n a t i o n p r o c e s s . Scheme 2 r e q u i r e s the s i m u l a t i o n of the e l i m i n a t i o n process and scheme 3 r e q u i r e s the s i m u l a t i o n of every f e a s i b l e a l t e r n a t i v e a t each step of the e l i m i n a t i o n process. I n t h i s t h e s i s scheme 2 i s chosen and the e l i m i n a t i o n process i s simulated i n the f o l l o w i n g way: A c o n n e c t i o n matrix . t h a t r e p r e s e n t s the topology of the system i s formed. T h i s matrix has a one i n p o s i t i o n ( i , j ) i f and only i f Y #0; otherwise i t has a zero i n t h a t p o s i t i o n . i j Since the p a t t e r n of non-zero elements i n the admittance and Jacobian matrices are the same, t h i s c o n n e c t i o n matrix can be used t o order the rows i n the Jacobian matrix. Because the value of each element i n t h i s matrix i s e i t h e r 1 or 0, only one b i t o f memory need be used to r e p r e s e n t each element. Thus, the connection matrix f o r a 1000 node system w i l l take o n l y 125k b y t e s . T h e r e f o r e , l a r g e power systems can be handled without r e q u i r i n g l a r g e computer memory. The e l i m i n a t i o n process i s s i m u l a t e d by u s i n g l o g i c a l AND, OR and EXCLUSIVE OR o p e r a t i o n s . A f t e r e l i m i n a t i n g one bus the non-zero elements of a l l the other rows are counted and the next bus to be e l i m i n a t e d i s chosen. 18 When the s i m u l a t i o n process i s f i n i s h e d and the order of buses has been s p e c i f i e d , the rows of the J a c o b i a n matrix are e l i m i n a t e d i n t h a t o r d e r . Each row i n the Jacobian matrix i s e i t h e r a s i n g l e row ( f o r (P-V) type buses) or a double row (for (P-Q) type buses). The o r d e r i n g of the equations need be determined only once, p r i o r to e l i m i n a t i o n , and i s used f o r each and every i t e r a t i o n t h a t the problem might take. Prgg r am m i n g_ t eg h n i gu es The scheme f o r s t o r i n g the [ Y] matrix i s the same as that used i n the Ward-Hale method. The data s t r u c t u r e and the l o g i c f o r forming and s t o r i n g the J a c o b i a n matrix i s e x p l a i n e d i n . Appendix C. Except f o r some changes i n the working-row scheme, i t i s e s s e n t i a l l y the same as that suggested by Tinney and Hart [ 9 ] , I t i s obvious t h a t the computer storage r e q u i r e d by t h i s method i s more than t h a t r e q u i r e d i n the Ward-Hale method because s t o r a g e i s needed f o r both the Jacobian and the [ Y ] m a t r i c e s , and the Jacobian matrix r e q u i r e s about twice as much st o r a g e as [ I ] . Consequently, f o r very l a r g e power systems, or whenever computer s t o r a g e i s very c r i t i c a l , the Ward-Hale method i s p r e f e r r e d . I f r e c t a n g u l a r c o - o r d i n a t e s are used i n s t e a d , then the J a c o b i a n matrix w i l l need even more computer storage space because two e q u a t i o n s d e s c r i b e every bus. However, the l o g i c 19 f o r s t o r i n g the J a c o b i a n matrix and f o r performing the e l i m i n a t i o n process w i l l be somewhat s i m p l e r . A l s o the need f o r u s i n g s i n e and c o s i n e f u n c t i o n s a t each i t e r a t i o n i s removed. O v e r a l l , both f o r m u l a t i o n s take about the same time to execute. Any d i f f e r e n c e i n convergence c h a r a c t e r i s t i c s i s i n f a v o r of p o l a r c o - o r d i n a t e s [ 9 ] . 20 3. PROPOSED METHOD 3 ,J_ I n t r o d u c t i o n A new approach to the s o l u t i o n of the l o a d - f l o w problem w i l l be d e s c r i b e d i n t h i s chapter. The proposed method i s based upon a d i f f e r e n t f o r m u l a t i o n of the load-flow equations. The r e s u l t a n t equations have only one n o n l i n e a r term each i n s t e a d of more n o n l i n e a r terms as i n p r e v i o u s l y reported, methods. The n o n l i n e a r term can be l i n e a r l y approximated, and the s o l u t i o n of the l i n e a r i z e d equations can be used i n an i t e r a t i v e procedure to determine the answers to the l o a d - f l o w problem. Thus the proposed method i s a d i r e c t method [1 ]. 3.2 D e r i v a t i o n ^ O f _ T h e Equations L e t the f i n a l values of v o l t a g e s , c u r r e n t s and powers be r e p r e s e n t e d , r e s p e c t i v e l y , by: [V] • • • > V } n [I] {ir v .... in> (3-1) [S] {s1, s 2, sn> and the present assumed or c a l c u l a t e d v a l u e s of these q u a n t i t i e s -be r e p r e s e n t e d by the c o r r e s p o n d i n g primed symbols: 21 [V] = (v ,^ v 2, v;> [i«] = (i«, r 2 , .... i;> ( 3 _ 2 ) [s1] {s;, sj, ..... s;> where n i s the number of buses i n the power system. The u s u a l f o r m u l a t i o n of the l o a d - f l o w equations i s d e r i v e d by working from the present assumed or c a l c u l a t e d values of (3-2) and c o r r e c t i n g them u n t i l the f i n a l v a l u e s of (3-1) are reached. The proposed method, on the other hand, i s based upon eguations that are d e r i v e d by assuming that the power system i s i n the f i n a l s t a t e d e f i n e d by the val u e s of (3-1) and f i n d i n g the changes t h a t w i l l b r i n g i t to the present s t a t e d e f i n e d by the v a l u e s of (3-2) . I n i t i a l l y assume a l l bus v o l t a g e s are h e l d c o n s t a n t at t h e i r f i n a l v a l u e s except f o r the v o l t a g e of bus i which i s changed to : V* = V. + AV. (3-3) T h i s w i l l cause a change i n the power a t bus i i t s e l f as w e l l as a l l the other buses i n t e r c o n n e c t e d to t h i s bus. The change i n the power at bus k can be c a l c u l a t e d as f o l l o w s : A S k " s i - *k k k k k - <Vk + AV k)(I k + M k ) * - V k I * (3-4) 22 or "\ - \ " k + * ( 3 . 5 ) but A V, =0 f o r k * i k hence AS = V.AI. + AV.I. + AV A l * fr, ^ 1 x i i i i i U-o) and AS k = V kAI* k ^ i ( 3 _ 7 ) Also i = I j ^ . V. j kj j (3-8) t h e r e f o r e f o r any k A I k " Y k i A V i (3-9) and s u b s t i t u t i n g i n (3-6) and (3-7), we get:. AS = V.Y*. AV* + AV.I* + AV.Y*.AV* c\ in>, i l i i i l i i i i " i U-IO; A S k = V k Y k i A V l k * 1 Equation (3-10) may a l s o be w r i t t e n as: AS = V' Y* AV* + AV.I* (3-12) 1 1 11 1 1 1 w S i m i l a r l y a l l the f i n a l (unprimed) values of the bus 23 v o l t a g e s are each changed i n t u r n to t h e i r present (primed) v a l u e s . At the end of t h i s process [ S ] w i l l be changed to [S • ]. I t i s obvious t h a t t h i s can be done i n any order. For convenience the bus v o l t a g e s w i l l be changed i n the order t h a t the buses are numbered and assume that bus 1 i s the s l a c k bus where and (3-13) For any bus i i n the system, the t o t a l change i n power may be d e f i n e d as the sum of three changes as f o l l o w s : I i} AS. -which i s the chancre of power at bus i due to changes i n the v o l t a g e s of buses 1 to ( i - 1 ) ; i i ) A?* 1 which i s the change of power a t bus i due to the change i n the v o l t a g e of bus i and; III i i i ) As.^ which i s the change of power a t bus i due to changes i n the v o l t a g e s of buses (i+1) to n. Using the above d e f i n i t i o n s and equations (3-11) and (3-12) we g e t : A S 1 = ± y 1 v Y * A V * 1 k=l 1 i k k ( 3 - 1 4 ) A_III S t t , A A 4 1 " V i k " \ . (3-16) 24 but i-1 n :* = £ **v*+ ,1 Y" v* * li y* ^ +1 y* v* S u b s t i t u t i n g f o r I i n (3-15) and a d d i n g (3-14) t h r o u g h (3-16) t o g e t h e r t o f i n d t h e t o t a l power c h a n g e we t h e r e f o r e have: AS* = AS* + A S 1 1 + A S 1 1 1 -L 1 1 X .or AS R -• -± = Z V! Y, .AV, + AV V y" -V , " 1 k = i i K l K i ^ x i k v k (3-19) Now S. = V Y Y* V* 1 1 k=l i k k (3-20) v * * S i t h e r e f o r e ) Y., V, = — k=l l k k V i (3-21) and (3-19) becomes A S i = I V i Y - i A V * + AV. - i 1 k=i i ik k i v. (3-22) N o t e t h a t o n l y t h e l a s t t erm on t h e r i g h t hand s i d e o f (3-22) i s n o n l i n e a r and can be l i n e a r i z e d by u s i n g some a p p r o x i m a t e v a l u e f o r V . F o r t h e c o m p l e t e s y s t e m , (n-1) 25 eguations, of the form of (3-22), would be obtained. Equation (3-22) was d e r i v e d by u s i n g the s e t of [ V ] as " f i n a l " v a l u e s and the s e t of [ V ] as " i n i t i a l " v a l u e s . But equation (3-22) i s s t i l l v a l i d when the [ V ] v a l u e s are " f i n a l " and the [ V ] values are " i n i t i a l " as d e f i n e d by (3-1) and (3-2), r e s p e c t i v e l y , because AS!: and A V ^ w i l l both change s i g n . Note, however, that equation (3-22) i s not d e r i v a b l e by using the l a t t e r approach and, i n f a c t , each of the equations obtained would have one l i n e a r term and (n-1) n o n l i n e a r terms. As expected, equation (3-22) does not depend upon the order i n which the bus vol t a g e s are changed t o t h e i r f i n a l v a l u e s . T h e r e f o r e , we can make use of t h i s f a c t and f o r any bus i we can assume t h a t i t i s the f i r s t bus to be changed to i t s f i n a l v a l u e with a l l the other bus v o l t a g e s s e t at t h e i r present v a l u e s . Equation (3-22) would r e s u l t . 3 i3__The_Algorithms The manipulation of equation (3-22) i n c o n j u n c t i o n with d i f f e r e n t assumptions y i e l d s v a r i o u s a l g o r i t h m s to s o l v e the loa d - f l o w problem. Some of the many p o s s i b l e a l g o r i t h m s w i l l now be d e s c r i b e d . i 26 Rew r i t i n g (3-22) as S i " S i " "VT S i " kl± V i Y l k AVk (3-23) V ! n we get vT S i " SI = J x Vl 4 AVk (3-24) I f we assume t h a t V! ~V., i i i e V! v. - 1 1 t h S n AS± J V' Y*_ AV* k=i i ik k (3-25) Equation (3-25) can be used as a l i n e a r approximation to (3-22) with the n o n l i n e a r term set to z e r o . T h i s seems reas o n a b l e s i n c e the c o e f f i c i e n t of AV. i n the n o n l i n e a r term i s i n the x order o f 1.0 P.U. and, t h e r e f o r e , much s m a l l e r than the c o e f f i c i e n t of AV± which i s v i Y i i • Equating r e a l and imaginary p a r t s of equation (3-25) w i l l r e s u l t i n 27 where A P i = l A i k \ + l \ A (3-26) A Q i = lB±k\~ K k B k (3-27) K. k , * A V! Y , , £ A.. + j B „ x ik ik J ik and A V k = ak + % For a (P-Q) type bus both equations (3-26) and (3-27) can be w r i t t e n . However, f o r a (P-V) type bus, Q i s not scheduled i and, t h e r e f o r e , (3-27) cannot be w r i t t e n , but the magnitude of voltage i s to be kept constant at such a bus. This c o n s t r a i n t can be w r i t t e n as: (V ! + A V . ) ( V ! + A V ) * = I I 2 x i T A V ° i • (3-28) or | V ' | 2 + V ' AV* + A V . v ' * + AV.AV* = U 2 -1- 1 X X 1 1 1 (3-29) V . + V 1« G o ff 2(ll{ -A___i A V * } « ^ _ | v , r (3-30) Equation (3-30) i s exact but no n l i n e a r . By using V* instead of we f i n d the l i n e a r approximation: 28 U? _ |V.|2 2-^" " ei«i + hh (3"31) where V i " e i + j f i W r i t i n g (3-26) and (3-27) f o r any (P-Q) type bus and (3-26) and (3-31) f o r any (P-V) type bus, and s o l v i n g the r e s u l t a n t s e t of e q u a t i o n s , the approximate v o l t a g e c o r r e c t i o n s can be found. I f V! i s used i n s t e a d of V. i n the n o n l i n e a r term of x 1 (3-22) then A S - i = I V* Y* AV* + AV - i i £ i xk f l Vk + A V i v' (3-32) Let A Vk = ak + % ( 3 _ 3 3 ) V i Y i k = A ± k + j B i k (3-34) and °i A V[ = a i + J b i (3-35) then equating r e a l and imaginary p a r t s of (3-32) we o b t a i n : 29 I V k + a i a i " Vi (3-36) I Aik ak + V i + V i (3-37) For a (P-Q) type bus both equations (3-36) and (3-37) can be w r i t t e n . However, f o r a (P-V) type bus Q± i s not scheduled and, t h e r e f o r e , (3-37) cannot be w r i t t e n and a. and b. cannot be i i c a l c u l a t e d . In a l g o r i t h m B, equations (3-36) and (3-37) are used i n the case of (P-Q) type buses and equations (3-26) and (3-31) i n the case of (P-V) type buses. A l g o r i t h m C T h i s a l g o r i t h m i s e s s e n t i a l l y the same as a l g o r i t h m B except t h a t f o r (P-V) type buses Q^  i s used i n s t e a d of i n (3-35) to c a l c u l a t e a. and b. . i i ~J:vT—~ = a i + j b i <3-38> Thus f o r (P-V) type buses, equations (3-36) and (3-31) apply while f o r (P-Q) type buses, equations (3-36) and (3-37) aga i n apply. Voltage a c c e l e r a t i o n f a c t o r s may be used i n t h i s a l g o r i t h m as i n the Ward-Hale method. A P i = £Aik°<k + k k 30 T h i s a l g o r i t h m i s a l s o e s s e n t i a l l y based upon a l g o r i t h m B„ By assuming that Di - | V* | i n equation (3-35) and r e a r r a n g i n g , we get: f. a i = ~ ^ h (3-39) where V! = e. + j f. ana i i J l i Equation (3-39) i s used to r e p l a c e a i n a l l the other i e q u a t i o n s . T h i s r e s u l t s i n a s e t of l i n e a r e q uations that have only one equation and one unknown f o r each (P-V) type bus. Let k^  and k^  r e p r e s e n t the s e t of a l l (P-Q) type buses and the s e t of a l l (P-V) type buses, r e s p e c t i v e l y . Then f o r any (P-Q) type bus we have: . f A P i = I <Aik\ + B i i A ) + I ( B i k - A i k i T ) p k k l k2 k + - b ± 3 i (3-40) ^1 = I <Blk"k " AilA> " I (Aik + Bik ^  > \ K l *2 ek where + b l 0 i + a iB i (3-41) and 31 = a. + 1 b. 1 J 1 and f o r any (P-V) type bus we have: AP f. ~ < bi + a i i 7 ) e i (3-42) where = a. + j b. x J x V! x thus, f o r each (P-Q) type bus we w i l l have two e q u a t i o n s , (3-40) and (3-41) and two unknowns, a and 8 . But f o r each (P-V) type bus t h e r e i s only one equation (3-42) and one unknown g . Rearranging equation (3-31) of a l g o r i t h m B, we have : and, as i n the case of a l g o r i t h m D, (3-43) can be s u b s t i t u t e d i n (3-36) and (3-37). Then f o r (P-Q) type buses we have: a (3-43) AP (3-44) AQ (3-45) 32 whereas f o r (P-V) type buses: U2 - IV!I i 1 x1 2e. 2 f. - (b, + a. — )B. i 1 e. x x (3-46) 3o_4 E£ o a i a i H I!lili2_ A . B . d.-.Storag:e As i n t h e case of Newton's method, the s o l u t i o n of a s e t o f l i n e a r s i m u l t a n e o u s e q u a t i o n s poses a computer s t o r a g e problem but t h i s can be overcome by the use of s u b - o p t i m a l o r d e r i n g o f the buses and s p a r s i t y programming. S t o r a g e r e q u i r e m e n t s f o r a l g o r i t h m s A, B and C a r e s l i g h t l y more than t h a t f o r Newton's method. For a l g o r i t h m s D and E t h e s t o r a g e r e q u i r e m e n t s a r e the same as f o r Newton's method. The s p a r s i t y of the m a t r i x of c o e f f i c i e n t s i n t h e s e a l g o r i t h m s i s t h e same as t h a t f o r the J a c o b i a n m a t r i x . So by u s i n g a scheme t o o r d e r the buses i n a s u b - o p t i m a l f a s h i o n , the amount o f s t o r a g e r e q u i r e d by a l g o r i t h m s D and E w i l l be the same as Newton's method i n p o l a r ; c o - o r d i n a t e s . For a l g o r i t h m s D and E, the s t o r a g e scheme f o r the m a t r i x of c o e f f i c i e n t s i s i d e n t i c a l t o t h a t used f o r the J a c o b i a n m a t r i x i n the p o l a r form of Newton's method, as d e s c r i b e d i n Appendix C. For a l g o r i t h m s A, B and C the l o g i c f o r s t o r i n g the ele m e n t s i s s i m p l e r but s t o r a g e r e q u i r e m e n t s a r e g r e a t e r . 33 4. NUMERICAL RESULTS U^ J _ I n t r o d u c t i o n The a l g o r i t h m s d e s c r i b e d i n Chapters 2 and 3 were programmed and compared with each other using s e v e r a l t e s t systems. The t e s t systems were of d i f f e r e n t s i z e s and c o n f i g u r a t i o n s , as shown i n T a b l e (4-1). The data given [ 2 2 ] f o r system 3 i s erroneous and incomplete: the leakage reactance of one t r a n s f o r m e r t h a t should be 0.034 P.U. i s not s p e c i f i e d and the reactance of one t r a n s m i s s i o n l i n e i s recorded as a n e g a t i v e v a l u e when i t should be p o s i t i v e . However, the system as d e s c r i b e d [ 2 2 ] with n e g l i g i b l e t r ansformer leakage (zero) and n e g a t i v e l i n e r e a c t a n c e i s used as t e s t system 3* and the c o r r e c t e d system as t e s t system 3. In a d d i t i o n to the o r i g i n a l systems mentioned i n Tab l e (4-1), new t e s t systems were c r e a t e d by changing the buses i n the o r i g i n a l system system to be a l l of e i t h e r (P-Q) type or (P-V) type. To d i f f e r e n t i a t e between t e s t systems with the same c o n f i g u r a t i o n the a d d i t i o n of the l e t t e r s a, b and c were used f o r the o r i g i n a l system, the a l l (P-Q) type system and the a l l (P-V) type system, r e s p e c t i v e l y . For example, system 1.a i s the o r i g i n a l system, system 1.b i s system 1 with a l l i t s buses of (P-Q) type and system 1.c i s the same system with a l l i t s buses of (P-V) type. The n o t a t i o n used to rep r e s e n t the d i f f e r e n t a l g o r i t h m s are summarized i n Table (4-2). Note t h a t the Ward-Hale method was 34 programmed using no a c c e l e r a t i o n technique. Algorithms A, B, C, D and E are the proposed a l g o r i t h m s d e s c r i b e d i n Chapter 3. Two s e t s of i n i t i a l v a l u e s are used f o r each case, the " f l a t - s t a r t " and the s o - c a l l e d " f i n a l - s t a r t " . For the f l a t - s t a r t a l l the v o l t a g e magnitudes are s e t to t h e i r scheduled v a l u e s or, i f not scheduled, to the magnitude of the s l a c k bus and a l l the angles are s e t equal to z e r o . The f i n a l - s t a r t i s the s e t of i n i t i a l v a l u e s c l o s e to the f i n a l answers, and may be e i t h e r the rounded f i n a l v a l u e s , the f i n a l s o l u t i o n of the system obtained with a bigger t o l e r a n c e or the f i n a l s o l u t i o n of a s i m i l a r system. Comparison of the r e s u l t s obtained by u s i n g the f l a t - s t a r t and the f i n a l - s t a r t w i l l i n d i c a t e the dependence of t h e convergence of an a l g o r i t h m upon the i n i t i a l v a l u e s . The stop c r i t e r i o n f o r a l l the a l g o r i t h m s i s the s l a c k bus power mismatch. 'The t o l e r a n c e of the power mismatch i s d i f f e r e n t f o r the v a r i o u s systems as f o l l o w s : 1) 0. 0005 p.u. f o r 2) 0. 00005 p.u. f o r 3) 0. 005 p.u. f o r *») 0. 0005 p. u. f o r 5) 0. 5 p.u. f o r systems 1.a, l . b and 1„c; systems 2. a, 2. b and 2.c; systems 3 and 3•; systems 4. a, 4. b and 4»c; system 5. j # OF # OF TOTAL# OF # OF # OF |NOTATION P-V P-Q BOS LINES TX. BR. SOURCE OF DATA I S y s t e m 1 I J S y s t e m 2 I I S y s t e m 3 I I S y s t e m 4 I I S y s t e m 5 | 1 | 3 | 5 1 6 I 0 1 6 I Ref [ 2 1 ] l 1 | 4 | 6 1 7 I 2 1 7 I Ref [ 2 ] | 2 | 1 8 | 2 1 | 2 9 | 3 1 3 2 | Ref [ 2 2 ] 1 4 | 2 8 j 3 3 | 2 3 | 11 | 3 4 | Ref [ 1 8 ] | 2 2 | 7 0 | 9 3 | 9 9 | 5 7 l 1 5 6 | Ref [ 1 9 ] i i i i A. TABLE ( 4 - D Test Systems. I | w | Ward-Hale method I H | N | Newton's method (polar form) | A | Algorithm A r +-j \ — 4 | B | Algorithm B I + I C | Algorithm C r 4-| D | Algorithm D I +• — ^ | E | Algorithm E L I i TABLE ( 4 - 2 ) A l g o r i t h m s . 36 {K2 R e s u l t s T a b l e s (4-3) and (4-4) show t h e e x e c u t i o n time and the number o f i t e r a t i o n s t a k e n by each a l g o r i t h m when the f l a t - s t a r t i s used; and T a b l e s (4-5) and (4-6) when t h e f i n a l - s t a r t i s used. S95P^F.J-§2n-Ql-£2Ry er2 eP9g_ C h a r a c t e r i s t i c s As can be seen from T a b l e s (4-4) and (4-6) , a l l the a l g o r i t h m s based upon t h e proposed method, e x c e p t a l g o r i t h m A, converge f o r a l l the c a s e s w i t h both t h e f l a t - s t a r t and t h e f i n a l - s t a r t . A l g o r i t h m A d i d not converge f o r system 5, even when t h e f i n a l - s t a r t was used. The reason f o r t h i s was found a f t e r a c a r e f u l s t u d y o f t h e d a t a o f system 5. For s e v e r a l buses o f t h i s system the net i n j e c t e d power i s about 40 P.O., and t h e branches c o n n e c t e d t o these buses have e i t h e r s m a l l S a d m i t t a n c e o r n e g a t i v e r e s i s t a n c e . Thus, t h e v a l u e of _ i i s * v i comparable t o and t h e r e f o r e (3-25) i s not a good a p p r o x i m a t i o n t o ( 3 - 2 2 ) . A l g o r i t h m B converges f o r a l l t h e c a s e s but i t s weakness r e g a r d i n g (P-V) t y p e buses i s n o t i c e a b l e . As a matter o f f a c t i f a l l the buses i n a system were (P-V) t y p e , then t h i s a l g o r i t h m would be e x a c t l y the same as a l g o r i t h m A. T h e r e f o r e , a l g o r i t h m B i s weak f o r systems w i t h too many (P-V) type buses, such a s system 5. S Y S T E M S A L G 0 R I T H M S | 1- a 1. b 1.c 2. a 2. b 2.c 3 3' 4. a 4. b 4.c 5 W |57 !59 t^9 1239 j 295 |239 |23418 | (**) j108045 |l08447 |l09064 ^8126 22 II | 38 |42 | 20 | 72' | 79 | 39 | 806 | 1434 | 1721 | (**) I 1054 | 24083 A |69 |63 |59 J191 |178 | 99 | 1041 | 741 | 27071 | 15594 i 3483 J (**) B |45 134 158 | 87 | 69 | 99 | 561 | 962 | 2536 | 2318 j 3550 J 46 3 9 31 C |33 137 J 42 | 68 | 68 | 66 | 823 | 969 | 2677 | • 2348 | 1483 | 30055 D J 32 135 | 26 | 67 | 73 | 46 | 935 11048 | 1951 1 3116 | 1262 1 25085 *" E |31 { 3 4 { 2 8 } 63 j 68 j 59 j 832 j 806 j 1783 { 2938 { 988 { 25832 I J I L . I L L I L L 1 J TABLE (4-3) Comparing the execution times taken by d i f f e r e n t algorithms when f l a t - s t a r t i s used. Times i n m i l l i s e c o n d s . (**) means no convergence S Y S T E M S |1.a 1.b 1.c 2.a 2-b 2.c 3 3' 4.a 4. b 4.c 5 r W !16 | l 7 | l 5 | 49 t 64 | 45 | >500 |{**) l >1000 | >1000 | >1000 | 1028 A r + +• + + + + + +-L | N | 3 | 3 | 2 | 4 | 4 | 3 | 3 | G J- + + + + + + + --+-0 | A | 5 | 5 | 4 | 10 | 10 | 5 | 4 | R |. x x x x x x x +-1 | E | 3 | 2 | 4 | 4 | 3 | 5 | 2 | T (- + + + + + + + 4-H | C | 2 | 2 | 2 | 3 | 3 | 3 | 3 | M L x f x + x x x x . S | D | 2 | 2 | 2 | 3 | 3 | 3 | 3 | I E I 2 | 2 | 2 i 3. | 3 | 3 | 3 | I J J L L I I L L . j 3 | <**) j 57 | 38 | 5 I 5 | 4 I 5 | " 5 | 4 | 3 | 5 1 4 | 3 | 5 | 3 I 3 8 1 (**) 8 I + ~ 3 I + -3 I +-2 j 49 3 3 -j TABLE (4-4) Comparing the number of i t e r a t i o n s taken by d i f f e r e n t a l g o r i t h m s when f l a t - s t a r t i s used. (**) means no convergence CD S Y S T E M S 1 W A h — L 1 N G (-— 0 1 A R j I 1 B T J - — H 1 c M j - — S 1 D 1 E a 1. b 1.c 2.a 2.b 2.c 3 3' 4.a 4.b 4.c 5 1*36 |39 |30 j 82 |l28 |~62 M1445 " [ ( * * ) I 4551 1 | 32712 |62998 M 9 6 8 1 5 1 |17 |18 |14 | 26 | 28 | 18 | 346 |(**) | 1260 I 2309 | 775 | 17386  J 47 |42 120 | 82 | 76 | 29 | 556 | 720 | 18605 | 12884 | 1490 | (**) {20 { 2 1 {20 } 29 j 30 | 29 | 338 } 533 | 711 | 632 | 1488 | 16120 |23 |24 |23 I 31 | 34 | 35 | 340 | 308 | 705 | • 661 1 1042 I 12661 I 2 0 {22 |l7 | 30 | 31 { 24 j 405 | 375 | 874 | 911 | 930 | 11265 |19 |20 |17 | 29 j 29 | 24 | 365 | 347 | 798 I 828 J 611 J 11705 J I J I I J I I I I L 1 TABLE (4-5) Comparing the execution times taken by d i f f e r e n t algorithms when f i n a l - s t a r t i s used. Times i n milliseconds. (**) means no convergence S Y S T E M S f l . a 1.b 1.c 2 . a 2.b 2.c 3 3* 4 .a 4.b 4.c 5 ] )~W | 8 | 1 1 | 7 | 15 | 27 | 11 | 250 I (*'*) | 415 | 298 | 578 | 250 i L |' N | 1 | 1 | 1 | 1 | 1 | 1 | 1 | (**) | 2 | 4 | 2 | 2 | 0 | A | 3 | 3 | 1 | 4 | 4 | 1 | 2 | 3 | 40 | 31 J 3 ] (**) | 1 I B | 1 | 1 | 1 J • 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 J 1 | H | C | 1 | 1 ] 1 | 1 | 1 | 1 | 1 | 1 | 1 I 1 I 2 1 1 | S | D | 1 | 1 | 1 | 1 | 1 J 1 | 1 I . 1 I 1 1 1 1 2 | 1 | i E | 1 I 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | TABLE (4-6) Comparing the number of i t e r a t i o n s taken by d i f f e r e n t algorithms when f i n a l - s t a r t i s used. (**) means no convergence o 41 Algorithms C, D and E have almost i d e n t i c a l convergence c h a r a c t e r i s t i c s . They converged f o r a l l the t e s t systems i n a few i t e r a t i o n s . However, f o r system 4.c a l g o r i t h m E converged f a s t e r than the other two. This s h o u l d be t r u e f o r any system with too many (P-V) type buses s i n c e the exact equation (3-43) i s used i n s t e a d of the approximate e q u a t i o n (3-39) i n a l g o r i t h m E. The Ward-Hale method converged f o r a l l the t e s t systems, except system 3', I t can be seen t h a t i t takes a l a r g e number of i t e r a t i o n s to converge for- l a r g e systems. Newton's method f a i l e d t o converge f o r system 4.b when the f l a t - s t a r t was used but converged when the f i n a l - s t a r t was used. T h i s method a l s o d i d not converge f o r system 3' even when the f i n a l - s t a r t was used, the f i n a l - s t a r t being the s o l u t i o n of system 3. These two cases i n d i c a t e that Newton's method i s more dependent upon the i n i t i a l v a l u e s than any of the a l g o r i t h m s based upon the proposed method. Dependency of ... Newton's method to a f a v o r a b l e i n i t i a l v a l u e has a l s o been d i s c u s s e d by o t h e r s [ 2 0 ] . 42 Comparison Of Computing Speed The e x e c u t i o n times mentioned i n T a b l e s (4-3) and (4-5) are not very a c c u r a t e , s i n c e a t i m e - s h a r i n g computer system was used to compare the a l g o r i t h m s . Some cases were run more than once and t h e average e x e c u t i o n time i s recorded i n T a b l e s (4-3) and (4-5). Experience has shown t h a t the f i g u r e s g i v e n i n these t a b l e s have a d e v i a t i o n o f , at most, 6%. A l g o r i t h m A i s the slowest among the a l g o r i t h m s of the proposed method. In the case of system 3', however, we can see that i t converges f a s t e r than any of the other a l g o r i t h m s . T h i s i s o n l y f o r t u i t o u s because of the t o l e r a n c e used. Al g o r i t h m B i s b e t t e r than a l g o r i t h m A. I t converges f o r a l l the t e s t systems and i s f a s t e r than a l g o r i t h m A. For system 3, t h i s a l g o r i t h m converged f a s t e r than any of the o t h e r a l g o r i t h m s , but t h i s again i s f o r t u i t o u s because of the t o l e r a n c e used. Algorithm C i s f a s t e r than a l g o r i t h m B but i t i s s t i l l weak as f a r as (P-V) type buses are concerned, the reason being t h a t i t has two eguations and two unknowns f o r a (P-V) type bus while the p o l a r form of the Newton's method has only one equation and one unknown f o r such a bus. T h e r e f o r e , a l g o r i t h m C s o l v e s a system of l i n e a r equations with a rank g r e a t e r than the rank of equations i n Newton's method and t h i s r e s u l t s i n a g r e a t e r e x e c u t i o n time per i t e r a t i o n . Algorithms D and E combine the two e q u a t i o n s of a (P-V) type bus i n t o one equation and, t h e r e f o r e , use a system of 43 l i n e a r equations with the same number of e q u a t i o n s and unknowns as i n the p o l a r form of Newton's method. T a b l e s (4-3) and (4-5) show t h a t these two a l g o r i t h m s are f a s t e r than Newton's method. Algorithm E uses the exact form of (3-43) and i s b e t t e r when a l l the buses of a l a r g e system are of (P-V) type. T h i s kind of system i s unusual and, t h e r e f o r e , we can assume that a l g o r i t h m D can always be c o n s i d e r e d as the best a l g o r i t h m from the p o i n t of view o f speed. Algorithms D and E r e q u i r e the same storage as the p o l a r form of Newton's method because the matrix of c o e f f i c i e n t s has e x a c t l y the same form as the Jacobian matrix. The data s t r u c t u r e and the l o g i c f o r s t o r i n g the matrix of c o e f f i c i e n t s i s i d e n t i c a l t o t h a t d e s c r i b e d i n Appendix C f o r s t o r i n g the Jac o b i a n matrix. The storage requirements of a l g o r i t h m s A, B and C i s s l i g h t l y h i g h e r , but these a l g o r i t h m s need l e s s s t o r a g e than the Newton's method i n r e c t a n g u l a r form. The storage requirements f o r the Ward-Hale method i s about h a l f of the st o r a g e needed f o r any of the other a l g o r i t h m s . T h i s i s the main advantage of the method and i s used when storage i s a problem. 44 4.3 Summary When the Ward-Hale method i s used with a c c e l e r a t i o n f a c t o r s i t i s more e f f i c i e n t . For example, i n the case of system 1.a u s i n g the a c c e l e r a t i o n f a c t o r of 1.2 w i l l reduce the number of i t e r a t i o n s by h a l f . The problem i s t h a t the b e s t a c c e l e r a t i o n f a c t o r s must be determined f o r each system i n d i v i d u a l l y and t h i s r e q u i r e s s e v e r a l runs of the program i n order to f i n d the best f a c t o r s . Use of a c c e l e r a t i o n f a c t o r s was a l s o t r i e d with a l g o r i t h m C but showed no advantage. A l l the r e s u l t s p r e v i o u s l y d i s c u s s e d i n t h i s chapter use no a c c e l e r a t i o n f a c t o r s . O v e r a l l , we can see t h a t a l g o r i t h m s D and E are f a s t e r and have b e t t e r convergence c h a r a c t e r i s t i c s than e i t h e r Newton's method or the Ward-Hale method. Also t h e i r s t o r a g e requirements i s about the same as the p o l a r form of Newton's method and about twice as much as the Ward-Hale method. I f s t o r a g e i s not a problem, these a l g o r i t h m s are p r e f e r a b l e to both the Ward-Hale method and Newton's method. 45 5. C O N C L U S I O N S A new f o r m u l a t i o n of the l o a d - f l o w problem has been presented i n t h i s t h e s i s . Some of the many p o s s i b l e a l g o r i t h m s t h a t can be d e r i v e d based upon t h i s new f o r m u l a t i o n were programmed and compared with the Ward-Hale method and Newton's method. S e v e r a l t e s t systems of d i f f e r e n t s i z e s and c o n f i g u r a t i o n s were used to compare the a l g o r i t h m s . I t was shown t h a t : 1) Algorithms based upon the new f o r m u l a t i o n converge over a wider range of i n i t i a l v a l u e s and t h e r e f o r e have b e t t e r convergence c h a r a c t e r i s t i c s than Newton's method and the Ward-Hale method. 2) Algorithms D and E of those based upon the new f o r m u l a t i o n are f a s t e r than the Uewton's method and the Ward-Hale method. 3) The s t o r a g e requirements f o r any of the proposed a l g o r i t h m s i s about the same as the p o l a r form of Newtcn's method and, t h e r e f o r e , by use of sub-optimal o r d e r i n g of the egua t i o n s and improved programming techniques s t o r a g e w i l l not be a major problem. 4) Use of a c c e l e r a t i o n f a c t o r s w i l l not improve the performance of the a l g o r i t h m s based upon the new f o r m u l a t i o n . 46 REFERENCES 1. M.A.Laughton and M.w.Humphrey Davies: "Numerical Techniques Of Power-System Load-Flow Problems", Proc. IEE, V c l . I l l , No. 9 pp. 1575-1588, September 1964. 2. J.B.ward and B.W.Hale: " D i g i t a l Computer S o l u t i o n Of Power-Flow Problems", AIEE Trans, on PAS V o l . 75, pp. 398-404, June 1957. 3. R.J.Brown and W.F.Tinney: " D i g i t a l S o l u t i o n s For Large Power Networks", AIFE Trans, on PAS V o l . 76, pp. 347, June 1957. 4. R.H.Jordan: "Rapid Converging D i g i t a l Lead-Flow", AIEE Trans, on PAS V c l . 76, pp. 1433-1438, February 1958. 5. A.F.Glimm and G.W.Stagg: "Automatic C a l c u l a t i o n Of Lead Flows", AIEE Trans, on PAS V o l . 76, pp. 817-825, October 1957. 6. J.E.Van Ness: " I t e r a t i o n Methods For D i g i t a l Load-Flow S t u d i e s " AIEE Trans, on PAS V o l . 78A. Pt. I l l , pp. 583-588, August 1959. 7. J.E.Van Ness and J . H . G r i f f i n : " E l i m i n a t i o n Methods For Load-Flow S t u d i e s " , AIEE Trans, on PAS V o l . 80, pp. 299-304, June 1961. 8. W.F.Tinney and J„W.Walker: " D i r e c t S o l u t i o n s Of Network Equations By Op t i m a l l y Ordered T r i a n g u l a r F a c t o r i z a t i o n " , Proc. Of the IEEE, V o l . 55, pp. 1801-1809, November 1967. 9. W.F.Tinney and C.E.Hart: "Power Flow S o l u t i o n Ey Newtcn's Method", IEEE Trans, on PAS V o l . 86, pp. 1449-1460, November 1967. 10. H.W.Dommel, W.F.Tinney and W.L.Powell: " F u r t h e r Developments In Newton's Method For Power System A p p l i c a t i o n " , Conference Paper. No. 70 CP 161, f o r the IEEE Winter Power Meeting, 1970. 11. J . P . B r i t t o n : "Improved Load-Flow Performance Through A More General Equation Form", IEEE Trans, on PAS V o l . 90, pp. 109-116, June 1971. 12. A.M.Sasson, C.Trevino and F.Aboytes: "Improved Newtcn's Load-Flow Through A M i n i m i z a t i o n Technique", IEEE Trans. On PAS, V o l . 90, pp. 1974-1981 September 1971. 47 13. N.Nabona and L . L . F r e r i s : "New Programming Approach To The Newton-Raphson load-Flow", Conference Paper, IEEE PES Winter Meeting, Jan 28-Feb 2, 1973. 14. R . B i l l i n t o n , K . E . B o l l i n g e r and S.B.Dhar, "A New M o d i f i e d Newton's Method For Load-Flow A n a l y s i s " , Conference Paper No. C72 004-5 f o r the IEEE Winter Power Meeting, 1972. 15. F.Gomez and A.Semlyen: "The Newton-Raphson Load-Flow With Complex V a r i a b l e s " , Conference Paper, IEEE PES, Suirmer Meeting, J u l y 1972. 16. A.M.Sasson, C.Trevino and F.Aboytes: "Optimal Load-Flow S o l u t i o n Using The Hessian M a t r i x " , Proc. Of the 1971 PICA, pp. 203-209 May 1971. 17. H.W.Domrael D i s c u s s i o n on Ref 9. 18. T.Nordstrom: " D i g i t a l Load-Flow", Report f o r EE 553 Course, EL EC. ENG. Dept. U.B.C. , 1967. 19. T.Nordstrom: "B.C. Hydro Sample Data" 20. W.F.Tinney and H.W.Doinmel D i s c u s s i o n on: Y.P.Dusonchet, S.N.Talukdar, H.E.Sinnot and A,H.El-Abiad: "Load-Flows Using A . Combination Cf P o i n t J a c o b i And Newton's Method", IEEE Trans, on PAS V o l . 90, pp. 941-949* May/June 1971. 21. W.D.Stevenson, J r : ^Po wer^Sy ste m_ .Analog is'J L (Book), 2nd E d i t i o n , McGraw-Hill, Tokyo, 1962, pp. 219. 22. J.E.Bonaparte and W.W.Maslin: " S i m p l i f i e d Lead Flow", Proc. of 1967 PICA, pp. 385-394, May 15-17,1967. 48 Appendix Assume t h a t a transformer admittance a i s connected between F i g , A. 1. A. with t u r n s - r a t i o n and leakage buses i and j , as shown i n n:1 i 1 < F i g . A. 1 T h i s can be represented by the e q u i v a l e n t ir c i r c u i t shown i n F i g . A.2 [2 ] -i h I t — — - I 1 I I I I Y ={1-n)a| I | |Y =n(n-1) a I I I I I I I I I I J 1 F i g . A . 2 For t r a n s m i s s i o n l i n e s the exact • TT e q u i v a l e n t can be found [ 2 1 ] , but t h i s i s not necessary unless the l i n e s are very long. Normally the approximate ir e q u i v a l e n t , c a l l e d the nominal TT e q u i v a l e n t , i s used and i s s u f f i c i e n t l y good. ( F i g . A.3) 49 The nodal admittance matrix i s formed using the e q u i v a l e n t i ^ = 1 / 2 ! 1 , 1 T-r - h I I Y = y/2 i I I |Y = y/2 I I 1 I 3 F i g . A. 3 c i r c u i t s of t r a n s m i s s i o n l i n e s and t r a n s f o r m e r s as f o l l o w s Y = Y . . = - Y , i j 1 Y. . =Y. . + Y „ i i i i 2 Y. . =Y. . + Y 0 33 33 3 50 Appendix B. The s t o r a g e scheme used to s t o r e the. nodal admittance matrix i s d e s c r i b e d here. Only the non-zero elements, along with the a p p r o p r i a t e p o i n t e r s are s t o r e d as shown i n F i g . B. 1 \ T TcTeT T~TTc[eT TTcleiT | a | b | - + H |a|bf~+-ia|bf~+--i I I |d|f| | | |d|f| | |d|f| Element 1 Element n | < Diagonals > | < O f f - D i a g o n a l s a= r e a l part of admittance c= column i n d i c a t o r e= row i n d i c a t o r F i g . B.1 Diagonal elements occupy the f i r s t n l o c a t i o n s of the v e c t o r . The p o i n t e r s are used to c h a i n together the elements t h a t are i n the same row or i n the same column. The l a s t element i n a row, or i n a column, p o i n t s to the f i r s t element (the d i a g o n a l element) a g a i n . I n s e r t i n g new elements i s done by breaking the c h a i n a f t e r the d i a g o n a l element and making a new chain v i a the new i n s e r t e d element. A garbage p o i n t e r i s used to i n d i c a t e the f i r s t empty l o c a t i o n i n the v e c t o r . The empty l o c a t i o n s are chained together and the garbage p o i n t e r i s modified a f t e r any i n s e r t i o n or d e l e t i o n . Note that the new empty l o c a t i o n s are always added to the top of the garbage c h a i n and are used f i r s t . b= imaginary p a r t of admittance d= next-in-row p o i n t e r f= next-in-column p o i n t e r 51 The above scheme has the f o l l o w i n g advantages: 1) Because of the l o c a t i o n of the d i a g o n a l elements, no scanning i s necessary to f i n d these elements. S i n c e the d i a g o n a l elements are used more o f t e n i n any a l g o r i t h m , t h i s reduces the ex e c u t i o n time. 2) To f i n d an o f f - d i a g o n a l element, the scanning e f f o r t can be minimized through scanning e i t h e r i n the a p p r o p r i a t e row or column whichever has l e s s elements. 3) Since the p o i n t e r s are s t o r e d c l o s e t o the va l u e of the elements, i n a t i m e - s h a r i n g computer system, the number of READ/WRITE o p e r a t i o n s on a secondary s t o r a g e d e v i c e w i l l be l e s s than t h a t of c o n v e n t i o n a l storage schemes. 4} The scheme can be e a s i l y programmed i n FORTRAN through the use of the EQUIVALENCE statement. 52 Appendix C. The data s t r u c t u r e and the l o g i c f o r forming and s t o r i n g the t r i a n g u l a r i z e d J a c o b i a n matrix i s d e s c r i b e d here. Except f o r some changes i n the working-row t h i s i s the scheme suggested by Tinney and Hart [9 ]. Each row of the Jacobian matrix i s formed i n the working-row and i s l a t e r s t o r e d i n the t a b l e of the J a c o b i a n elements when the e l i m i n a t i o n process i s performed upon t h a t . The t a b l e of the Jacobian elements i s a v e c t o r t h a t c o n t a i n s a l l the elements of the t r i a n g u l a r i z e d J a c o b i a n matrix and t h e i r p o i n t e r s . The p o i n t e r s are i n t e g e r numbers which are p o s i t i v e f o r (P-Q) type buses and ne g a t i v e f o r (P-V) type buses. Another v e c t o r i s used t o s t o r e the i n t e g e r s that p o i n t to the s t a r t i n g l o c a t i o n of each row. These p o i n t e r s a l s o f o l l o w the same r u l e f o r (P-Q) type and (P-V) type buses. The r e s i d u a l s and the d i a g o n a l elements are s t o r e d a t the f i r s t l o c a t i o n s where a row s t a r t s . For example, i f the s t a r t i n g p o i n t e r of row i i s a n e g a t i v e i n t e g e r , then i i s a s i n g l e row (bus i i s (P-V) t y p e ) . In t h i s case the f i r s t e n t r y i n the t a b l e of the Jacobian elements i s AP., f o l l o w e d by an i n t e g e r number j . I f j i s n e g a t i v e i t r e p r e s e n t s a (P-V) type bus and i t i s f o l l o w e d by H^ and another i n t e g e r number r e p r e s e n t i n g another bus, while i f j i s p o s i t i v e i t r e p r e s e n t s a (P-Q) type bus and i s f o l l o w e d by H , N and another i n t e g e r number. T h i s n o t a t i o n i j i j i s used f o r a l l the p o i n t e r s i n a s i n g l e row. However, i f row i i s a double row (bus i i s (P-Q) type) then the f i r s t t h r e e l o c a t i o n s of t h i s row w i l l be AP , AQ and N followed by the 1 i i i 53 f i r s t i n t e g e r p o i n t e r i n t h a t row. In the case of double rows each negative p o i n t e r j i n the row i s f o l l o w e d by H , J and another p o i n t e r while each p o s i t i v e p o i n t e r i s f o l l o w e d by It.. , J , N . . , L. . and another p o i n t e r . The end of each row i s i j i J i J i m p l i e d by the s t a r t of the next row. Each row i s formed and t r i a n g u l a r i z e d i n the working-row, p r i o r to being s t o r e d i n the t a b l e of the J a c o b i a n elements. Each element of the working-row c o n s i s t s of f o u r l o c a t i o n s , i n which H, J , N and L are s t o r e d . I t a l s o has a column p o i n t e r . I t s l e n g t h i s s u f f i c i e n t to s t o r e a f u l l J a c o b i a n row. Once the row i s formed, a l i n e a r combination of the p r e v i o u s l y s t o r e d rows i s added to i t such that i t w i l l not have any elements i n the column range t h a t has been processed b e f o r e . During t h i s process, new elements may be c r e a t e d and some of the elements may be d e l e t e d and modified. New elements are added to the end of the working-row and d e l e t e d elements w i l l have a zero p o i n t e r . At the end of the process only the elements with non-zero column p o i n t e r s w i l l be s t o r e d i n the compact t a b l e of the J a c o b i a n elements. 

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