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MINIMIZATION OF POWER LOSSES OVER ELECTRIC
POWER TRANSMISSION LINES
By
OKE, Michael Olufemi
B.Sc. (Benin), P.G.D. Eng. (Ado-Ekiti), M.Sc. (Ilorin)
Matric. No.: 01/68EV002
A THESIS SUBMITTED TO THE DEPARTMENT OF
MATHEMATICS, FACULTY OF SCIENCE, UNIVERSITY OF ILORIN,
ILORIN, NIGERIA, IN PARTIAL FULFILMENT OF THE
REQUIREMENTS FOR THE AWARD OF THE DEGREE OF
DOCTOR OF PHILOSOPHY (Ph.D.) IN MATHEMATICS.
JULY, 2012.
i
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CERTIFICATION
This is to certify that the research work reported in this thesis was car-
ried out by OKE, Michael Olufemi with matriculation number 01/68EV002
in the Department of Mathematics, Faculty of Science, University of Ilorin,
Ilorin, Nigeria.
........................................
Professor O.M. Bamigbola
(Supervisor)
........................................
Professor M.O. Ibrahim
(Head of Department)
........................................
(External Examiner)
ii
.......................................
Date .......................................
Date .......................................
Date
-
DEDICATION
This work is dedicated to my late father: Pa David Eniola Oke.
iii
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ACKNOWLEDGEMENTS
To God be the glory for the great and marvellous things he has done in
my life. I will forever be grateful to God almighty, the King of Kings, the
Lion of Judah, my messiah and everlasting Father, for giving me the grace
to complete this research work. His protection over me throughout my so-
journ in this university and the manifestation of his invisible hands made
the whole work a success.
I am very grateful for the unrivalled support I enjoyed from my amiable
and indefatigable supervisor, Prof. O.M. Bamigbola. His guidance, en-
couragement and constructive criticisms of the research work at every stage
made it a success.
I will like to thank Engr. (Prof.) I.E. Owolabi, Engr. (Prof.) S.B.
Adeyemo, Engr. (Prof.) J.O. Aribisala, Prof. O. Olaofe, Engr. (Dr.) E.A.
Okunade and Engr. A.A. Adegbemile for their fatherly advice and encour-
agement.
I will like to appreciate Engr. (Prof.) O.S. Onohaebi for the data on
empirical modelling, Engr. D.L. Atandare for the materials on electrical
power systems and some engineers of the Power Holding Company of Nige-
ria who have contributed in one way or the other to the success of this
research work. They include Engr. P.O. Falana, Engr. G.O. Ajayi, Engr.
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N.O. Emeka and Engr. A. Adekogba of Ado-Ekiti district headquarters.
Others include Engr. E.O. Bello of Akure business unit, Engr. P. Atuluku
of Kabba district headquarters and Engr A. Falana of Ilorin business unit.
My special thanks go to all members of sta of the Department of Math-
ematics, University of Ilorin, particularly Professors M.O. Ibrahim, J.A.
Gbadeyan, T.M. Adeniran, T.O. Opoola and J.S. Sadiku, Drs. O.A. Taiwo,
R.B. Adeniyi, J.O. Omolehin, S.O. Makanjuola, M.S. Dada, A.S. Idowu,
E.O. Titiloye , K. Rauf and K.O. Babalola as well as Dr (Mrs) O.A. Fadipe-
Joseph and Dr (Mrs) C.N. Ejieji.
I cannot but mention the support and encouragement I enjoyed from
Dr (Mrs) Y.O. Aderinto. I will also like to mention the encouragements
from my friends and colleagues who are still on the Ph.D. programme, their
camaraderie made the tension bearable.
I am also grateful to my parents, Late Pa D.E. Oke and Mrs. E.O. Oke,
for the basic education they gave me which qualies me for the postgradu-
ate work. I thank the authority of Ekiti State University, Ado-Ekiti for the
study leave which they gave me to undertake the programme.
Finally, I thank my wife, Olubunmi, and my children, Victor and Peace,
for their understanding and cooperation throughout the period of this re-
search work.
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TABLE OF CONTENT
page
TITLE PAGE
CERTIFICATION
DEDICATION
ACKNOWLEDGEMENTS
TABLE OF CONTENT
LIST OF TABLES
LIST OF FIGURES
ABSTRACT
CHAPTER ONE : GENERAL INTRODUCTION
1.1 BACKGROUND TO THE STUDY
1.2 GOAL AND OBJECTIVES OF THE STUDY
1.3 SIGNIFICANCE OF THE STUDY
1.4 ORGANIZATION OF THE THESIS
1.5 NOTATIONS
1.6 DEFINITION OF SOME BASIC TERMS
i
ii
iii
iv
vi
x
xi
xiii
1
4
5
5
6
7
CHAPTER TWO : ELECTRIC POWER TRANSMISSION SYS-
TEMS
2.1 ELECTRIC POWER SYSTEMS
2.1.1 Historical Developments
2.1.2 Importance of Electric Power System
2.1.3 Electric Power Systems in Nigeria
vi
11 11 12 13
-
2.2 ELECTRIC SUPPLY SYSTEMS
2.2.1 Alternating Current and Direct Current Transmission Systems
2.2.2 Overhead and Underground Systems
2.3 MECHANICAL REQUIREMENTS FOR OVERHEAD LINES
2.4 MAIN COMPONENTS OF OVERHEAD LINES
2.4.1 Conductors
2.4.2 Line Supports
2.4.3 Insulators
2.4.4 Cross-arms
2.4.5 Stays
2.4.6 Miscellaneous Components of Overhead Lines
2.5 TRANSMISSION LINE CONSTANTS
2.5.1 Line Resistance
2.5.2 Line Inductance
2.5.3 Line Capacitance
2.5.4 Shunt Conductance
2.6 SKIN EFFECT
2.7 ECONOMICS OF POWER TRANSMISSION
2.7.1 Economic Choice of Conductor Size
2.7.2 Economic Choice of Transmission Voltage
2.8 CORONA PHENOMENON
2.8.1 Factors Aecting Corona
2.8.2 Advantages and Disadvantages of Corona
2.8.3 Methods of Reducing Corona
vii
19 20 21 23 23 24 25 26 26 27 27 28 28 28 29 29 29 30 31 31 31 32 33 33
-
CHAPTER THREE : MATHEMATICAL MODELS FOR POWER
FLOW OVER TRANSMISSION LINES
3.1 MATHEMATICAL PRELIMINARIES
3.1.1 Modelling
3.1.2 Dierential Equations
3.1.3 Laplace Transformation
3.2 KIRCHOFFS CIRCUIT LAWS
3.2.1 Kircho s Current Law
3.2.2 Kircho s Voltage Law
34 34 35 36 37 37 37
3.3 MATHEMATICAL MODEL FOR ELECTRIC POWER FLOW ALONG
LOSSY TRANSMISSION LINES
3.3.1 Model Formulation
3.3.2 Model Solution
38 38 40
3.4 MATHEMATICAL MODEL ALONG TRANSMISSION LINES WHEN
LEAKAGE TO GROUND IS SMALL
3.4.1 Model Formulation
3.4.2 Model Solution
3.5 ANALYSIS OF MATHEMATICAL MODELS
43 43 44 46
CHAPTER FOUR : MINIMIZATION OF POWER LOSSES OVER
TRANSMISSION LINES
4.1 OHMIC AND CORONA LOSSES
4.1.1 Ohmic Loss
4.1.2 Corona Loss
4.2 MATHEMATICAL MODELS FOR POWER LOSSES
viii
47 47 48 48
-
4.2.1 Model Based on Ohmic and Corona Losses
4.2.2 Empirical Models as Functions of Distance
48 50
4.3 MULTIVARIABLE OPTIMIZATION WITHOUT CONSTRAINTS 71
4.3.1 Properties of Hessian Matrix
71
4.3.2 Necessary and Sucient Conditions for the Existence of Extremal
Points
4.4 MINIMIZATION OF POWER LOSSES
4.5 DISCUSSION ON RESULTS
CHAPTER FIVE : GENERAL CONCLUSION
5.1 SUMMARY OF THESIS
5.2 SUMMARY OF RESULTS
5.3 CONCLUSION
5.4 RECOMMENDATION
REFERENCES
ix
72 78 79 80 80 81 82 83
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LIST OF TABLES
Table 2.1: Per Capital Consumption of Electricity in some Countries
15
Table 4.1: Simulated Results of Power Losses on 330 KV Single Circuit of
the Nigerian Transmission Network
51
Table 4.2: Simulated Results of Power Losses on 330 KV Double Circuit of
the Nigerian Transmission Network
Table 4.3: Summations for a Load of 100 MW on Single Circuit
Table 4.4: Summations for a Load of 200 MW on Single Circuit
Table 4.5: Summations for a Load of 300 MW on Single Circuit
Table 4.6: Summations for a Load of 100 MW on Double Circuit
Table 4.7: Summations for a Load of 200 MW on Double Circuit
Table 4.8: Summations for a Load of 300 MW on Double Circuit
x
52 55 57 61 65 67 70
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LIST OF FIGURES
Figure 2.1: Pictorial view of 330 KV double circuit transmission line tower
of the Nigerian transmission network. 17
Figure 2.2: Pictorial view of 330 KV single circuit transmission line tower
of the Nigerian transmission network.
Figure 3.1: Equivalent Circuit of a Transmission Line
18 38
Figure 4.1: Scatter Diagram for Power Losses in MW for a load of 100
MW on Single Circuit
53
Figure 4.2: Graph of Power Losses in MW for a load of 100 MW on Single
Circuit
53
Figure 4.3: Scatter Diagram for Power Losses in MW for a load of 200
MW on Single Circuit
56
Figure 4.4: Graph of Power Losses in MW for a load of 200 MW on Single
Circuit
56
Figure 4.5: Scatter Diagram for Power Losses in MW for a load of 300
MW on Single Circuit
59
Figure 4.6: Graph of Power Losses in MW for a load of 300 MW on Single
Circuit
59
Figure 4.7: Scatter Diagram for Power Losses in MW for a load of 100
MW on Double Circuit
Figure 4.8: Graph of Power Losses in MW for a load of 100 MW on
Double Circuit
63 63
Figure 4.9: Scatter Diagram for Power Losses in MW for a load of 200
xi
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MW on Double Circuit
Figure 4.10: Graph of Power Losses in MW for a load of 200 MW on
Double Circuit
66 66
Figure 4.11: Scatter Diagram for Power Losses in MW for a load of 300
MW on Double Circuit
Figure 4.12: Graph of Power Losses in MW for a load of 300 MW on
Double Circuit
xii
69 69
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ABSTRACT
Availability of electric power has been the most powerful vehicle for fa-
cilitating economic, industrial and social developments of any nation. Elec-
tric power is transmitted by means of transmission lines which deliver bulk
power from generating stations to load centres and consumers. For electric
power to get to the nal consumers in proper form and quality, losses along
the lines must be reduced to the barest minimum. A lot of research has been
carried out on analysis and computation of losses on transmission lines us-
ing reliability indices, but hardly any on the minimization of losses using
analytical methods. In another vein, a large body of literature exists for the
solution of optimal power ow problems using evolutionary methods, but
none of them has employed the versatile tool of mathematical modelling.
Thus, the goal of this work is to use the classical optimization approach
coupled with the mathematical modelling technique to minimize the trans-
mission power losses. Specically, the objectives of the study were to:
(i.) develop mathematical models for power ow and power losses along
electric power transmission lines and solve the mathematical models
for electric power ow along transmission lines using an analytical
method;
(ii.) develop empirical models of power losses as functions of distance; and
(iii.) minimize the power losses using the classical optimization technique.
In the research, I employed Kircho s circuit laws and a combination
xiii
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of corona and ohmic losses in obtaining the mathematical models for the
power ow and power losses respectively. Empirical models of the power
losses were developed using regression analysis.
The ndings of this study were:
(i.) the models for power ow along transmission lines evolved as homo-
geneous second-order partial dierential equations which were solved
analytically using the method of Laplace transform;
(ii.) the model for power losses over the transmission lines was obtained
as the sum of the ohmic and corona losses;
(iii.) the empirical models developed are monotonic increasing functions of
distance. Thus, establishing that power losses increases with distance;
(iv.) power losses are minimized when the operating transmission voltage
is equal to the critical disruptive voltage.
With the above results, a workable strategy can be formulated to reduce
to the barest minimum electric power losses along transmission lines so as
to ensure availability of electric power, in proper form and quality, to con-
sumers. Hence, this research work has addressed the problem of minimizing
electric power losses during transmission.
Xiv
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MINIMIZATION OF POWER LOSSES OVER ELECTRIC POWER TRANSMISSION LINES
-
1
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Abstract
-
Chapter 1
GENERAL INTRODUCTION
1.1
BACKGROUND TO THE STUDY
Energy is a basic necessity for the economic development of a nation. There are dierent
forms of energy, but the most important form is the electrical energy, Gupta (2008) and
Mehta and Mehta (2008). A modern and civilized society is so much dependent on the use
of electrical energy. Activities relating to the generation, transmission and distribution of
electrical energy have to be given the highest priority in the national planning process of any
nation because of the importance of electrical energy to the economic and social development
of the society. In fact, the greater the per capital consumption of electrical energy in a
country, the higher the standard of living of its people. Therefore, the advancement of
a country is measured in terms of its per capital consumption of electrical energy, Gupta
(2008) and Mehta and Mehta (2008).
Power plants planning in a way to meet the power network load demand is one of
the most important and essential issues in power systems. Since transmission lines connect
generating plants and substations in power network, the analysis, computation and reduction
of transmission losses in these power networks are of great concern to scientists and engineers.
A lot of research works have been carried out on the above listed aspects. Zakariya
(2010) made a comparison between the corona power loss associated with HVDC trans-
mission lines and the ohmic power loss. The corona power loss and ohmic power loss were
measured and computed for dierent transmission line congurations and under fair weather
and rainy conditions. It was pointed out in the work that the general trend of neglecting
the corona power loss is not always valid. It was found from the comparison that, when
1
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the transmission line is moderately or lightly loaded, the percentage of corona power loss to
ohmic power loss could reach up to one hundred percent especially if the transmission line
is operating at a voltage well above the corona onset value. This percentage is also found to
increase substantially under rainy conditions. Finally, it was also discovered that, the ratio
of corona to ohmic power loss, decreases with increasing number of bundles. Numphetch
et al. (2011) worked on loss minimization using optimal power ow based on swarm in-
telligences. Thabendra et al. (2009) considered multi-objective optimization methods for
power loss minimization and voltage stability while Abdullah et al. (2010) looked at trans-
mission loss minimization and power installation cost using evolutionary computation for
improvement of voltage stability. Bagriyanik et al. (2003) used a fuzzy multi-objective
optimization and genetic algorithm-based method to nd optimum power system operating
conditions. In addition to active power losses, series reactive power losses of transmission
system were also considered as one of the multiple objectives. Onohaebi and Odiase (2010)
considered the relationship between distance and loadings on power losses using the exist-
ing 330 KV Nigerian transmission network as a case study in his empirical modelling of
power losses as a function of line loadings and lengths in the Nigeria 330 KV transmission
lines while Moghadam and Berahmandpour (2010) developed a new method for calculating
transmission power losses based on exact modelling of ohmic loss. Ramesh et al. (2009)
looked at minimization of power loss in distribution networks by using feeder restructuring,
implementation of distributed generation and capacitor placement method. Lo and Gers
(2006) considered feeder reconguration for losses reduction in distribution systems. Others
who researched into power losses include Rugthaicharoencheep and Sirisumrannukul (2009),
Crombie (2006), Marwan and Imad (2002), Ayman (2004), Sarajcev et al. (2003) and Daniel
(2005), to mention a few.
Various researchers have also worked on the ow of power on electrical networks. Pandya
and Joshi (2008) presents a comprehensive survey of various optimization methods for solving
optimal power ow problems. The methods considered in the work include linear program-
ming, Newton-Raphson, quadratic programming, nonlinear programming, interior point and
articial intelligence. Under the articial intelligence method, the following were also con-
sidered articial neural network method, fuzzy logic method, genetic algorithm method,
evolutionary programming method, ant colony optimization method and particle swarm
optimization method. It was found in the paper that the classical methods have a lot of
2
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limitations. In most cases, mathematical formulations have to be simplied to get the solu-
tions because of the extremely limited capability to solve real-world large-scale power system
problems. The classical methods are weak in handling qualitative constraints and they have
very poor convergence. The methods are also very slow and computationally expensive in
handling large-scale optimal power ow problems. It was also discovered in the paper that
the articial intelligence methods are relatively versatile for handling various qualitative
constraints and that the methods can nd multiple optimal solutions in a single simulation.
They are therefore suitable in solving multi-objective optimization problems. William and
Jose (2002) looked at alternative optimal power ow formulations while Claudio et al. (2001)
worked on comparison of voltage security constraint using optimal power ow techniques.
Roya et al. (2008) considered power ow modelling for power systems with dynamic ow
controller. Other researchers who also worked on power ow include Bouktir et al. (2004),
Swarup (2006), Tarjei (2006), Bouktir and Slimani (2005), Burchett et al. (1982), Dommel
and Tinney (1968), Heinkenschloss and Vicente (1994) and Taiyou and Robert (2006).
In addition, several researchers have also worked on electric power systems. Aderinto
(2011) worked on an optimal control model of the electric power generating system. In
the research work, she developed a mathematical model for the electric power generating
system using the optimal control approach and characterized the mathematical model by
prescribing the conditions for the optimality of the electric power generating system and the
analytic requirements for the existence and uniqueness of the solution to the system. The
optimality condition for the model was determined and the model was solved analytically
and numerically. In the study, two control variables were identied, the rst for load shed-
ding among the generators in the system and the second for restriction on the capacity of
the generators. The problem was formulated based on the second control variable since the
rst control variable can only be on or o as the case may be. The optimality conditions
for the system were imposed implicitly on the controls and the mathematical model repre-
sents a stable loss-free generating system. From the work, it was shown that the generation
loss can be controlled and stabilized. Oke et al. (2007) considered the perspectives on
electricity supply and demand in Nigeria while Ibe and Okedu (2007) looked at optimized
electricity generation in Nigeria. Bamigbola and Aderinto (2009) characterized an optimal
control model of electric power generating system. Karamitsos and Orfanidis (2006) con-
sidered an analysis of blackout for electric power transmission systems while Aderinto et
3
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al. (2010) looked at optimal control of air pollution with application to power generating
system model. Others whose researches touched on electric power systems include Savenkov
(2008), Youssef and Hackum (1989), Williams and John (2006), Anderson (2008), Bansal
(2005), Nanda et al. (1989), Aribia and Abdallah (2007), Vaisakh and Rao (2008), Kamin-
skyi (2009), Billinton (1994), Schenk and Ahsan (1985), Jocic et al. (1983), Doraiswami et
al (1995), Caprio (1984), Dandeno (1982), Miroslav et al. (2001), Bockarjova et al. (2003)
Okafor and Adebanji (2009), Dmytro et al. (2007), Grigsby (1998), Komolafe et al. (2009),
Kundur (1994), Kusko (1968), Lee et al. (1986), Rajput (2003), Shahildehpour and Labudda
(2005), Thomas and Martin (2002), Wayne (2001), Youssef and Hackum (1989), Authur and
Connie (1988), Branimir and Radivo (1993), Hicks (1966), Joe et al. (2004), Baskaran and
Palanisamy (2005), Ayodele et al. (2008) and Lee et al. (1988), to list a few. As such, much
emphasis has been on proper design of electrical power systems and reduction of losses using
feeder reconguration and evolutionary techniques.
Loss minimization is a critical component for ecient electric power supply systems.
Losses in an electric power system should be around 3 percent to 6 percent, Ramesh et al.
(2009). In developed countries, it is not greater than 10 percent. However, in developing
countries it is still over 20 percent, Ramesh et al. (2009). Therefore stakeholders in the power
sector are currently interested in reducing the losses on electric power lines to a desired and
economic level. The purpose of this research work, therefore, is to develop mathematical
models for power losses along transmission lines and to minimize the losses using classical
optimization techniques.
1.2
GOAL AND OBJECTIVES OF THE STUDY
Power losses result in lower power availability to the consumers, leading to inadequate
power to operate their appliances. High eciency of power system is determined by its
low power losses. The goal of this research work therefore is to use classical optimization
techniques to minimize the transmission power losses on transmission lines. The objectives
of the research work are to:
(i.) Develop mathematical models for electric power ow and power losses along electric
power transmission lines;
4
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(ii.) Solve the mathematical models for electric power ow along transmission lines analyt-
ically;
(iii.) Develop empirical models of power losses as functions of distance; and
(iv.) Minimize power losses using the classical optimization technique.
1.3
SIGNIFICANCE OF THE STUDY
The mathematical representation of power ow along transmission lines provides a bet-
ter understanding of the ow of electric power on transmission lines and the evolution of
voltage and current along the lines. The mathematical representation of power losses along
transmission lines gives an insight into the major problems on electric power transmission.
The minimization of losses on electric power transmission lines using classical optimization
technique provides a solution, in a compact form, to the major problem encontered in power
transmission.
1.4
ORGANIZATION OF THE THESIS
The remaining part of this thesis are organised as follows:
Various notations used in the thesis are listed in section 1.5 while section 1.6 gives the
denition of some basic terms used in the thesis. Chapter two focuses on electric power
transmission systems detailing on requirements for transmitivity. Chapter three is devoted
to the development of mathematical models for power ow over transmission lines. Mathe-
matical preliminaries were considered in section 3.1. In section 3.2, we formulated and solved
the model for electric power ow along lossy transmission lines, while in section 3.3, we de-
rived and solved the model for electric power ow along transmission lines when leakage to
ground along the line is small. We then analysed the models in section 3.4.
In chapter four, we treated minimization of power losses over transmission lines. Specif-
ically, secion 4.1 is on preamble where we detailed the requirements for the existence of
an extemum of a function of several variables. In this section, we also discussed ohmic
and corona losses which we now used in subsection 4.2.1 for the development of a model for
power losses along transmission lines and in subsection 4.2.2, we developed empirical models
5
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of power losses as functions of distance. In Section 4.3, we considered the power loss func-
tion as a multivariable optimization without constraints and minimized it using the classical
optimization technique while in section 4.4, we looked at the minimization of power losses
using dierential calculus. Discussion on results is what we have in section 4.5. The thesis
is rounded up in chapter ve with general conclusion. Section 5.1 treated a summary of the
work reported in the thesis and summarized the results obtained in section 5.2. Section 5.3
is on conclusion while section 5.4 suggests outstanding issues for further research work.
1.5
NOTATIONS
We made use of the following notations in this thesis:
(Ik) represents current along the kth branch.
(Vk) represents voltage along the kth branch.
represents summation.
L represents Laplace transform.
L1 represents inverse Laplace transform.
Isc(x) represents complementary function.
Isp(x) represents particular solution.
I represents current along the conductor.
R represents resistance of the conductor.
f represents frequency of transmission.
represents air density factor.
r represents radius of conductors.
d represents space between the transmission lines.
q represents charge on the transmission line.
v represents potential dierence between the conductors.
V represents operating voltage.
V0 represents distruptive voltage.
represents resistivity of the conductor.
represents ux leakage.
L represents length of the conductor.
A represents cross-sectional area of the conductor.
6
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represents conductivity of the conductor.
TLoss represents total loss on transmission lines.
LOhmic represents ohmic loss.
LCorona represents corona loss.
1.6
DEFINITION OF SOME BASIC TERMS
In this section, we give the denition of some basic terms used in the thesis.
1. Optimization
Optimization is the act of getting the best result under given circumstances, Rao
(1998). It can therefore be dened as the process of obtaining the optimal (best)
solution to certain mathematical problems, which are often models of physical reality,
Minoux (1986). Many problems in engineering, management and planning lead to
mathematical models requiring the idea of optimization for solution, Craven (1995).
2. Classical Optimization
The classical optimization techniques are methods used in nding the optimum of
continuous and dierentiable functions. It is an analytical method that makes use of
dierential calculus techniques in nding the optimum points. The classical optimiza-
tion method forms the basis for the development of most of the numerical optimization
techniques.
3. Hessian Matrix
An Hessian matrix is a square matrix of second order partial derivatives of a function
of several variables. It was developed in the 19th century by a German mathematician
called Ludwig Otto Hesse.
4. Degenerate and Non-degenerate Critical Point
If the derivative of a function f is equal to zero at some point x, then f has a critical
or stationary value at x. The determinant of the Hessian matrix at x is called the
discriminant. If this discriminant is equal to zero then, the point x is called a degener-
7
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ate or non-morse critical point of f. Otherwise it is a non-degenerate or morse critical
point of f.
5. Positive Denite Matrix
A matrix A of order n is said to be positive denite if all its eigenvalues are positive.
That is, if all values of which satises the determinant equation
|A I | = 0
are positive, Rao (1998).
Another test of the positive deniteness of a matrix A of order n is the evaluation of
its determinants:
A1 =
a11
A2 =
a11 a12
a21 a22
a11 a12 a13
A3 = a21 a22 a23
a31 a32 a33
....
a11 a12 a13.....a1n
a21 a22 a23.....a2n
An = a31 a32 a33.....a3n
an1 an2 an3.....ann
A matrix A of order n will therefore be positive denite if and only if all values of A1,
A2, A3, ....., An are positive.
6. Negative Denite Matrix
A matrix A of order n is said to be negative denite if and only if the signs of Ai in
(5) above is (1)i for i = 1,2,3,4,.....,n.
8
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dx
dx
7. Positive Semidenite Matrix
A matrix A of order n is said to be positive semidenite if and only if some of the Ais
in (5) above are positive and the remaining ones are zero.
8. Eigenvalues
Eigenvalues of a matrix A are all values of which satises the determinantal equation
det (A I ) = |A I | = 0
(1.1)
where I is an identity matrix of the same order as A
9. Initial Value Problem
An initial value problem (IVP) is a dierential equation in which the solution y(x)
satises prescribed side conditions imposed on the unknown y(x) or its derivatives at
an initial point x0 , Dennis and Michael (2005) and Eagleeld (1989). An initial value
problem is of the form
Solve
subject to
dny n
= f (x, y, y , y , ....., y(n1)) (1.2)
y(x0) = y0, y (x0) = y1, y (x0) = y2, ....., y(n1)(x0) = yn1
(1.3)
where y0, y1, y2, ..., yn1. are arbitrarily specied real constants.
The values of y(x) and its rst (n - 1) derivatives at a single point x0 , that is y(x0) =
y0, y (x0) = y1, y (x0) = y2, ....., y(n1)(x0) = yn1 are called the initial conditions.
10. Boundary Value Problem
A boundary value problem (BVP) is a dierential equation in which the solution y(x)
satises prescribed conditions imposed on the unknown y(x) or its derivatives at more
than one point. A dierential equation of the form:
Solve
a2(x)
d2y 2
+ a1(x)
dy dx
+ a0(x)y = g(x).
(1.4)
subject to
y(a) = ya, y(b) = yb,
9
(1.5)
-
dx dx
dx dx
is called a boundary value problem. The prescibed values y(a) = ya, y(b) = yb are
called boundary conditions, Dennis and Michael (2005), Etgen (1999) and Kreyszig,
(1987).
11. Homogeneous and Nonhomogeneous Dierential Equations
An nth-order linear dierential equation of the form in (1.6) below is said to be non-
homogeneous if g(x) is not identically zero, Dennis and Michael (2005).
an(x)
dny
n
+ a(n1)(x)
d(n1)y
(n1)
+ ... + a1
dy dx
+ a0(x)y = g(x).
(1.6)
If g(x) is equal to zero, then the nth-order dierential equation is called homogeneous
and we have
an(x)
dny
n
+ a(n1)(x)
d(n1)y
(n1)
+ ... + a1
dy dx
+ a0(x)y = 0.
(1.7)
This explanation also holds for partial dierential equations.
12. Critical Disruptive Voltage
The critical disruptive voltage (V0) is the minimum voltage at which corona occurs.
13. Node or Junction
This is a point where two or more branches meet.
14. Ohmic Loss
Ohmic loss is a loss of power on transmission lines which occurs as a result of the
resistance of conductors against the ow of current.
15. Corona Loss
Corona loss is a loss of power on transmission lines which normally occurs as a result
of the ionization of thin layer of air around the line. This ionization of air is experi-
enced when the applied voltage exceeds the critical disruptive voltage in high voltage
transmission lines.
10
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Chapter 2
ELECTRIC POWER
TRANSMISSION SYSTEMS
2.1
2.1.1
ELECTRIC POWER SYSTEMS Historical Developments
Before 1800, researches on electrical and magnetic phenomena were only carried out by
very few scientists. As at that time, no real applications were known. People illuminated
their homes with candles , whale oil lamps and kerosine lamps, Atandare (2007) and Duncan
and Muluktla (1986). Between 1800 and 1810, commercial illuminating gas companies were
formed. It was rst formed in Europe and later in the United States of America. Scientic
research increase in the area of electrical and magnetic phenomena throughout the 19th
century. Two independent researchers Michael Faraday and Joseph Henry Ampere had
already observed that magnetic elds were created by electric currents but no one had
discovered how electrical currents could be produced from magnetic elds. Faraday worked
on such problems between 1821 and 1831 and nally succeeded in formulating a law on
it that bears his name. He subsequently built a machine that generated voltage based
on the principle of magnetic induction. Between 1840 and 1877 several people including
Charles Wheatstone, Carl Siemens and Gramme, applied the principle of induction for the
construction of primitive electrical generators, Atandare (2007), Charles (1986) and Duncan
and Muluktla (1986).
11
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In 1878, a 29-year old inventor named Thomas Edison worked on a number of projects
including the development of an incandescent electric lamp. In October 1879, after several
unsuccessful trials and experiments, an enclosed evacuated bulb was energised. In 1882 the
rst system installed to sell electrical energy for incandescent lighting in the United States of
America began operations. The system was DC, three wire, 220/110 volts. The early days
electrical companies referred to themselves as illuminating companies because lighting
was their only service. In 1890, the newly formed Westinghouse Company (WC) developed
another form of electricity name Alternating Current. With this, most of the problems
associated with DC generators were eliminated, Atandare (2007), Olle (1987) and Duncan
and Muluktla (1986).
2.1.2
Importance of Electric Power System
It is no doubt that the civilization of mankind are closely interwoven with energy. Electri-
cal energy occupies a top position in the energy hierarchy because of its usefulness at home,
industry, agriculture and even in the transportation sector. Electrical energy can be gener-
ated centrally in bulk and transmitted economically over long distance. The advancement
in science and technology has made it possible to convert electrical energy into any desired
form like heat, light, motive power etc. This has given electrical energy a place of pride
in the modern world. The social structures and the industrial development of any country
depends primarily upon low cost and uninterrupted supply of electrical energy, Mehta and
Mehta (2008). Availability of electricity has been the most powerful vehicle of introducing
economic development and social change throughout the world. The process of moderni-
sation, increase in productivity, agriculture and industry basically depend upon adequate
supply of electrical energy. The annual per capital consumption of electrical energy is a very
important yardstick for measuring the development of a nation, Gupta (2008).
Generation of electrical energy is the conversion of energy available in dierent forms
in nature to electrical energy. The ever increasing use of electrical energy for industrial,
domestic and commercial purposes necessitated the bulk production of electrical energy.
This bulk production is achieved with the help of suitable power production stations which
are generally referred to as electric power generating stations or electric power plants. A
generating station usually employs a prime mover coupled with an alternator to produce
electric power.
12
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Electrical energy is generated at power stations which are usually situated far away
from load centres. Hence an extensive network of conductors between the power stations
and the consumers is required. This network of conductors may be divided into two main
components, called the transmission system and the distribution system. The transmission
system is to deliver bulk power from power stations to load centres and large indusrial
consumers while the distribution system is to deliver power from substations to various
consumers.
Electrical energy produced must be transmitted and distributed to the point of use as
soon as it is needed. Transmission lines and other materials are needed to achieve this pur-
pose. Transmission lines are materials or media that are used to transmit electric energy and
signals from one point to another, specically from a source to a load. They can be regarded
as a set of conductors being run from one place to another and supported on transmission
towers. This involves connections between an electric generating plant and a substation
which is several hundred kilometers away. The transmission and distribution stages are
very important to electric power system, because without these stages the generated power
cannot get to the load centres not to talk of getting to the nal consumers. Power losses
along these stages should be reduced to the bearest minimum so that the nal consumer
will get the normal power to operate their appliances, Mehta and Mehta (2008), Wadhwa
(2009) and Atandare (2007).
Power plants planning in a way to meet the power network load demand is one of
the most important and essential issues in power systems. Since transmission lines con-
nect generating plants and substations in power network, the analysis and computation of
transmission losses of these power networks are of great concern to scientists and engineers.
Another issue of great importance to scientists and engineers is nding methods to reduce
the losses on electric power lines to a desired and economic level.
2.1.3
Electric Power Systems in Nigeria
Source of electric power was rst known in Nigeria in 1896 when a 30 KW, 80 Hz, single
phase locomotive generator was installed in Ijora, Lagos, the then seat of British colony. The
operation, maintenance and distribution of this generator was solely the responsibility of the
Power Works Department (PWD). In 1924, with the increasing population, a three phase,
50 Hz system of power system became known and electric power were been distributed in
13
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few cities of the country by some isolated generating stations like Cameroons Development
Corporation (CDC), African Timber and Polywood Company (ATPC) and Nigeria Electrical
Supply Corporation (NESCO), Atandare (2007).
In 1946, the management of electrical power supply in the country was taken over by the
Nigeria Government Electricity Undertaking (NGEU). This new organ of government took
care of electricty distribution and expansion in the country. In 1952, Electricity Corpora-
tion of Nigeria (ECN) was establised and this gave birth to the Ijora Power Station which
had 10 MW coal-red turbo-generators, Atandare (2007). Some investigations for possible
siting of hydro electric power stations ware carried out in 1953 by Netherlands Engineering
Consultants on behalf of Electricity Corporation of Nigeria. This now resulted in the con-
struction of Kainji Dam and the associated hydro-generators for power production. With
the construction of Kainji Dam, Niger Dam Authority (NDA) was established in 1964 with
the responsibility of further constructing the dam, power station and the associated 330
KV transmission lines between Kainji and the national control centre at Osogbo, Atandare
(2007), Manafa (1978).
In Nigeria, there cannot be any successful survey on generation, transmission and distri-
bution of electricity without reference to National Electric Power Authority (NEPA) which
was established by Decree 24 of 1st April, 1972, with the almalgamation of Electricity Cor-
poration of Nigeria (ECN) and Niger Dams Authority (NDA). The decree gave NEPA the
mandate to maintain and co-ordinate an ecient electricity supply to all parts of the coun-
try. NEPA was also empowered to manage and maintain electrical power undertakings,
establish new electric power undertakings, generate, transmit and distribute electric power
to every part of the country, Power Sector Reforms (2005) and Atandare (2007).
However, in March, 2006 NEPA was renamed Power Holding Company of Nigeria (PHCN)
with eighteen business units. NEPA (now PHCN) has eight major generating stations lo-
cated nationwide. These stations are connected by transmission substations to form the
National Grid System with the control centre at Osogbo, Osun State. These stations in-
clude three hydropower stations and ve thermal stations. The total installed capacity of
the existing government-owned generating stations in Nigeria is 6200MW. Although the
stations produced below the actual installed capacity of 6200MW, Power Sector Reforms
(2005). In order to improve the power generation in the country, the Federal government has
seven new on-going thermal power projects in the Niger Delta Area. The total generating
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Country Per Capital Consumption (in KW)
United State of America 3.2
Cuba 0.38
United Kingdom 1.33
Ukraine 1.33
Iraq 0.42
South Korea 1.09
Nigeria 0.03
Egypt 0.27
capacity of these on-going thermal projects is 2250MW, Popopla et al. (2008). There are
some existing independent power producers in the country with total generating capacity of
2552MW. These independent power producers also have on-going projects with a generating
capacity of 378MW. If all the existing and on-going power generating stations are producing
at optimum level, Nigeria will be generating a total of 11380MW, Atandare (2007).
The per capital consumption of electricity in a country is one of the strongest and most
reliable indices for measuring the degree of development of that nation. The per capital
consumption of electricity in Nigeria is 0.03 KW. This is very low compared to the per
capital consumption of electricity in other countries. We can see this in Table 2.1 which
gives the per capital consumption of electricity in some selected countries as given by the
International Energy Institutes comparative analysis of the per capital consumption of
electricity worldwide, Atandare (2007).
Table 2.1: Per Capital Consumption of Electricity in some Countries, Atandare
(2007).
Improvement in the quality and quantity of infrastructural services, especially electricity,
is fundamental to rapid and sustainable economic growth in any country. But inadequate
quantity, quality and access to electricity services have been a regular feature in the Nigerian
power sector, Iwayemi (2008), Adeniyi (2008) and Adeyemo (2008). The Transmission
15
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Company of Nigeria PLC (TCN) manages Nigerians power grid. TCN ensures that power
is transmitted eciently over the national grid and delivered to the distribution companies
in their designated franchise areas, TCN Reports (2006). The Transmission Company of
Nigeria (TCN) is subdivided into ve zones for management and operational purposes. It
is managed from a national control centre at Osogbo, Osun State and a secondary control
centre at Shiroro, Niger State. It has six regional oces and several satellite work centres,
TCN Reports (2006), Atandare (2007),Fasina (2008) and Onohaebi and Odiase (2010)
The Nigerian 330KV transmission network employed 350mm2 aluminium conductor steel
re-inforced (ACSR). Single and double circuits are used in the trasmission network. The
double circuit has the advantage that it ensures continuity of power supply. In case there is
breakdown of one circuit, the continuity of supply can be maintained by the other circuit.
The supporting structures are made of steel towers and are spanned at an average distance
of 500m apart. The towers have heights of 75 metres for double circuits and 54 metres for
single circuits, Onohaebi and Odiase (2010). Figures 2.1 and 2.2 show the 330 KV double
circuit and single circuit transmission line towers respectively.
16
-
.
17
-
.
18
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The Nigerian transmission network comprises of over 11000km of transmission lines. (i.e
over 5000km of 330KV transmission lines and 6000km of 132KV transmission lines). It
also has about 24000km of 33KV subtransmission lines and 19000km of 11KV distribution
lines together with 22500 substations all over the country, Atandare (2007) and Onohaebi
and Odiase (2010). The National Electric Power Authority (NEPA), now Power Holding
Company of Nigeria (PHCN), had built twenty three 330KV and ninety 132KV transmission
substations as at 1992 and all these trasmission lines and substations are put into operation
nationwide, TCN Reports, (2006).
With all these in place, there are still a lot of problems with the transmission of electricity
in Nigeria. Loss of power on transmission lines is a global problem and this is a major
problem we have with the transmission of electricity in Nigeria. The Nigerian 330 KV
transmission grid is characterized by high power losses. Most of these power losses are
due to very long transmission lines. Some of these lines include, Benin to Ikeja West (280
km), Osogbo to Benin (251 km), Osogbo to Jebba (249 km), Jebba to Shiroro (244 km),
Birnin Kebbi to Kainji (301 km), Jos to Gombe (265 km) and Kaduna to Kano (230 km),
Onohaebi and Odiase (2010). Distance is not the only factor responsible for loss of power on
transmission lines. Other factors include, the type and size of the conductor, enviromental
factors such as temperature, air density factor etc.
The power loss in Nigerian transmission system was estimated at 337.5 GWH in 2005.
High power losses in an electrical system imply high nancial losses to the nation. The nan-
cial loss associated with the loss in power in 2005 was estimated at 2.6 billion Naira, Kuale
and Onohaebi (2007). In order to maintain a good electric power system, the power losses
on transmission lines must be minimal. Minimal losses will help to ensure that generators,
transformers, lines, etc are subjected to less stresses, Onohaebi and Odiase (2010). Power
generation in a system and the cost involved in the generation will be reduced if the total
losses in transmission are minimal. This is because power generation must meet with load
demands as well as losses, Mehta and Mehta (2008), Wadhwa (2009) and Atandare (2007).
2.2
ELECTRIC SUPPLY SYSTEMS
The convayance of electric power from a power station to consumwers premises is known
as electric supply system. Therefore an electric supply system consists of three main compo-
19
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nents which include the power stations, the transmission system and the distribution system.
Electric power is produced at power stations which are usually located far away from con-
sumers. It is then stepped up and transmitted over long distances from the power stations
to load centres by means of conductors known as transmission lines. We have primary and
secondary (or sub-) transmission stages. Finally, power is distributed to a large number
of consumers through a distribution network. We also have primary and secondary (sub-)
distribution stages. The electric supply system can be broadly classied into:
i. Alternating Current and Direct Current Systems
ii. Overhead and Underground Systems.
2.2.1
Alternating Current and Direct Current Transmission Sys-
tems
Electrical power can be transmitted and distributed by either alternating current (AC) or
direct current (DC) systems but in practice 3-phase, 3-wire AC system is generally used
for transmission of large blocks of power and 3-phase, 4-wire AC system is used for the
distribution of electric power. The main advantage of AC transmission system is that voltage
can be stepped up at generating end by means of step up transformers to a desired value for
transmission purposes and then stepped down at the distributing end by means of step down
transformers for distribution purposes. This permits the transmission of electric power at
high voltage. Apart from this, the maintenance of AC sub-stations is easier and cheaper.
Also in AC transmission system, electric power can be generated at high voltages easily,
Gupta (2008), Mehta and Mehta (2008). The AC system also has its own disadvantages
which include the following:
i. An AC line requires more copper than a DC line
ii. In overhead transmission lines, spacing between the conductors is always kept more in
order to provide adequate insulation and avoid corona loss.
iii. The construction of an AC transmission line is more complicated than the one for a
DC transmission line.
iv. The eective resistance of the transmission line is increased because of skin eect in
AC line.
20
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v. AC transmission line has capacitance. Therefore there is a continuous loss of power
due to charging current even when the line is open
Transmission of electric power by high voltage DC system is superior to that of AC system
because of the following reasions.
i. There is no skin eect in a DC system. This enables the entire cross-section of the
conductor to be utilized.
ii. It requires only two conductors for transmission as against three for the AC system.
iii. There is less corona loss in a DC line. Therefore there is less interference with com-
munication circuits.
iv. For the same operating voltage, the stress on the insulation is less in a DC line than
in an AC line. This implies that a DC system requires less insulation.
v. There is no inductance, capacitance and surge problems in a DC transmission.
A major disadvantage of a DC system is that the DC voltage cannot be stepped up for
transmission of power at high voltages. Another disadvantage is that electric power cannot
be generated at high DC voltage.
It is clear from the above explanations that high voltage DC transmission is better than
high voltage AC transmission even though transmission of electricity is being done at present
in most countries by AC system. Therefore there is an increasing interest by engineers in
DC high voltage transmission of electricity. The introduction of mercury arc rectiers and
thyratrons have made it possible to convert AC to DC and vice versa. This arangement
will now enable generation and distribution of electricity to be done by AC system and high
voltage transmission of electricity to be done by DC system.
2.2.2
Overhead and Underground Systems
Electric power can be transmitted or distributted either by means of overhead lines or by
underground cables. The underground cables are rarely used for power transmission because
of the following reasons. In the rst place, power is generally transmitted over long distances
to load centres so the installation costs for underground transmission will be very high. The
initial installation costs of underground system is almost double that of overhead system.
21
-
Secondly electric power has to be transmitted at high voltages for economic reasons. It
will therefore be very dicult to provide proper insulation for the cables to withstand the
high pressures. The underground system cannot be operated above 66 KV because of the
insulation problem whereas overhead transmission system can be designed to operate at 400
KV or above, Gupta (2008). With the continuous rise in voltage level as a result of increase
in power demand, power transmission by overhead transmission lines is now the order of
the day. Another advantage of overhead transmission system over underground system is
that overhead system is more exible than underground system. In overhead system, new
conductors can be laid along with the existing ones for load expansion. In underground
transmission systems such new conductors needed for load expansion will be laid in new
channels. Though there are very rare chances of faults occuring in undergroung systems, if
it occurs it is always very dicult to locate and more expensive to repair than in overhead
systems. The underground system also has its own advantage over the overhead system
which include the following:
i. The underground system is safer than the overhead system.
ii. The maintenance cost of underground system is very low compared to that of overhead
system.
iii. In underground systems there is no interference to communication circuits.
iv. Because of less spacing between conductors in underground systems, the inductance
on the line is very low and therefore voltage drop is low in underground cables than
overhead cables.
v. Underground transmission and distribution systems are neater because no wire is vis-
ible outside.
vi. There are very few chances of faults in underground system.
vii. Underground system is free from interruption of services on account of thunder storm,
lightning or objects falling across the wires.
22
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2.3
MECHANICAL REQUIREMENTS FOR OVER-
HEAD LINES
Transmission line is a very important link between generating stations and major load
centres because power from generating stations is transmitted at high voltage over long
distances to these load centres. It has now become imperative that transmission of power
is carried out with minimum loss and disturbance because of the increase in the demand
for power as a result of industrial growth. To achive this goal, the transmission line should
be designed and constructed in such a way that the current carring capacity would be high
so as to transmit the required power over a given distance without much voltage drop and
overheating. The losses on the line should be small and the insulation of the line should
be enough to cope with the high voltage in the system. An overhead transmission line is
subjected to uncertain weather conditions and other external interference. This now calls for
the use of proper mechanical factors to give the transmission system sucient mechanical
strength so that it will be technically sound, reliable and ecient. In general, the strength
of the line should be such as to cope with the worst probable weather conditions and provide
satisfactory service over a long period of time without too much maintenance.
2.4
MAIN COMPONENTS OF OVERHEAD LINES
The main components of overhead lines are:
i. Conductors
ii. Line supports
iii. Insulators
iv. Cross-arms
v. Guys and Stays
vi. Miscellaneous Components of Overhead Lines which include: lightning arrestors, fuses
and isolating switches, barbed wires, danger plates, continuous earth wires, vee-guards,
guard wires and bird guards.
23
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2.4.1
Conductors
The conductor is one of the most important items in the transmission of electric power.
Therefore proper choice of material and size of the conductor is of considerable importance.
The conductor materials used in the transmission of electricity should have the following
properties:
i. high electrical conductivity;
ii. high tensile strength (in order to withstand mechanical stresses);
iii. low specic gravity (so that weigth per unit volume is small); and
iv. low cost (so that it can be used for long distances).
All the above properties are not found in a single material. Therefore, while selecting the
conductor material for a particular transmission purpose, a compromise is made between
the cost and the required mechanical and electrical properties.
All conductors used for overhead transmission lines are preferably stranded in order to
increase its exibility. Solid wires are only used as conductors when the cross-sectional area
needed is small and the conductor is for a short distance. If solid wires are used for larger
cross-section and very long distances, continuous vibrations and swinging would produce
mechanical fatigue and the wire would fracture at the point of support, Mehta and Mehta
(2008). In stranded conductors, there is generally one central wire and round this wire we
have successive layers of wires containing 6, 12, 18, 24, 30, ....... wires.
Copper is an ideal material for the transmission of electric power because of its high
electrical conductivity, lower electrical resistivity, high current density and greater tensile
strength. However, because of its high cost and non-availability, it is rarely used for the
purpose.
Aluminium is cheap, light and has a lower electrical conductivity, higher electrical re-
sistivity, lower current density and tensile strength as compared to copper. Aluminium is
also available for use in abundance. The smaller conductivity of aluminium implies that, for
any particular transmission eciency, the cross-sectional area of conductor must be greater
in aluminium than in copper. In fact, the diameter of aluminium conductor will be about
1.26 times the diameter of copper conductor, Mehta and Mehta (2008). The specic gravity
24
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of aluminium (2.71 gm/cc) is less than that of copper (8.9 gm/cc). The increased cross-
sectional area of aluminium exposes a greater surface of it to wind pressure and its lightness
made it liable to greater swings and hence larger cross-arms are required. Due to lower
tensile strength and higher co-ecient of linear expansion of aluminium, the sag is greater
in aluminium conductors than copper.
Considering the combined properties of cost, resistivity, conductivity, availability, ten-
sile strength, weight etc., aluminium has an edge over copper. Therefore, aluminium is
widely used as a conductor material for transmission purposes. But due to its low tensile
strength, aluminium conductors generally produce greater sag. In order to increase the
tensile strength, aluminium conductors are normally reinforced with a core of galvanised
steel wires. The composite conductor that is formed with this reinforcement is known as
Aluminium Conductor Steel Reinforced, (ACSR). It will now comprise of central core of
galvanised steel wires surrounded by a number of aluminium strands. For better tensile
strength, the diameters of both steel and aluminium wires are the same and the cross
section of the two metals are generally in the ratio of between 1:6 and 1:4. With this ar-
rangement, the steel core takes greater percentage of mechanical strength while aluminium
strands carries the bulk of current. The Nigerian 330 KV transmission network employed
350mm2 aluminium conductor steel reinforced (ACSR).
2.4.2
Line Supports
The main function of line support is to assist the conductors in a way to keep them at an
appropriate level above the ground. Line support must be capable of carrying insulator and
conductors load as well as loads due to wind. The line support for long distance transmission
at higher voltage is usually steel towers. This is because of its high mechanical strength and
longer life span than any other line supports. Also, it can withstand most of the severe
climatic conditions and it permits the use of longer spans. Therefore the risk of interrupted
service due to broken insulation is drastically reduced because of the longer span. The heigth
of steel towers depends on line voltage and the length of span. In Nigeria, double circuit
and single circuit steel towers are used with heights of 75 metres and 54 metres respectively,
Onohaebi and Odiase (2010). Reinforced Concrete (RCC) poles, steel poles and wooden
poles are used as supports for distribution of low voltage of up to 11 KV.
25
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2.4.3
Insulators
The current along the conductors in the overhead transmission lines should not be allowed
to ow to the earth through the line supports. This implies that the conductors should be
properly insulated from the line supports. The insulators provides appropriate insulation
between the conductors and the line support. It therefore prevents any leakage of current
from the conductors to the earth. Air is a general insulator for overhead lines. The most
commonly used material for the insulation of overhead lines is porcelain. Glass and steatite
are occassionally used as insulator materials. Porcelain is stronger mechanically than glass
and steatite. It is also less aected by temperature changes. To be able to function eectively,
a very good insulator should have the following properties:
i. An insulator should have high electrical resistance in order to prevent leakages of
current to the earth.
ii. It should have high mechanical strength in order to withstand wind load and conductor
load.
iii. It should have high relative permittivity so that the dielectric strength will be high.
iv. The insulator materials should be non-porous in order not to lower the permittivity.
2.4.4
Cross-arms
The function of cross-arms is to keep the conductors at a safe distance from each other and
also from the poles. It is a cross-piece tted to the end portion of the top of the pole by means
of brackets. These brackets are known as pole brackets and are general used for supporting
insulators. Steel cross-arms are generally used for steel poles because they are stronger than
any other cross-arms. There are various other types of cross-arms like MS channel, angle
iron or angle wooden which are used for 11 KV and 33 KV lines. Cross-arms are also of
various shapes which include U-shape, V-shape, straigth or zig-zag shape. The length of
cross-arms should be suitable enough for the spacing of the conductors. The cross-arms
should also be strong enough to withstand the resultant forces caused by insulators.
26
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2.4.5
Stays
These are braces or cables that are fastened to the pole, at the terminal end, at a very good
angle to resist forces. This becomes essential in order to enable the overhead line supports
to stay at a very good position to withstand the pull by conductors and other lateral forces.
The theoretical angle between the stay and the pole should be 450. But in general practice,
it is not always possible to achive this, so stay designs are based on a minimum angle of 300
between the pole and the stay.
2.4.6
Miscellaneous Components of Overhead Lines
Other components of overhead lines which include lightning arrestors, fuses and isolating
switches, barbed wires, danger plates, continuous earth wires and guard wires, are discussed
below.
i. Lightning Arrestors - This is a device to discharge excessive voltages due to lightning
built upon the line to the earth.
ii. Fuses and Isolating Switches - These are to isolate dierent parts of the overhead
system
iii. Barbed Wires - Barbed wires are wrapped on poles at a height of about 2.5 metres
from the ground. This will prevent climbing of the poles by unauthorised people.
iv. Danger Plates - It is provided on poles as a warning measure to indicate the working
voltage of the line together with the word danger. It is posted at a heigth of about
2.5 metres above the ground.
v. Continuous Earth Wire - Countinuous earth wire is generally run on top of the towers
to protect the transmission line against lightning discharges.
vi. Guard Wires - Guard wires, which are solidly connected to the earth, are provided
above and below power lines while crossing telephone or telegraph lines.
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2.5
TRANSMISSION LINE CONSTANTS
Transmission lines are basically electrical circuits having distributed constants (or parame-
ters). These constants includes:
i. Line Resistance
ii. Line Inductance
iii. Line Capacitance
iv Shunt Conductance
The performance of a transmission line depends upon these constants to a considerable
extent.
2.5.1
Line Resistance
Every electric conductor oers opposition to the ow of current and this opposition is called
the resistance (R) of the conductor. The resistance is distributed uniformly along the whole
length of the line. The resistance of transmission line conductors, against current ow, is
the most important cause of power loss in transmission line and this aects the transmission
eciency of the line, Mehta and Mehta (2008) and Wadhwa (2009). The resistance of a line
conductor having resisitivity (), length (L) and cross-sectional area (A) is given by
L R = [ ]
A
(2.1)
2.5.2
Line Inductance
Series inductance (L) mainly governs the power transmission capacity of the line. When
an alternating current ows through a conductor, a charging ux is set up which links the
conductor, Mehta and Mehta (2008). The conductors therefore posses inductance due to
these ux leakages. The inductance is also uniformly distributed along the whole length of
the transmission line. Inductance oers opposition to the ow of varying current in a circuit,
Mehta and Mehta (2008). This is dierent from resistance which oers opposition to the
ow of both steady (direct) and varying (alternating) current. The opposition to the ow
28
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of varying current, as a result of inductance, is called voltage drop. Inductance is generally
dened as ux per unit current. That is
where represents Flux leakage
and
I represents Current
2.5.3 Line Capacitance
L =
I
(2.2)
Shunt capacitance (C) causes a charging current to ow in the transmission line. Any two
conductors separated by an insulating medium constitute a capacitor or a condenser, Mehta
and Mehta (2008) and Wadhwa (2009). As we know, any two conductors of an overhead
transmission line are separated by air which acts as insulation, therefore, capacitance exists
between any two overhead line conductors. The capacitance is uniformly distributed over
the total length of the transmission line. It may therefore be regarded as a uniform series
of condensers that are connected between the conductors. Capacitance is generally dened
as charge per unit potential dierence. That is,
C = q v
(2.3)
where
q represents charge on the transmission line
and
v represents Potential dierence between the conductors
2.5.4
Shunt Conductance
The shunt conductance (G) is mostly due to leakages over the insulator and is always very
small, Mehta and Mehta (2008). Just like any other transmission parameters, it is also
uniformly distributed over the total length of the transmission line.
2.6
SKIN EFFECT
Current is uniformly distributed over the whole cross-section of the conductor when a
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conductor is carrying steady direct current (DC). But in alternating current (AC) the ow
of current is not unformly distributed. In fact, in an AC system, no current ows through
the core of the conductor as most current concentrates near the surface of the conductor as
frequency of transmission increases. This is as a result of the fact that a solid conductor
usually consists of a large number of strands each carrying a small part of the current.
Normally, the inductance of each strand will vary with its position. Therefore, the strand
near the centre is surrounded by greater magnetic ux than the one at the surface. Hence
the strand at the centre has greater inductance than the one at the surface. The high
reactance of the inner strands causes the alternating current to ow near the surface of the
conductor particularlly when the transmission frequency is high, Mehta and Mehta (2008),
Gupta (2008).
When an electromagnetic wave interacts with a conductive material, mobile charges
within the material are made to oscillate. The movement of these mobile charges (which are
usually electrons) constitute an alternating electric current. As the frequency of the current
increases, current density tends to decrease in the central axis of the conductor and increase
near the surface of the conductor. That is, the electric current tends to ow at the skin of
the conductor at an average depth called the skin depth. The skin depth is a measure of the
distace over which the current falls to 1 e (about 0.37) of its original value. This phenomenon
is known as skin eect. Skin eect will cause a decrease in the eective cross-sectional
area of the conductor and hence increase the resistance of the conductor. An increase in
the resistance of the conductor will consequently increase the ohmic or line losses of the
transmission line.
2.7
ECONOMICS OF POWER TRANSMISSION
The commercial aspect of the design of power transmission is very essential to an electrical
engineer. He must design the various aspect of the transmission scheme in a way to achieve
maximum economy. Two fundamental economic principles which inuences the electrical
design of a transmission line are:
i. Economic choice of conductor size
ii. Economic choice of transmission voltage
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2.7.1
Economic Choice of Conductor Size
The determination of proper size of conductor for the transmission line is of great importance
because the cost of conductor material is a very considerable part of the total cost of a
transmission line. The most economical area of conductor is that for which the total annual
cost of transmission line is minimum. This is known as the Kelvins law, Mehta and Mehta
(2008). The total annual cost of transmission line is a function of the annual charge on
capital outlay and annual cost of energy wasted in the conductor.
2.7.2
Economic Choice of Transmission Voltage
We all know that if transmission voltage is increased, the volume of conductor material
required is reduced and this will denitely decrease the expenditure on the conductor ma-
terial. It should also be noted that, an increase in the transmission voltage will lead to
a rise in the cost of transformers, switchgear, insulation materials for the conductor and
other terminal apparatus of the line. Therefore, there is an optimum transmission voltage
for every transmission line beyond which there is nothing to gain in terms of economy. The
transmission voltage where the costs of conductors, insulators, switchgear, transformer and
other terminal apparatus is minimum is called Economical Transmission Voltage (ETV).
2.8
CORONA PHENOMENON
When an alternating potential dierence is applied across two conductors whose spacing is
large as compared to their diameters, then the atmospheric air surrounding the conductor
is subjected to electro-static stresses. At low voltage there is no apparent change in the
condition of the atmospheric air around the conductors. However, when the applied voltage
is gradually increased and it exceeds a certain value called the critical disruptive voltage then
the conductors are surrounded by a faint violet glow. This phenomenon is called corona and
is accompanied by the production of ozone, hissing sound, power loss and radio interference.
The higher the voltage is raised, the higher and larger the luminous envelops become and
the greater the hissing noise, the power loss and the radio interference. The production of
ozone is readily detected because of its characteristic odour. The glow is due to the fact that
the atmospheric air around the conductor becomes conducting due to electro-static stresses.
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The phenomenon is very much evident in transmission lines of 100 KV and above. If the
conductors are polished and smooth, the corona glow will be uniform throughout the length
of the conductors, otherwise the rough points will appear brighter.
2.8.1
Factors Aecting Corona
Since corona occurs as a result of the ionization of the air surrounding the line conductors,
it is aected by the physical state of the atmosphere as well as by the condition of the line.
The following are the factors upon which corona depends
2.8.1.1 Atmosphere
Since corona is caused by the bombardment of molecules with subseqent dislodging
of electrons by ionised particles, it will denitely be aected by the physical state of
the atmosphere. The voltage gradient for the breakdown of the air is proportional
to its density. In the stormy weather, the number of ions will be more than normal,
therefore corona may occur at much less voltage than in fair weather.
2.8.1.2 Conductors Size, Shape and Condition
The corona is greatly aected by the size, shape and surface condition of the conductor.
An irregular or rough surface will give rise to more corona. Therefore a stranded
conductor will have more corona eects than a solid conductor because of its irregular
surface. The corona decreases with increasing diameter of conductor.
2.8.1.3 Spacing between Conductors
An increase in the spacing between conductors reduces the electro-static stresses. This
therefore reduces the corona eect. If the spacing between the conductors is made very
large as compared to their diameter, there may not be any corona eect.
2.8.1.4 Line Voltage
The line voltage considerably aects corona. If it is low, there is no change in the
condition of air surrounding the conductors and hence no corona is formed. But when
the line voltage is increased to such a value that electro-static stresses developed at the
conductor surfaces, then corona will occur because the atmospheric air surrounding
the conductor will start conducting.
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2.8.2
Advantages and Disadvantages of Corona
Corona eect has advantages and disadvantages. An electrical engineer has to strike a
balance between the advantages and the disadvantages in order to design a very good high
voltage transsmission line. The advantages include
i. Corona usually reduces the eects of transients produced by surges.
ii. As a result of corona formation, the air surrounding the conductor becomes conducting
and hence the diameter of the conductor is increased. This increase in diameter reduces
electro-static stresses between the conductors.
Corona eect also has the following disadvantages
i. Ozone is produced by corona and this may cause corrosion of the conductor due to
chemical action.
ii. Corona is accompanied by a loss of energy and this greatly aects the transmission
eciency of the line.
2.8.3
Methods of Reducing Corona
Intense corona eects are observed at an operating voltage of 33 KV and above. Therefore
careful design should be made to avoid corona on the sub-station rated for 33 KV and higher
voltages. The following methods can be used to reduce corona
i. By increasing conductors size so that the voltage at which corona occurs is raised.
This will reduce the eect of corona
ii. By increasing the spacing between conductors, the voltage at which corona occurs is
also raised to reduce corona eects. It is to be noted that there is a limit to which we
can increase the spacing between conductor as this may cause an increase in the cost
of supporting structures considerably.
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Chapter 3
MATHEMATICAL MODELS FOR
POWER FLOW OVER
TRANSMISSION LINES
3.1
3.1.1
MATHEMATICAL PRELIMINARIES Modelling
A model can be described as a representation of real life problems in a simplied form.
A mathematical model is a model developed using mathematical concepts like equations,
variables, operators, etc, Dilwyn and Hamson (1993), Ruhul and Charles (2008). It is often
desirable to describe the behavior of some real life phenomenon or system, whether physical,
sociological, ecological, scientical, technological or even economical, in mathematical terms.
The mathematical desciption of a system or phenomenon is called a mathematical model and
is constructed with certain goals in mind, Ruhul and Charles (2008), Dennis and Michael
(2005). Thus, mathematical modelling is the art of translating real life problems from an
application area into tractable mathematical formulations whose theoretical and numerical
analysis provides insight, answers and guidance useful for the originating application, Arnold
(2003). Hence, mathematical modelling serves as a bridge between the study of mathematics
and the applications of mathematics to various elds of human endeavous, and is an essential
part of the process of solving real life problem optimally, Ruhul and Charles (2008). An
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empirical model is a model developed from and based entirely on data. In this kind of model,
relationships between variables are derived by looking at the data available on the variables
and developing a mathematical form which is a compromise between accuracy of t and
simplicity of mathematical representation, Dilwyn and Hamson (1993). Empirical models
are not based on physical laws or principles neither are they derived from assumptions
concerning the variables, Dilwyn and Hamson (1993).
In this chapter, we developed mathematical models of electric power ow along trans-
mission lines. We developed a mathematical model for power losses along tranmission lines
in chapter four. Also in chapter four, we developed empirical models of power losses for
dierent loads along transmission lines as functions of distance.
3.1.2
Dierential Equations
A Dierential Equation (DE) is an equation containing the derivatives of one or more depen-
dent variables, with respect to one or more independent variables. Dierential equations are
of fundamental importance in engineering because many physical laws and relations appear
mathematically in the form of dierential equations, Kreyszig (1987), Khorasani and Adibi
(2003). The order of a DE is the order of the highest dierential coecient contained in it.
The power to which the highest derivative is raised is called the degree of the DE.
An Ordinary Dierential Equation(ODE) is an equation containing derivatives of one
or more dependent variables with respect to a single independent variable. An equation
involving partial derivatives of one or more dependent variables with respect to one or
more independent variables is called a Partial Dierential Equation(PDE). The independent
variables can be anything such as time, velocity, distance, etc. In most of the applications
of control systems engineering, the independent variable is time, Matilde, Jose and Sanchez
(2009), Otarod and Khodakarim (2008).
An nth-order ordinary dierential equation given by
F (x, y, y , y , ..., yn) = 0
is said to be linear if F is linear in y, y , y , ..., yn. This implies that the dependent variable y
and all its derivatives are of the rst degree. Also for linearity of the dierential equation, the
coecients of the dierential equation must depend at most on the independent variable. A
non linear ordinary dierential equation is just an ordinary dierential equation that is not
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linear. In this case, non linear functions of the dependent variable or its derivatives can occur
in the equation and the coecients can be functions of both dependent and independent
variables.
An nth-order ODE is said to be nonhomogeneous if
F (x, y, y , y , ..., yn) = g(x).
. If g(x) = 0 then the dierential equation is said to be homogeneous. The models of
the electric power ow along a transmission line are in form of homogeneous second order
partial dierential equations, which are then transformed into a non-homogeneous ordinary
dierential equation by making use of Laplace transformation.
3.1.3
Laplace Transformation
A function F(s) dened by the integral
F (s) = f (t)estdt 0
is called the Laplace transform of the function f(t) and is usually denoted by
F (s) = L[f (t)].
The Laplace transform of f(t) is said to exist if
f (t)estdt 0
converges for some values of s. f(t) is called the inverse Laplace transform of F(s) and is
usually denoted by
f (t) = L1[F (s)].
The Laplace transformation is a method for solving dierential equations and corresponding
initial and boundary value problems. It will transform initial and boundary value ordinary
dierential equations into algebraic equations, Gupta (2009), Stroud and Dexter (2003),
Kreyszig (1987) and Binoy (2009). It will also transform initial and boundary value partial
dierential equations into ordinary dierential equations, Kreyszig (1987), Murray (1967)
and Luke (1982). The Laplace transform method is widely used in engineering. We applied
it to solve the model for electric power ow along transmission lines.
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3.2
KIRCHOFFS CIRCUIT LAWS
In 1845, a German physicist, Gustav Kircho, rst described two laws that became central
to electrical engineering. The laws were generalized from the work of George Ohm. The
laws can also be derived from Maxwells equations, but were developed prior to Maxwells
work. The Kircho s circuit laws, or simply Kircho s rules, deal with the conservation of
charge and energy in electrical circuits. The two laws are the Kircho s current law and
Kircho s voltage law which are described below.
In this chapter, we applied these two Kircho laws to the equivalent circuit of transmis-
sion lines and then we formulated the model for power ow along transmission lines.
3.2.1
Kircho s Current Law
Kircho s current law (KCL), also known as Kircho s Junction Law, Kircho s Point
Rule, Kircho s Nodal Law or Kircho s First Law, denes the way that electrical current
is distributed when it crosses through a junction. Specically, the law states that: The
algebraic sum of currents in a network of conductors meeting at a junction is zero. That is,
n
(Ik) = 0 k=0
where n is the total number of branches in which current is owing. Since current is the ow
of electrons through a conductor, it cannot build up at a jun