DEDICATED TO
M Y P A R E N TS & TE A C H E R S
DIGITAL SOLUTION OF LOAD-FLOW
OPTIMAL LOAD-FLOW AND. STATIC STATE ESTIMATION PHOBLEMS IN ILL-CONDITIONED POWEI! SYSTEMS
BY
G.S.S.S.K. DURGAPRASAD
THESIS SU B M ITTE D IN PARTIAL FULFILMENT OF THE RBQUIREMENTS FOR TH E AW ARD OF TH E DEGREE OF
DOCTOR OF PHILOSOPHY IN
ELECTRICAL ENGINEERING
DEPARTMENT OF ELECTRICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY, DELHI
NOVEMBER 1980
CERTIFICATE
Th is i s to c e r t i f y that the th e s is e n t i t l e d , ’’D i g i t a l Solu
t i o n of Load-flow Optimal L o a d -flo u and S t a t i c S ta te Estim ation Problems in I l l - c o n d i t i o n e d Power Systems", which is-'beihg, sub
m itte d by Mr. G . S . S . S . K . Durga Prasad for the award o f the.,Degree o f Doctor o f Philosophy o f the In dian I n s t i t u t e of Technology, New D e l h i , i s a bonafide re co rd o f research work c a r r i e d out by him over the l a s t three years under my guidance and s u p e r v is io n .
The candidate has f u l f i l l e d the requirements o f a l l th8 r e g u la tio n s r e l a t i n g to the degree. The r e s u lt s obtained in the th e s is have not been submitted to any o th e r U n i v e r s i t y or I n s t i t u t e f o r the award o f a degree o r diplom a.
D r . Sa ra t C. T r i p a t n y y Professor
Department o f E l e c t r i c a l Engineering Indian I n s t i t u t e o f Technology
New D e l h i 110 016, INDIA
ACKNOWLEDGEMENTS
I am indebted to my s u p e r v i s o r , P rofessor S . C . T r i p a t h y , f o r suggesting t h i s problem and f o r h is v a lu a b le guidance . His keen i n t e r e s t in t h i s f i e l d has been a g re a t source o f encouragement.
I owe a debt o f g r a t it u d e f o r h is system atic o rg a n is a tio n o f the e n t i r e work and f o r his c r i t i c a l comments and u s e f u l suggestions i n the p re p a ra tio n of t h i s t h e s i s .
I am also indebted to P rofessor C . S . In d u lk a r f o r his v a lu a b le suggestions and guidance du rin g the course of t h i s t h e s i s .
I express my g ra te fu ln e s s to P r o f . l/.S. Rajamani, Head, Depart
ment of E l e c t r i c a l Engineering f o r p r o v id in g the f a c i l i t i e s * I am a ls o g r a t e f u l to P r o f . P . S . Satsangi and S . S . Lamba f o r t h e i r v a lu a b le d iscu ssio ns and suggestions*
I would l i k e to thank D r . S . Iwamoto, Department of E l e c t r i c a l E n g in e e rin g , Tokai U n i v e r s i t y ; Japan, f o r p r o v id in g me the 11 and 43 bus i l l - c o n d i t i o n e d t e s t systems da ta.
My thanks are due to P r o f . P .G » Reddy, Operations i n —charge, Computer C e n tre , and also to the o perating s t a f f o f the Centre f o r the help and f a c i l i t i e s provid ed to me.
I thank many of my fr ie n d s f o r t h e i r coop eration and encoura
gement rendered at v a rio u s stages of my research work.
F i n a l l y I express my s in c e re thanks to M r. P*M. Padmanabhan Nambiar f o r the neat ty p in g and Mr. R. Kapoor fo r n e a tly drawing the. diagrams.
G . S . S . S . K . DURGAPRASAD
ABSTRACT
Th is th e s is presents a p p l ic a t i o n o f mathematician K .Ft .Brown's method to the s o l u t i o n o f l o a d -f l o w , optim al load- f l o w and s t a t i c s t a t e estim ation problems. The method is p a r t i c u l a r l y e f f e c t i v e
f o r s o lu t io n of i l l - c o n d i t i o n e d systems o f n o n lin e a r a lg e b ra ic e q u a tio n s . I t i s a v a r i a t i o n o f Newton's method in c o r p o ra t in g Gaussian e lim in a t io n in such a way t h a t the most recent informa
t i o n i s always used at each step of the a lg o rith m , s i m i l a r to u/hat i s done in the G au ss-Se id el process. The i t e r a t i o n converges l o c a l l y and the convergence i s q u a d ra tic in n a tu re . A general d is c u s s io n o f i l l - c o n d i t i o n i n g o f a system o f a lg e b ra ic equations i s g iv e n . I t i s also shown by f i x e d - p o i n t form ulation th a t the proposed method f a l l s in the general category o f succe ssive approxi
mation methods. D i g i t a l computer solutions by the proposed Brown's method are given fo r s e v e ra l i l l - c o n d i t i o n e d power systems for which the standard l o a d -f lo w methods f a i l e d to converge. Even i f much progress has been made in l o a d - f l o w a n a l y s i s , there are s it u a t i o n s which cause d i f f i c u l t i e s in o b ta in in g s o lu tio n s w ith some o f the standard methods. Some o f the causes of i n s t a b i l i t y and d i v e r gence in lo a d -f lo w s o l u t i o n methods are
( a ) the p o s i t io n o f the re fe r e n o e -s la c k bus, ( b ) the choice of a c ce le ra tio n f a c t o r s ,
( c ) the existence o f ne gative l i n e reactance,
i
ii
( d ) c e r t a in types of r a d i a l systems, and
( e ) a la r g e r a t i o of l o n g - t o - s h o r t l i n e reactance fo r l i n e s te rm in a tin g i n the same bus.
Networks having the above features are known as i l l - c o n d i tio n e d systems and are described by a system of n o n lin e a r a lg e b r a i c e q u atio n s. The s o o lu tio n s of these equations are ve ry s e n s it i v e to sm all changes in s p e c i f ie d parameters such as bus data* In o th e r words, sm all changes in parameter produce a la rg e change in the s o l u t i o n .
A new a lg o rith m i s presented in t h i s th e s is f o r the l o a d - flow s o l u t i o n of iJ , l - c o n d it io n e d power systems using B ro u »fi's method and the nodal admittance m a t r ix . In Brown's method, each equation i s expanded in approximate T a y l o r ' s s e r i e s j houower, the most recent in fo rm a tio n a v a il a b l e i s immediately used in the con
s t r u c t i o n o f the next fu n c tio n argument, s i m i l a r t o the procedure i n the Gauss—S e id e l process f o r the s o l u t i o n of sets of n o n lin e a r e q u a tio n s . Th is c o n tra s ts s h a r p ly w ith Newton's method i n which a l l equations are tr e a te d s im u lta n e o u s ly . I t has been proved th a t Brown's method i s not e q u iv a le n t to Newton's method. D i g i t a l computer r e s u l t s o f the proposed l o a d -f l o w are presented
iii
f o r s e v e ra l i l , ^ c o n d i t i o n e d power systems, fo r which the standard l o a d -f l o w methods, namely, G a u s s -S e id e l, Newton's, F le tc h e r-P o w e ll and Fast Decoupled algorithm s f a i l e d to converge w ith usual f l a t s t a r t . Although the B ram eller and Denmead method works for i l l - c o n d itio n e d problems, i t shows poor convergence. A comparison o f t h i s method with the standard l o a d -f l o w methods i s also presented fo r the w e l l -c o n d it io n e d AEP 14, 30 and 57 bus systems.
The optim al r e a l power flow has been solved with approximate lo s s formulas by s e v e ra l a u th o rs. Approximate methods also e x is t f o r the optim al r e a c t iv e power flo w . R ecently, attempts have been made to so lve the optim al r e a l and r e a c t iv e power flow e x a c t l y . The general problem o f optimal l o a d -f l o w s u b je c t t o e q u a lity and i n e q u a l i t y c o n s tr a in ts were formulated by Dommel and Tinney using , the Lagrangian m u l t i p l i e r s method and Newton-Raphson l o a d -f l o w .
L a t e r , Sasson applie d the F le tc h e r-P o w e ll a lg o rith m to the s o lu t i o n o f lo a d -f lo w and optimal l o a d -f l o w problems w ith Z a n g w il l 's f o r m u la t io n . However, these well-known methods f o r optimal lo a d - flow do not work fo r the i l l - c o n d i t i o n e d power systems. I t i s shown in t h i s th e s is th a t the a p p lic a t io n o f Brown's a lg o rith m to the s e t o f necessary c o n d itio n s fo r a minimum o f the s c a l a r o b je c tiv e
fu n c tio n s u b je c t to the e q u a lit y and i n e q u a l i t y c o n s tra in ts r e s u lt s i n fa s te r convergence to the s o l u t i o n o f the i l l - c o n d i t i o n e d power systems comparod to the other methods. D i g i t a l computer re s u lt s by the proposed Brown's optim al l o a d -f l o w a lg o rithm are presented f o r 13-bus i l l - c o n d i t i o n e d , 5 - and 2 5 - bus w e ll-c o n d it io n e d t e s t
iv
systems. Also the th e s is presents a comparative assessment o f con
vergence c h a r a c t e r i s t i c s of a l l the th re e methods (Dommel and T i n n e y , Sasson, and Brown) fo r optim al lo a d -f lo w s o l u t i o n o f w e l l - co n d itio n e d powor systems. For the a p p l ic a t i o n o f Brown’ s method, th e optimal l o a d -f l o w problem form ulation i s such th a t no move
from one fe a s ib le s o l u t i o n to another i s to be made fo r achie
v i n g the optim al s o l u t i o n u n lik n in Qommel and T i n n a y 's g ra d ie n t te ch n iq u e .
The s t a t i c s ta te e s tim atio n has also been solved by the pro
posed Brown’s method. I t i s shown th a t the method converges fa s te r than the Gauss-Newton method and F le tc h e r-P o w e ll method fo r i l l - c o n d itio n e d systems. However, a l l these three methods are based on weighted le a s t square fo rm u la tio n , th a t i s , m in im iza tio n o f the sum o f squared r e s id u a ls (mismatches) o f bus i n j e c t i o n s and the l i n e flo w s .
CONTENTS
Page
A bs tra ct i
Chapter l i In tr o d u c t io n 1
1.1 Load-Flow
1
1.2 Optimal Load-Flow 3
1.3 S t a t i c S tate Estim ation 4
1.4 O u t lin e of the Thesis 4
Chapter I I ; Review of Load-Flow, Optimal Load-Flow 8
and S t a t i c S ta te Estim a tion Methods
2.1 Review o f Load-Flow Methods 8
2. 1. 1 Basic Load-Flow Equ a tio n s, Bus Types, 9 Mismatches and S o lu t io n Accuracy C r i t e r i a
2 . 1 . 2 Y -M a t r ix I t e r a t i v e Methods 1:3
2 . 1 . 3 Z -M a t r ix Methods 17
2 . 1 . 4 Newton Methods 21
2 . 1 . 5 Decoupled Methods 24
2 . 1 . 6 M inim iza tio n Methods 26
2 . 1 . 7 Load-Flaw S o lu t io n fo r I l l - c o n d i t i o n e d 28 Systems
2.2 Review o f Optimal Load Flow Methods 29
2. 2. 1 Optimal Load-Flow S o lu t io n fo r 33
I l l - c o n d i t i o n e d Systems
2.3 Review of S t a t i c S ta te Estim a tion 33 Methods
Page 2. 3. 1 Weighted Least Square S t a t i c S ta te 36
Estim ation
2 . 3 . 2 S ta te E stim a tio n Using F le tc h e r-P o w e ll 38 A lg o rithm
2 . 3 . 3 S t a t i c S ta te E stim a tio n fo r 40 I l l - c o n d i t i o n e d Systems
Chapter I l l s I l l - c o n d i t i o n e d Systems, Eigenvalue 42 A n a ly s is and K.M. Brawn’ s Method
3.1 I l l - c o n d i t i o n e d Systems 42
3 .2 Eigenvalue A n a ly s is and C o nd itio n Number 43
3. 3. 1 K»M. Brown’ s Method 46
3 . 3 . 2 Choice o f hR i n the Computer 52
Implementation
3 . 3 . 3 Computational E f f i c i e n c y o f Brown's 52 Method
3 . 3 . 4 Brown's Method I t e r a t i o n Function 53 3 . 3 . 5 M atrix Formulation o f Brown's Method 55
3 . 4 F ix e d -P o in t Formulation 56
3 .5 A p p lic a t io n of Brown's Method 57
Chapter I V% L o a d -F lo w -S o lu tio n by Brown's Method 60 (Stu dy o f Sample Systems)
4.1 In tr o d u c t io n 60
4.2 Load-Flow Problem Formulation 62
4 .3 L o a d -F lo w -S o lu tio n A lg o rith m 63 4 .4 Brown's Load-Flow Results and Comparison 65
4 .5 Summary 80
Chapter
Chapter
Chapter
\Js Optimal Load Flow S o lu t io n by Brown's
Method ■
5.1 In tr o d u c t io n
5.2 Basic Equations fo r Optimal Load-Flow 5.3 Optimal Load-Flow Problem Formulation 5.4 Optimal Load-Flow S o lu tio n
5. 4. 1 S o lu t io n A lg o rith m
5 . 4 . 2 In e q u a lit y C o n s tr a in ts on C o n tro l Parameters 5.5 Optimal Load-Flow Results and Comparison
5.6 Summary
\JIs S t a t i c S tate E stim a tio n by Brown's Method 6.1 In tr o d u c t io n
6 .2 S t a t i c St at e E stim a tio n Problem Formulation 6.3 S t a t i c State E stim a tio n S o lu tio n A lgorithm 6 .4 S t a t i c State E stim a tio n Results and Comparison
6.5 Summary
1/11 - Conclusions
Re ferences
Page 81
81 82 83 86
86 88 91 102
104 104 105 108 110
114
115
118
