I 
AN IMPROVED DC LOAD FLOW TECHNIQUE FOR 
RELIABILITY STUDIES OF POWER SYSTEMS 
A dissertation submitted to 
THE UNIVERSITY OF MORATUWA 
for 
1>i I ' % 
H i 
/H 1 
THE DEGREE OF MASTER IN ENGINEERING 
by. 
P E R M A N E N T 
R t i ! • L K L AG E 
M f TC cfc 
R E M . - V t . u HsCM 
TH£- L I B R A R Y . 
ALEXANDER S. MALEMBEKA 
ELECTRICAL ENGINEERING DEPARTMENT 
UNIVERSITY OF MORATUWA 
SRI LANKA 
JUNE,1983 
3952V 
7 ^  - 7 
(i) 
CONTENTS 
Page 
ACKNOWLEDGEMENT (ii) 
1.0 INTRODUCTION , 1 
2.0 BRIEF BACKGROUND AND HISTORICAL DEVELOPMENT OF POWER 
SYSTEM RELIABILITY STUDIES 5 
3.0 METHODS FOR QUANTITATIVE EVALUATION OF RELIABILITY OF 
POWER SYSTEMS 8 
3.1 AVERAGE INTERRUPTION RATE 8 
3.2 FREQUENCY AND DURATION 9 
3.3 MARKOV PROCESS 16 
3.4 APPROXIMATE 19 
3.5 CONTINGENCY ANALYSIS 2.1 
3.6 COMPOSITE SYSTEM RELIABILITY .22 
4.0 TECHNIQUES FOR LOAD FLOW ANALYSIS 29 
4.1 AC FAST DECOUPLED LOAD FLOW 31 
4.2 DC LOAD FLOW FOR USE WITH A NETWORK ANALYSER 35 
4.3 IMPROVED DC LOAD FLOW FOR USE WITH A DIGITAL 
COMPUTER 40 
4.4 STOCHASTIC LOAD FLOW 45 
5.0 CALCULATION OF RELIABILITY INDICES 61 
5.1 PROBABILITY OF FAILURE OF CONSUMER LOAD POINTS 61 
5.2 SENSITIVITY FACTORS 61 
5.3 EFFECT OF. GENERATOR OUTAGE ON THE FAILURE OF 
CONSUMER POINTS 65 
5.4 OVERALL SYSTEM PERFORMANCE 65 
5.5 SERVICE FAILURE PROBABILITY 65 
5.6 EXPECTED UNSERVED ENERGY 67 
6.0 R E S U L T S AND C O N C L U S I O N S 
6.1 A P P L I C A T I O N OF THE METHOD AND R E S U L T S 
6.2 C O N C L U S I O N S 
A P P E N D I X 1 
A P P E N D I X 2 
A P P E N D I X 3 
R E F E R E N C E S 
(ii) 
ACKNOWLEDGEMENT 
The author is highly indebted to Dr. O.P. Kulshreshtha, now the UNESCO 
Chief Technical Advisor to the Open University of Sri Lanka who in 
1975 made arrangements for the award of the UNESCO Fellowship while 
he was working in the United Republic of Tanzania. He wishes to 
express his gratitude to Professor S. Karunaratne of the University of 
Moratuwa for providing the facilities in the Electrical Engineering 
Department and for his tireless assistance in problems relating to 
Electrical Engineering in general. 
The author is also highly indebted to his Supervisor, Dr. A.S. Induruwa 
for his invaluable assistance and encouragement without which this work 
would have not been realised. 
He wishes to thank Dr. J.R. Lucas of the University of Moratuwa for his 
help in computer programming and Drs. H. Sriyananda and M.P. Dias for 
reading the manuscript and for their suggestions in presenting the 
material in this dissertation. 
Much thanks go to the staff members of the Computer Centre at the 
University of Moratuwa, Open University and University of Colombo for 
the good service provided by them. 
The author takes this opportunity to extend his sincere appreciation 
to the United Nations Educational, Scientific and Cultural 
organization for granting him the Fellowship and to the United Nations 
Development Programme in Sri Lanka for administering the fellowship. 
Finally he wishes to thank his employer, the Principal Secretary of 
the Ministry of National Education of the United Republic of Tanzania 
for granting him study leave to undertake this course of study. 
(iii) 
LIST OF .SYMBOLS 
» P - deterministic vectors which distribute the system 
load to individual buses. 
H - susceptance of series element of transmission line (i,k) 
D - number of days of study 
d. - mean duration of a contingency state 
e - mean duration of system load being in a peak state 
* =
V e J 9 i
 • i 1 1 - complex voltage at node I 
E* - complex conjugate of voltage E^ 
erf - the error function .. 
ex]j - exponent 
*i current through line (i,k) at node i 
i,k - node identification 
j - complex operator -• 
Lcj - critical load of a contingency network state 
L 
system load being in a peak state 
L 
0 - system load being in a low state 
^ - failure rate of a component 
- repair rate of a component 
e 
ik - voltage angle difference between nodes i and k, (0 - 0 ) 
i k 
0 
1 - voltage angle at node i (reference to slack node) 
(iv) 
U - duration of normal weather 
N_. - contingency state of a transmission network 
W - normal state of a transmission network 
P - real power flow in line (i,k) at node i 
Pr - probability 
Plr"' - the probability of overload of line mn due to line pq 
mn 
V 
i 
being out of service 
f} - reactive power flow in line (i,k) at node i 
0 - modified reactive power in line (i.k) 
m 
q - reactive power generated in line (i,k) 
P - random variable for system load 
r ~ series resistance of line (i,k) 
R - sensitivity factor 
J'i. - complex power flow in line (i,k) at node i, (S. = P. - jo ) 
1 1 1 1 
voltage magnitude a^ . node i 
x - series reactance of line (i,k) 
i_ K. 
x - inverse of the series susceptance of line (i,k), (x = 1) 
sh 
y^ k - shunt susceptance of line (i,ki 
z . . - series impedance of line (i,k) , z. = (r + jx ) 
ik ik ik ik