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
CERTIFICATION
This is to certify that the research work reported in this thesis was carried 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
Date
(Supervisor)
........................................
.......................................
Professor M.O. Ibrahim
Date
(Head of Department)
........................................
.......................................
(External Examiner)
Date
ii
DEDICATION
This work is dedicated to my late father: Pa David Eniola Oke.
iii
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 sojourn 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, encouragement 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 encouragement.
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 Nigeria 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.
iv
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 staff of the Department of Mathematics, 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. FadipeJoseph 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 qualifies me for the postgraduate 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 research work.
v
TABLE OF CONTENT
page
TITLE PAGE
i
CERTIFICATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
TABLE OF CONTENT
vi
LIST OF TABLES
x
LIST OF FIGURES
xi
ABSTRACT
xiii
CHAPTER ONE : GENERAL INTRODUCTION
1.1 BACKGROUND TO THE STUDY
1
1.2 GOAL AND OBJECTIVES OF THE STUDY
4
1.3 SIGNIFICANCE OF THE STUDY
5
1.4 ORGANIZATION OF THE THESIS
5
1.5 NOTATIONS
6
1.6 DEFINITION OF SOME BASIC TERMS
7
CHAPTER TWO : ELECTRIC POWER TRANSMISSION SYSTEMS
2.1 ELECTRIC POWER SYSTEMS
11
2.1.1 Historical Developments
11
2.1.2 Importance of Electric Power System
12
2.1.3 Electric Power Systems in Nigeria
13
vi
2.2 ELECTRIC SUPPLY SYSTEMS
19
2.2.1 Alternating Current and Direct Current Transmission Systems
20
2.2.2 Overhead and Underground Systems
21
2.3 MECHANICAL REQUIREMENTS FOR OVERHEAD LINES
23
2.4 MAIN COMPONENTS OF OVERHEAD LINES
23
2.4.1 Conductors
24
2.4.2 Line Supports
25
2.4.3 Insulators
26
2.4.4 Cross-arms
26
2.4.5 Stays
27
2.4.6 Miscellaneous Components of Overhead Lines
27
2.5 TRANSMISSION LINE CONSTANTS
28
2.5.1 Line Resistance
28
2.5.2 Line Inductance
28
2.5.3 Line Capacitance
29
2.5.4 Shunt Conductance
29
2.6 SKIN EFFECT
29
2.7 ECONOMICS OF POWER TRANSMISSION
30
2.7.1 Economic Choice of Conductor Size
31
2.7.2 Economic Choice of Transmission Voltage
31
2.8 CORONA PHENOMENON
31
2.8.1 Factors Affecting Corona
32
2.8.2 Advantages and Disadvantages of Corona
33
2.8.3 Methods of Reducing Corona
33
vii
CHAPTER THREE : MATHEMATICAL MODELS FOR POWER
FLOW OVER TRANSMISSION LINES
3.1 MATHEMATICAL PRELIMINARIES
34
3.1.1 Modelling
34
3.1.2 Differential Equations
35
3.1.3 Laplace Transformation
36
3.2 KIRCHOFF’S CIRCUIT LAWS
37
3.2.1 Kirchoff ’s Current Law
37
3.2.2 Kirchoff ’s Voltage Law
37
3.3 MATHEMATICAL MODEL FOR ELECTRIC POWER FLOW ALONG
LOSSY TRANSMISSION LINES
38
3.3.1 Model Formulation
38
3.3.2 Model Solution
40
3.4 MATHEMATICAL MODEL ALONG TRANSMISSION LINES WHEN
LEAKAGE TO GROUND IS SMALL
43
3.4.1 Model Formulation
43
3.4.2 Model Solution
44
3.5 ANALYSIS OF MATHEMATICAL MODELS
46
CHAPTER FOUR : MINIMIZATION OF POWER LOSSES OVER
TRANSMISSION LINES
4.1 OHMIC AND CORONA LOSSES
47
4.1.1 Ohmic Loss
47
4.1.2 Corona Loss
48
4.2 MATHEMATICAL MODELS FOR POWER LOSSES
48
viii
4.2.1 Model Based on Ohmic and Corona Losses
48
4.2.2 Empirical Models as Functions of Distance
50
4.3 MULTIVARIABLE OPTIMIZATION WITHOUT CONSTRAINTS 71
4.3.1 Properties of Hessian Matrix
71
4.3.2 Necessary and Sufficient Conditions for the Existence of Extremal
Points
72
4.4 MINIMIZATION OF POWER LOSSES
78
4.5 DISCUSSION ON RESULTS
79
CHAPTER FIVE : GENERAL CONCLUSION
5.1 SUMMARY OF THESIS
80
5.2 SUMMARY OF RESULTS
80
5.3 CONCLUSION
81
5.4 RECOMMENDATION
82
REFERENCES
83
ix
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
52
Table 4.3: Summations for a Load of 100 MW on Single Circuit
55
Table 4.4: Summations for a Load of 200 MW on Single Circuit
57
Table 4.5: Summations for a Load of 300 MW on Single Circuit
61
Table 4.6: Summations for a Load of 100 MW on Double Circuit
65
Table 4.7: Summations for a Load of 200 MW on Double Circuit
67
Table 4.8: Summations for a Load of 300 MW on Double Circuit
70
x
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.
18
Figure 3.1: Equivalent Circuit of a Transmission Line
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
63
Figure 4.8: Graph of Power Losses in MW for a load of 100 MW on
Double Circuit
63
Figure 4.9: Scatter Diagram for Power Losses in MW for a load of 200
xi
MW on Double Circuit
66
Figure 4.10: Graph of Power Losses in MW for a load of 200 MW on
Double Circuit
66
Figure 4.11: Scatter Diagram for Power Losses in MW for a load of 300
MW on Double Circuit
69
Figure 4.12: Graph of Power Losses in MW for a load of 300 MW on
Double Circuit
69
xii
ABSTRACT
Availability of electric power has been the most powerful vehicle for facilitating economic, industrial and social developments of any nation. Electric 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 final 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 using 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 flow 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 transmission power losses. Specifically, the objectives of the study were to:
(i.) develop mathematical models for power flow and power losses along
electric power transmission lines and solve the mathematical models
for electric power flow 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 Kirchoff ’s circuit laws and a combination
xiii
of corona and ohmic losses in obtaining the mathematical models for the
power flow and power losses respectively. Empirical models of the power
losses were developed using regression analysis.
The findings of this study were:
(i.) the models for power flow along transmission lines evolved as homogeneous second-order partial differential 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 consumers. Hence, this research work has addressed the problem of minimizing
electric power losses during transmission.
Xiv
MINIMIZATION OF POWER LOSSES OVER ELECTRIC
POWER TRANSMISSION LINES
1
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 different
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 transmission lines and the ohmic power loss. The corona power loss and ohmic power loss were
measured and computed for different transmission line configurations 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
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 flow based on swarm intelligences. Thabendra et al. (2009) considered multi-objective optimization methods for
power loss minimization and voltage stability while Abdullah et al. (2010) looked at transmission 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 find 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 existing 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 reconfiguration 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 flow of power on electrical networks. Pandya
and Joshi (2008) presents a comprehensive survey of various optimization methods for solving
optimal power flow problems. The methods considered in the work include linear programming, Newton-Raphson, quadratic programming, nonlinear programming, interior point and
artificial intelligence. Under the artificial intelligence method, the following were also considered artificial 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
limitations. In most cases, mathematical formulations have to be simplified to get the solutions 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 flow problems. It was also discovered in the paper that
the artificial intelligence methods are relatively versatile for handling various qualitative
constraints and that the methods can find 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 flow formulations while Claudio et al. (2001)
worked on comparison of voltage security constraint using optimal power flow techniques.
Roya et al. (2008) considered power flow modelling for power systems with dynamic flow
controller. Other researchers who also worked on power flow 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 identified, the first for load shedding 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
first control variable can only be on or off as the case may be. The optimality conditions
for the system were imposed implicitly on the controls and the mathematical model represents 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) considered an analysis of blackout for electric power transmission systems while Aderinto et
3
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), Kaminskyi (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 reconfiguration and evolutionary techniques.
Loss minimization is a critical component for efficient 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 efficiency 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 flow and power losses along electric
power transmission lines;
4
(ii.) Solve the mathematical models for electric power flow along transmission lines analytically;
(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 flow along transmission lines provides a better understanding of the flow 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
definition 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 flow over transmission lines. Mathematical preliminaries were considered in section 3.1. In section 3.2, we formulated and solved
the model for electric power flow along lossy transmission lines, while in section 3.3, we derived and solved the model for electric power flow 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. Specifically, 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
of power losses as functions of distance. In Section 4.3, we considered the power loss function 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 differential calculus. Discussion on results is what we have in section 4.5. The thesis
is rounded up in chapter five 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.
L−1 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.
δ 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 difference between the conductors.
V represents operating voltage.
V0 represents distruptive voltage.
ρ represents resistivity of the conductor.
ψ represents flux leakage.
L represents length of the conductor.
A represents cross-sectional area of the conductor.
6
σ 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 definition 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 defined 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 finding the optimum of
continuous and differentiable functions. It is an analytical method that makes use of
differential calculus techniques in finding the optimum points. The classical optimization 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
ate or non-morse critical point of f. Otherwise it is a non-degenerate or morse critical
point of f.
5. Positive Definite Matrix
A matrix A of order n is said to be positive definite if all its eigenvalues are positive.
That is, if all values of λ which satisfies the determinant equation
|A − λI | = 0
are positive, Rao (1998).
Another test of the positive definiteness of a matrix A of order n is the evaluation of
its determinants:
A1 =
a11
a11 a12
A2 =
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 definite if and only if all values of A1,
A2, A3, ....., An are positive.
6. Negative Definite Matrix
A matrix A of order n is said to be negative definite if and only if the signs of Ai in
(5) above is (−1)i for i = 1,2,3,4,.....,n.
8
7. Positive Semidefinite Matrix
A matrix A of order n is said to be positive semidefinite 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 λ which satisfies the determinantal equation
det (A − λI ) = |A − λI | = 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 differential equation in which the solution y(x)
satisfies prescribed side conditions imposed on the unknown y(x) or its derivatives at
an initial point x0 , Dennis and Michael (2005) and Eaglefield (1989). An initial value
problem is of the form
Solve
dn y
dx n
= f (x, y, y , y , ....., y(n−1))
(1.2)
subject to
y(x0) = y0, y (x0) = y1, y (x0) = y2, ....., y(n−1)(x0) = yn−1
(1.3)
where y0, y1, y2, ..., yn−1. are arbitrarily specified real constants.
The values of y(x) and its first (n - 1) derivatives at a single point x0 , that is y(x0) =
y0, y (x0) = y1, y (x0) = y2, ....., y(n−1)(x0) = yn−1 are called the initial conditions.
10. Boundary Value Problem
A boundary value problem (BVP) is a differential equation in which the solution y(x)
satisfies prescribed conditions imposed on the unknown y(x) or its derivatives at more
than one point. A differential equation of the form:
Solve
a2(x)
d2y
dx
2
dy
+ a0(x)y = g(x).
dx
+ a1(x)
(1.4)
subject to
y(a) = ya, y(b) = yb,
9
(1.5)
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 Differential Equations
An nth-order linear differential equation of the form in (1.6) below is said to be nonhomogeneous if g(x) is not identically zero, Dennis and Michael (2005).
an(x)
d ny
dx
n
+ a(n−1)(x)
d(n−1)y
dx
(n−1)
+ ... + a1
dy
+ a0(x)y = g(x).
dx
(1.6)
If g(x) is equal to zero, then the nth-order differential equation is called homogeneous
and we have
an(x)
dny
dx
n
+ a(n−1)(x)
d(n−1)y
dx
(n−1)
+ ... + a1
dy
+ a0(x)y = 0.
dx
(1.7)
This explanation also holds for partial differential 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 flow 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 experienced when the applied voltage exceeds the critical disruptive voltage in high voltage
transmission lines.
10
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 first formed in Europe and later in the United States of America. Scientific
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 fields were created by electric currents but no one had
discovered how electrical currents could be produced from magnetic fields. Faraday worked
on such problems between 1821 and 1831 and finally 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
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
first 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. Electrical 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 generated 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 modernisation, 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 different 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
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 purpose. Transmission lines are materials or media that are used to transmit electric energy and
signals from one point to another, specifically 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 final consumers. Power losses
along these stages should be reduced to the bearest minimum so that the final 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 connect 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 finding 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 first 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
few cities of the country by some isolated generating stations like Cameroon’s 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 Corporation of Nigeria (ECN) was establised and this gave birth to the Ijora Power Station which
had 10 MW coal-fired 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 construction 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 distribution 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 Corporation of Nigeria (ECN) and Niger Dams Authority (NDA). The decree gave NEPA the
mandate to maintain and co-ordinate an efficient electricity supply to all parts of the country. 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 located nationwide. These stations are connected by transmission substations to form the
National Grid System with the control centre at Osogbo, Osun State. These stations include three hydropower stations and five 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
14
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 Institute’s 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).
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
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
Company of Nigeria PLC (TCN) manages Nigerian’s power grid. TCN ensures that power
is transmitted efficiently 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 five 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 offices 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
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 financial losses to the nation. The financial 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 consumwer’s premises is known
as electric supply system. Therefore an electric supply system consists of three main compo-
19
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 consumers. 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 classified into:
i. Alternating Current and Direct Current Systems
ii. Overhead and Underground Systems.
2.2.1
Alternating Current and Direct Current Transmission Systems
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 effective resistance of the transmission line is increased because of skin effect in
AC line.
20
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 effect 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 communication 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 rectifiers 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 first 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 difficult 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 flexible 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 difficult 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 visible 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
2.3
MECHANICAL REQUIREMENTS FOR OVERHEAD 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 sufficient mechanical
strength so that it will be technically sound, reliable and efficient. 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
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 specific 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 flexibility. 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 resistivity, 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 efficiency, 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 specific gravity
24
of aluminium (2.71 gm/cc) is less than that of copper (8.9 gm/cc). The increased crosssectional 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-efficient of linear expansion of aluminium, the sag is greater
in aluminium conductors than copper.
Considering the combined properties of cost, resistivity, conductivity, availability, tensile 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 arrangement, 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
conductor’s 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
2.4.3
Insulators
The current along the conductors in the overhead transmission lines should not be allowed
to flow 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 affected by temperature changes. To be able to function effectively,
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 fitted 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
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 different 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.
27
2.5
TRANSMISSION LINE CONSTANTS
Transmission lines are basically electrical circuits having distributed constants (or parameters). 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 offers opposition to the flow 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 flow, is
the most important cause of power loss in transmission line and this affects the transmission
efficiency of the line, Mehta and Mehta (2008) and Wadhwa (2009). The resistance of a line
conductor having resisitivity (ρ), length (L) and cross-sectional area (A) is given by
L
R = ρ[ ]
A
2.5.2
(2.1)
Line Inductance
Series inductance (L) mainly governs the power transmission capacity of the line. When
an alternating current flows through a conductor, a charging flux is set up which links the
conductor, Mehta and Mehta (2008). The conductors therefore posses inductance due to
these flux leakages. The inductance is also uniformly distributed along the whole length of
the transmission line. Inductance offers opposition to the flow of varying current in a circuit,
Mehta and Mehta (2008). This is different from resistance which offers opposition to the
flow of both steady (direct) and varying (alternating) current. The opposition to the flow
28
of varying current, as a result of inductance, is called voltage drop. Inductance is generally
defined as flux per unit current. That is
L=
ψ
I
(2.2)
where ψ represents Flux leakage
and
I represents Current
2.5.3
Line Capacitance
Shunt capacitance (C) causes a charging current to flow 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 defined
as charge per unit potential difference. That is,
C=
q
v
where
q represents charge on the transmission line
and
v represents Potential difference 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
29
(2.3)
conductor is carrying steady direct current (DC). But in alternating current (AC) the flow
of current is not unformly distributed. In fact, in an AC system, no current flows 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 flux 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 flow 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 flow 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 effect”. Skin effect will cause a decrease in the effective 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 influences the electrical
design of a transmission line are:
i. Economic choice of conductor size
ii. Economic choice of transmission voltage
30
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 Kelvin’s 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 definitely decrease the expenditure on the conductor material. 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 difference 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.
31
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 Affecting Corona
Since corona occurs as a result of the ionization of the air surrounding the line conductors,
it is affected 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 definitely be affected 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 Conductor’s Size, Shape and Condition
The corona is greatly affected 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 effects 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 effect. If the spacing between the conductors is made very
large as compared to their diameter, there may not be any corona effect.
2.8.1.4 Line Voltage
The line voltage considerably affects 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.
32
2.8.2
Advantages and Disadvantages of Corona
Corona effect 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 effects 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 effect 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 affects the transmission
efficiency of the line.
2.8.3
Methods of Reducing Corona
Intense corona effects 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 conductor’s size so that the voltage at which corona occurs is raised.
This will reduce the effect of corona
ii. By increasing the spacing between conductors, the voltage at which corona occurs is
also raised to reduce corona effects. 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.
33
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 simplified 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, scientifical, 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 fields of human endeavous, and is an essential
part of the process of solving real life problem optimally, Ruhul and Charles (2008). An
34
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 fit 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 flow along transmission 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
different loads along transmission lines as functions of distance.
3.1.2
Differential Equations
A Differential Equation (DE) is an equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables. Differential equations are
of fundamental importance in engineering because many physical laws and relations appear
mathematically in the form of differential equations, Kreyszig (1987), Khorasani and Adibi
(2003). The order of a DE is the order of the highest differential coefficient contained in it.
The power to which the highest derivative is raised is called the degree of the DE.
An Ordinary Differential 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 Differential 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 differential 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 first degree. Also for linearity of the differential equation, the
coefficients of the differential equation must depend at most on the independent variable. A
non linear ordinary differential equation is just an ordinary differential equation that is not
35
linear. In this case, non linear functions of the dependent variable or its derivatives can occur
in the equation and the coefficients 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 differential equation is said to be homogeneous. The models of
the electric power flow along a transmission line are in form of homogeneous second order
partial differential equations, which are then transformed into a non-homogeneous ordinary
differential equation by making use of Laplace transformation.
3.1.3
Laplace Transformation
A function F(s) defined by the integral
∞
F (s) =
f (t)e−stdt
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)e−stdt
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) = L−1[F (s)].
The Laplace transformation is a method for solving differential equations and corresponding
initial and boundary value problems. It will transform initial and boundary value ordinary
differential equations into algebraic equations, Gupta (2009), Stroud and Dexter (2003),
Kreyszig (1987) and Binoy (2009). It will also transform initial and boundary value partial
differential equations into ordinary differential 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 flow along transmission lines.
36
3.2
KIRCHOFF’S CIRCUIT LAWS
In 1845, a German physicist, Gustav Kirchoff, first 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 Maxwell’s equations, but were developed prior to Maxwell’s
work. The Kirchoff ’s circuit laws, or simply Kirchoff ’s rules, deal with the conservation of
charge and energy in electrical circuits. The two laws are the Kirchoff ’s current law and
Kirchoff ’s voltage law which are described below.
In this chapter, we applied these two Kirchoff laws to the equivalent circuit of transmission lines and then we formulated the model for power flow along transmission lines.
3.2.1
Kirchoff ’s Current Law
Kirchoff ’s current law (KCL), also known as Kirchoff ’s Junction Law, Kirchoff ’s Point
Rule, Kirchoff ’s Nodal Law or Kirchoff ’s First Law, defines the way that electrical current
is distributed when it crosses through a junction. Specifically, 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 flowing. Since current is the flow
of electrons through a conductor, it cannot build up at a junction, meaning that current is
conserved: what comes in must go out. When performing calculations, current flowing into
and out of the junction typically have opposite signs. This allows Kirchoff ’s current law to
be restated as: The sum of current flowing into a node equals the sum of current flowing
out of the node.
3.2.2
Kirchoff ’s Voltage Law
Kirchoff ’s voltage law (KVL), also known as Kirchoff ’s Second Law, Kirchoff ’s Loop
Law, or Kirchoff ’s Mesh Rule, describes the distribution of voltage within a loop, or a closed
conducting path of an electrical circuit. Specifically, Kirchoff ’s Voltage Law states that:
The algebraic sum of the voltage (potential) differences in any closed loop must equal zero.
37
That is,
n
(Vk) = 0
k=0
where n is the total number of voltages measured. The voltage differences include those
associated with electromagnetic fields (emfs) and resistive elements, such as resistors, power
sources (i.e. batteries) or devices (i.e. lamps, televisions, blenders, etc.) plugged into the
circuit.
3.3
MATHEMATICAL MODEL FOR ELECTRIC POWER
FLOW ALONG LOSSY TRANSMISSION LINES
3.3.1
Model Formulation
In this work, we are interested in determining the extent to which the output voltage and
current differs from their input values as the length of the transmission line approaches a
very small value. To this end, we consider an equivalent circuit of a transmission line of
length ∆x containing resistance R∆x, capacitance C∆x, inductance L∆x and conductance
G∆x as shown in Figure 3.1.
38
Applying the Kirchoff ’s Voltage Law on the equivalent circuit of the transmission line,
we have
1
1 ∂i
v = Ri∆x + L
2
2 ∂t
1
∆x + L
2
∂i
∂∆i
+
∂t
∂t
1
∆x + R [i + ∆i] ∆x + v + ∆v,
2
which on simplification gives
∆i
]∆x − L[
2
∆v = −R[i +
∂i
1 ∂∆i
+
∂t
2 ∂t
]∆x
(3.1)
Dividing through equation (3.1) by ∆x and simplifying the equation, we have
∆v
∆x
L
∂ 2i
∂t∂x
= − Ri + L
∂i
∂t
∂i
1
+
∂t
2
= − Ri + L
+R
∂i
∂x
∆x
(3.2)
Taking limits as ∆x tends to zero, we have
∂v
∂x
.
(3.3)
Applying the Kirchoff ’s Current Law on the equivalent circuit of the transmission line and
simplifying as above, we have
∂i
= − Gv +
∂x
G ∂v
2 ∂x
∆x + C
∂v
∂t
+
C ∂ 2v
2 ∂t∂x
∆x .
(3.4)
Taking limits as ∆x tends to zero, we have
∂i
= − Gv + C
∂x
∂v
∂t
.
(3.5)
The differential equations in (3.3) and (3.5) above describe the evolution of current and
voltage in a lossy transmission line. Differentiating equation (3.3) with respect to x, we
have
∂ 2v
= −L
2
∂ 2i
∂x∂t
−R
∂i
.
∂x
(3.6)
∂ 2v
∂t2
−G
∂v
.
∂t
(3.7)
∂x
Differentiating equation (3.5) with respect to t, we have
∂ 2i
∂t∂x
= −C
Substituting equations (3.5) and (3.7) in equation (3.6) we have
∂ 2v
∂x
= LC
2
∂ 2v
∂t
2
+ LG
∂v
∂t
+ RGv + RC
∂v
.
∂t
(3.8)
Differentiating equation (3.3) with respect to t and equation (3.5) with respect to x and
simplifying as above, we have
∂ 2i
∂x
2
= LC
∂ 2i
∂t
2
+ CR
∂i
+ RGi + GL
∂t
39
∂i
.
∂t
(3.9)
Equations (3.8) and (3.9) are hyperbolic partial differential equations for lossy transmission
lines. Dividing equation (3.9) by CL we have
∂ 2i
+
∂t2
G
C
Let λ =
, β=
R
L
G
R
+
C
L
1
and φ =
CL
∂ 2i
∂t2
∂i
+
∂t
GR
.
CL
1 ∂ 2i
CL ∂x2
i(x, t) =
.
(3.10)
, so that equation (3.10) now becomes
+ (λ + β)
∂i
∂t
+ (λ.β) i(x, t) = φ
∂ 2i
∂x2
.
(3.11)
Equation (3.11) can now be solved together with the folllowing initial conditions below
(3.12)
i(x, 0) = f (x), it(x, 0) = g(x).
where i is the current through the conductor, f(x) is the initial value of the current and g(x)
is the initial speed of current.
3.3.2
Model Solution
The model in (3.11 - 3.12) is a second order initial-value partial differential equation. There
are so many analytical methods of solving second order partial differential equations. These
include the methods of separation of variables, change of variable, Fourier transfom, Laplace
transform, to name a few. The Laplace transform method is chosen for the solution because
the model is an initial-value problem and the initial conditions are nonhomogeneous.
Taking the Laplace transform of equation (3.11) with respect to t and substituting the
initial conditions in (3.12), we have
s2Is(x) − sf (x) − g(x) + (λ + β) sIs(x) − (λ + β) f (x) + λ.βIs(x) = φ
d2Is(x)
dx2
.
(3.13)
where Is(x) = I (x, s)
That is,
d2Is(x)
φ
dx 2
− s2 + (λ + β) s + λ.β Is(x) = − [g(x) + (s + λ + β) f (x)] ,
(3.14)
and which can be rewritten as
d2Is(x)
dx2
where b2 =
s2+(λ+β)S+(λ)(β)
φ
− b2Is(x) = cy(x),
(3.15)
and cy(x) = − [g(x)+(s+φλ+β)f (x)] . The general solution of equation
(3.15) is
Is(x) = Isc(x) + Isp(x),
-43(3.16)
40
where Isc(x) is the complementary function and Isp(x) is the particular solution. Solving
the associated homogeneous differential equation for (3.15), we have the complementary
function as
Isc(x) = k1Isc1 (x) + k2Isc2 (x) = k1ebx + k2e−bx.
(3.17)
Using the method of variaton of parameters, Dennis and Michael (2005), Kreyszig (1987),
Riley et al. (2002) and Dass and Verma (2011), we seek a particular solution of the form
(3.18)
Isp(x) = U1(x)Isc1 (x) + U2(x)Isc2 (x).
w1
where Isc1 (x) = ebx, Isc2 (x) = e−bx, U1(x) =
w
dx
and U2(x) =
w2
w
dx.
Thus,
ebx
w=
e−bx
,
bebx −be−bx
0
w1 =
e−bx
cy(x) −be−bx
and
ebx
w2 =
0
bebx cy(x)
So we can easily see from the determinants above that w = −2b, w1 = −cy(x)e−bx and
cy(x)e−bx
w2 = cy(x)ebx. Therefore U1(x) =
2b
dx and U2(x) = −
cy(x)ebx
dx.
2b
Substituting the values of U1(x), U2(x), Isc1 (x) and Isc2 (x) in (3.18), we have the particular
solution as
Isp(x) =
ebx
2b
cy(x)e−bx dx −
e−bx
2b
cy(x)ebx dx.
(3.19)
The general solution to (3.15) is therefore
Is(x) = k1ebx + k2e−bx +
ebx
2b
e−bx
2b
cy(x)e−bx dx −
cy(x)ebx dx.
(3.20)
Substituting the values of b and cy(x) in (3.20), we have
√
Is(x) = k1e
x
√
s2 +(λ+β√)s+(λ)(β )
φ
+ k2e
−x
s2 +(λ+β
√ )s+(λ)(β )
φ
√
−
e
x
s2 +(λ+β
√ )S +(λ)(β )
φ
2 φ(s2 + (λ + β)S + (λ)(β))
(g(x) + (s + λ + β) f (x)) e
41
−x
√
s2 +(λ+β
√ )S +(λ)(β )
φ
dx
√
+
e
−x
s2 +(λ+β )S +(λ)(β )
√
φ
x
(g(x) + (s + λ + β) f (x)) e
2 φ(s2 + (λ + β)S + (λ)(β))
√
s2 +(λ+β )S +(λ)(β )
√
φ
dx, (3.21)
Simplifying the equation above and taking λ = β we have
x(s+β )
√
φ
Is(x) = k1e
+ k2 e
e
Is(x) = k1e
−x(s+β )
√
φ
+k2e
e
−x(s+β )
√
φ
−
e
√
2(s + β) φ
[g(x)e
x(s+β )
√
φ
−x(s+β )
√
φ
[g(x)e
(3.22)
(s + 2β)e
x(s+β )
√
φ
√
2(s + β) φ
]dx−
]dx
]dx.
−x(s+β )
√
φ
(s + 2β)e
√
]dx +
2(s + β) φ
x(s+β )
√
φ
−x(s+β )
√
φ
[g(x) + (s + 2β)f (x)e
[g(x) + (s + 2β)f (x)e
x(s+β )
√
φ
−x(s+β )
√
φ
e
√
+
2(s + β) φ
x(s+β )
√
φ
√
2(s + β) φ
−
√
2(s + β) φ
+
x(s+β )
√
φ
−x(s+β )
√
φ
x(s+β )
√
φ
[f (x)e
[f (x)e
]dx.
−x(s+β )
√
φ
]dx
(3.23)
Suppose that g(x) and f(x) are polynomials of degree n, then equation (3.23) becomes
Is(x) = k1e
x(s+β )
√
φ
+ k2 e
−x(s+β )
√
φ
e
x(s+β )
√
φ
n
x(s+β )
√
n
φ
(s + 2β)e
√
−
[ (−1)if i(x)[ [e
2(s + β) φ
i=0
e
−x(s+β )
√
φ
−x(s+β )
√
φ
n
x(s+β )
√
φ
(−1)igi(x)[ [e
√[
2(s + β) φ i=0
+
−x(s+β )
√
−x(s+β )
√
φ
(−1)igi(x)[ [e
√[
2(s + β) φ i=0
−
]dx]i+1
]dx]i+1
]dx]i+1
n
φ
(s + 2β)e
√
+
2(s + β) φ
x(s+β )
√
φ
(−1)if i(x)[ [e
[
]dx]i+1.
(3.24)
i=0
, Yusuf and Sani (2006). The solution of the original problem (3.11) - (3.12) will now be
it(x) = k1e
x(β )
√
φ
x
δ(t+ √ )+k2e
φ
−x(β )
√
φ
n
1
−√[
2 φ i=0
(−1)if i(x)e
1
+√[
2 φ i=0
1
+√[
2 φ i=0
x
1
δ(t− √ )− √ [
φ 2 φ i=0
x(β
√ )
φ
L−1[
n
−x(β
√)
φ
(−1)igi(x)e
n
(−1)if i(x)e
−x(β
√)
φ
n
(−1)igi(x)e
(s + 2β)e
(s + β)
xs
√
φ
[
e
−xs
√
L−1[
e φ
(s + β)
(s + 2β)e
L−1[
(s + β)
42
[
e
−xs
√
φ
xs
√
e φ
(s + β)
x(β )
√
L−1[
φ
−x(s+β
√ )
φ
x(s+β
√)
φ
[
e
[
e
−x(s+β )
√
φ
]dx]i+1
]dx]i+1
]dx]i+1
x(s+β
√)
φ
]dx]i+1
(3.25)
Taking β = 0 and applying the convolution theorem of Laplace transformation,we have the
final result as
x
x
1
it(x) = k1δ(t + √ ) + k2δ(t − √ ) − √ [
φ
φ
2 φ i=0
1
−√[
2 φ i=0
1
+√[
2 φ i=0
n
n
(−1)igi(x)[
[
0
n
δ(τ φ− √ )
τ (i+1)
δ(τ φ+ √ )
t
(−1)igi(x)[
[
0
[
0
t
(−1)if i(x)[
δ(τ φ− √ )
t
τ (i+1)
τ (i+1)
.u[t
.δ[t
x
− (τ + √ )]dτ ]
φ
.u[t
x
− (τ − √ )]dτ ]
φ
x
− (τ + √ )]dτ ]
φ
n
t
δ(τ φ+ √ )
x
1
+√[
(−1)if i(x)[
[
.δ[t − (τ − √ )]dτ ]
(3.26)
τ (i+1)
φ
0
2 φ i=0
where δ(t) is the Dirac delta function and u(t) represents the Heaviside step function.
3.4
MATHEMATICAL MODEL ALONG TRANSMISSION LINES WHEN LEAKAGE TO GROUND
IS SMALL
3.4.1
Model Formulation
If the leakage to ground on the transmission line is small, the conductance (G) and the
inductance (L) are very negligible. Setting G and L to zero in equation (3.8) we have
∂ 2v
∂x 2
∂v
= CR
.
∂t
(3.27)
Similarly setting G and L to zero in equation (3.9) we have
∂ 2i
∂x 2
∂i
= CR
.
∂t
(3.28)
Let α = CR, so that equation (3.28) now becomes
∂ 2i
∂x 2
=α
∂i
∂t
.
(3.29)
Equation (3.29) can now be solved together with the folllowing initial conditions below
i(x, 0) = f (x), it(x, 0) = g(x).
(3.30)
where i is the current through the conductor, f(x) is the initial value of the current and g(x)
is the initial speed of current.
43
3.4.2
Model Solution
Taking the Laplace transform of equation (3.29) with respect to t and substituting the initial
conditions in (3.30), we have
d2Is(x)
dx2
.
(3.31)
− s2Is(x) = − [g(x) + sf (x)] .
(3.32)
s2Is(x) − sf (x) − g(x) = α
That is,
α
d2Is(x)
dx2
This now gives
d2Is(x)
dx 2
where b2 =
s2
α
− b2Is(x) = cy(x).
(3.33)
and cy(x) = − [g(x)+αsf (x)] . The general solution of equation (3.33) is
(3.34)
Is(x) = Isc(x) + Isp(x),
where Isc(x) is the complementary function and Isp(x) is the particular solution. Solving
the associated homogeneous differential equation for (3.33), we have the complementary
function as
Isc(x) = k1Isc1 (x) + k2Isc2 (x) = k1ebx + k2e−bx.
(3.35)
Using the method of variaton of parameters, Dennis and Michael (2005), Kreyszig (1987),
Riley et al. (2002) and Dass and Verma (2011), we seek a particular solution of the form
(3.36)
Isp(x) = U1(x)Isc1 (x) + U2(x)Isc2 (x).
w1
where Isc1 (x) = ebx, Isc2 (x) = e−bx, U1(x) =
w
dx
and U2(x) =
Thus,
w=
ebx
e−bx
bx
be
w1 =
0
−be
−bx
e−bx
cy(x) −be−bx
and
w2 =
ebx
0
bebx cy(x)
44
,
w2
w
dx.
So we can easily see from the determinants above that w = −2b, w1 = −cy(x)e−bx and
cy(x)e−bx
w2 = cy(x)ebx. Therefore U1(x) =
cy(x)ebx
dx and U2(x) = −
2b
dx.
2b
Substituting the values of U1(x), U2(x), Isc1 (x) and Isc2 (x) in (3.36), we have the particular
solution as
ebx
2b
Isp(x) =
e−bx
2b
cy(x)e−bx dx −
cy(x)ebx dx.
(3.37)
The general solution to (3.33) is therefore
ebx
Is(x) = k1ebx + k2e−bx +
e−bx
cy(x)e−bx dx −
2b
cy(x)ebx dx.
2b
(3.38)
Substituting the values of b and cy(x) in (3.38), we have
Is(x) = k1e
√xs
α
−
− √xsα
+k2e
Is(x) = k1e
e
xs
√
xs
√
α
(g(x) + sf (x)) e
√
2s α
α
+ k2 e
−xs
√
α
−xs
√
xs
√
α
e α
√
−
2s α
−xs
√
[g(x)e
−xs
√
e
dx+
−xs
√
α
(g(x) + sf (x)) e
√
2s α
e α
]dx − √
2α
−xs
√
[f (x)e
α
]dx
−xs
√
xs
xs
e α
e α
√
√
α
α
+
[g(x)e
[f (x)e
]dx.
√
]dx + √
2s α
2α
Suppose that g(x) and f(x) are polynomials of degree n, then equation (3.40) becomes
Is(x) = k1e
xs
√
e α
−√[
2 α i=0
xs
√
α
dx.
(3.39)
xs
√
α
√xs
α
−xs
√
+ k2e
n
(−1)if i(x)[ [e
−xs
√
α
−
e
xs
√
α
n
e
−xs
√
α
]dx]i+1 +
−xs
√
(−1)igi(x)[ [e
√[
2s α i=0
−xs
√
α
α
]dx]i+1
n
√[
2s α i=0
(3.40)
(−1)igi(x)[ [e
xs
√
α
]dx]i+1
n
xs
e α
√
(3.41)
(−1)if i(x)[ [e α ]dx]i+1.
+√[
2 α i=0
, Yusuf and Sani (2006). Applying the convolution theorem of Laplace transformation,we
have the solution of the original problem (3.29) - (3.30) as
x
x
1
it(x) = k1δ(t + √ ) + k2δ(t − √ ) − √ [
α
α
2 α i=0
1
−√[
2 α i=0
1
+√[
2 α i=0
n
n
(−1)igi(x)[
[
0
n
δ(τ −α √ )
τ
(i+1)
t
(−1)igi(x)[
[
0
[
0
t
(−1)if i(x)[
δ(τ α− √ )
t
δ(τ α+ √ )
τ (i+1)
n
τ
(i+1)
.u[t − (τ
.δ[t
x
− (τ + √ )]dτ ]
α
.u[t
x
− (τ − √ )]dτ ]
α
x
+ √ )]dτ ]
α
t
δ(τ α+ √ )
x
1
+√[
(−1)if i(x)[
[
.δ[t
(3.42)
−
(τ
−
√
)]dτ ]
α
τ (i+1)
0
2 α i=0
where δ(t) is the Dirac delta function and u(t) represents the Heaviside step function.
45
3.5
ANALYSIS OF MATHEMATICAL MODELS
The mathematical models describe the evolution of current and voltage on transmission lines.
It also expresses the flow of power in form of partial differential equations and thereby giving
the equations which the voltage or current must satisfy on a uniform transmission line. The
model for a lossy transmission line contains all the primary constants or parameters of the
transmission line which include the resistance (R), The conductance (G), the inductance
(L) and the capacitance (C). Values of all these constants are specified per unit length.
The conductance is used to model leakage current through the dielectric that may occur
throughout the line length and the inductance is flux leakage per unit current. So in the
second model, conductance and inductance have negligible effects and are therefore set to
zero.
46
Chapter 4
MINIMIZATION OF POWER
LOSSES OVER TRANSMISSION
LINES
In this chapter, we shall develop mathematical models for power losses over transmission
lines and use the classical optimization techniques to find an optimal strategy for minimizing
power losses over the transmission lines. The classical optimization methods are used in
finding the optimum of functions that are continuous and differentiable. It is an analytical
method that makes use of the differential calculus in finding the optimum points.
4.1
4.1.1
OHMIC AND CORONA POWER LOSSES
Ohmic Loss
Ohmic loss otherwise known as line loss on power transmission occurs as a result of resistance
of conductors against current flow. The effective resistance of the transmission line is a
function of the current on the line. This is because of the heat produced in the conductor
resulting from current flow, and this leads to a rise in the conductor’s temperature. This
rise in temperature increases the resistance of the conductor and consequently the losses on
the line.
47
4.1.2
Corona Loss
When the applied voltage exceeds the critical disruptive voltage in high voltage transmission
lines, the thin layer of air around the transmission line ionizes. This ionization process results
in corona power loss, hissing noise, production of ozone and radio interference. The higher
the voltage is raised, the larger the luminous envelope becomes and the greater the power
loss, the radio interference and audible noise. The ohmic and corona losses on transmission
lines depend on both physical and enviromental factors. The physical factors include the
type of the conductor, the dimension of the conductor, the shape of the conductor, the type
of insulator, etc. The enviromental factors include air density factor, temperature, soluble
and non-soluble contaminations, etc.
4.2
MATHEMATICAL MODELS FOR POWER LOSSES
4.2.1
Model Based on Ohmic and Corona Losses
The main reason for losses in transmission and sub-transmission lines is the resistance of
the conductor against current flow. As a result, heat is produced in the conductor resulting
from the flow of current and this increases the temperature of the conductor. The rise in
the conductor’s temperature further increases the resistance of the conductor and this will
consequently increase the losses. This implies that ohmic power loss is the main component of
losses in transmission and sub-transmission lines, Mehta and Mehta (2008), Wadhwa (2009),
Moghadam and Berahmandpour (2010), Daniel (2005) and Gupta (2008). The value of the
ohmic power loss is given as
LOhmic = I 2RKW/Km/P hase.
(4.1)
where
I denotes current along the conductor and
R represents resistance of the conductor.
The formation of corona on transmission line is associated with a loss of power, which
will have some effect on the efficiency of the trasmission line. The corona power loss has the
48
value
LCorona = 242
f + 25
η
r
(V − V0)2.(10)−5KW/Km/P hase.
d
.
(4.2)
where
f represents the frequency of transmission,
η denotes the air density factor,
r is radius of the conductor,
d represents the space between the transmission lines,
V is the operating voltage and
V0 denotes the distruptive voltage.
The relation in (4.2) above was derived for a fair weather condition. The approximate
corona loss under a foul weather condition was obtained by taking V0 to be 0.8 times the fair
weather value, Mehta and Mehta (2008), Wadhwa (2009), James (2005) and Gupta (2008).
This relation will give accurate results if the supply frequency lies between 25 to 120 HZ and
the radius of the conductor is greater than 0.25 cm. The space between conductors must be
greater than 15 times the radius of the conductor and the operating voltage must be greater
than 1.8 times the disruptive voltage, Wadhwa (2009). The total loss on a transmission line
is given as
(4.3)
TLoss = LOhmic + LCorona.
i.e
TLoss = I 2R + 242
f + 25
η
.
r
(V − V0)2.(10)−5KW/Km/P hase.
d
(4.4)
The resistance of a conductor is given by
R=
ρL
A
.
(4.5)
where
R represents the resistance of the conductor,
ρ is the resistivity of the conductor,
L denotes the length of the conductor and
A is the cross-sectional area of the conductor.
To get the general form of equation (4.4), we substitute R =
ρL
A
for the value of resistance
and express the radius of the conductor in terms of its cross sectional area to get
TLoss = I 2
ρL
A
+ 242
f + 25 4 A
(V − V0)2.(10)−5KW/Km/P hase.
.
η
πd2
49
(4.6)
Equation (4.6) is a general power loss function for transmission lines. We can also use
conductivity of the material instead of resistivity in the above equation by just applying
ρ=
1
.
σ
(4.7)
where
ρ denotes the resistivity of the conductor and
σ represents the conductivity of the conductor.
Equation (4.6) will then become
TLoss = I 2
4.2.2
L
+ 242
σA
f + 25 4 A
(V − V0)2.(10)−5KW/Km/P hase.
.
η
πd2
(4.8)
Empirical Models of Power Losses as Functions of Distance
In order to study the effects of distances and loadings on power losses, Onohaebi and Odiase
(2010) computed the impedances of various lengths of line from the source of power based on
the 350mm2 cross-sectional area of Aluminium Conductor Steel Re-enforced (ACSR) used
for the Nigerian 330 KV transmission network. The longest line in the Nigerian 330 KV
transmission network is 310 km from Birnin Kebbi to Kainji. Based on this longest line,
Onohaebi and Odiase (2010) simulated loads of 100 MW, 200 MW and 300 MW with the
associated impedances for various lengths from 10 km to 340 km with power world simulator
and got the results reported in Tables 4.1 and 4.2. The loads were simulated for both single
and double circuits. Scatter diagrams and the graphs of losses as functions of distance are
shown in Figures 4.1 to 4.12.
Based on Tables 4.1 and 4.2, we will need six empirical models to represent losses as
functions of distance for various loads.
4.2.2.1 Empirical Model for Losses on Single Circuit 330 KV Nigerian Transmission Network
with a Load of 100 MW.
We are going to use regression analysis to determine the model. The scatter diagram
of losses against distance for a load of 100 MW on a single circuit network is as shown
in Figure 4.1 and the graph is in Figure 4.2.
50
Table 4.1: Simulated Results of Power Losses on 330 KV Single Circuit of the
Nigerian Transmission Network.
Length of Line in KM
Power Losses (in MW)
Power Losses (in MW)
Power Losses (in MW)
for a Load of 100 MW
for a Load of 200 MW
for a Load of 300 MW
10
0.05
0.18
0.43
20
0.09
0.37
0.87
40
0.18
0.73
1.75
60
0.26
1.10
2.84
80
0.34
1.46
3.64
100
0.41
1.85
4.66
120
0.49
2.22
5.71
140
0.56
2.52
6.84
160
0.63
2.98
8.07
180
0.70
3.37
9.39
200
0.76
3.77
10.86
220
0.83
4.17
12.47
240
0.90
4.57
14.40
260
0.97
5.06
16.74
280
1.00
5.40
18.69
300
1.10
5.85
24.40
320
1.17
6.30
Blackout
340
1.32
7.10
Blackout
51
Table 4.2: Simulated Results of Power Losses on 330 KV Double Circuit of the
Nigerian Transmission Network.
Length of Line in KM
Power Losses (in MW)
Power Losses (in MW)
Power Losses (in MW)
for a Load of 100 MW
for a Load of 200 MW
for a Load of 300 MW
10
0.02
0.09
0.01
20
0.04
0.18
0.41
40
0.08
0.35
0.81
60
0.12
0.51
1.20
80
0.15
0.68
1.61
100
0.18
0.81
1.96
120
0.22
0.95
2.32
140
0.25
1.09
2.68
160
0.29
1.23
3.04
180
0.34
1.37
3.40
200
0.39
1.50
3.74
220
0.45
1.63
4.09
240
0.52
1.77
4.43
260
0.60
1.90
4.76
280
0.70
2.04
5.06
300
0.77
2.19
5.48
320
0.95
2.34
5.97
340
1.23
2.53
5.97
52
.
53
From the scatter diagram and the graph, we can easily see that the function can be
represented approximately by a straight line. The estimated regression equation for
the best line is given by
Y = a + bX.
where
a=
n
k=1
(
Yi)(
and
b=
n
n
2
k=1(Xi) ) − (
n
n( k=1(Xi) 2)
n
n
nk=1(Xi)(Yi)) − (
n
n( k=1 (Xi)2)
n(
n
k=1
−(
−
k=1
n
(k=1
(4.9)
Xi)(
k=1 Xi)
Xi)(
n
k=1(Xi)(Yi))
2
k=1
Yi)
Xi)2
.
.
(4.10)
(4.11)
Harper (1984). Table 4.3 gives all the summations for the determination of the values
of a and b.
Substituting the summations from Table 4.3 in equations (4.10) and (4.11) we have
a=
(11.76)(714100) − (3070)(2693.3)
18(714100) − (3070)2
.
=
129385
3428900
.
= 0.037733675
. and
b=
18(2693.3) − (3070)(11.76)
18(714100) − (3070)2
=
12376.2
3428900
= 0.003609379101
The estimated regression equation is therefore given by
Y=
[12376.2X + 129385]
3428900
(4.12)
4.2.2.2 Empirical Model for Losses on Single Circuit 330 KV Nigerian Transmission Network
with a Load of 200 MW.
The scatter diagram of losses against distance for a load of 200 MW on a single circuit
network is as shown in Figure 4.3 and the graph is in Figure 4.4.
54
Table 4.3:
Summations
for
Distance(Xi) Losses(Yi)
n
i=1
a
Load
of
100
MW
XiYi
Xi2
on
Yi2
10
0.05
100
0.5
0.0025
20
0.09
400
1.8
0.0081
40
0.18
1600
7.2
0.0324
60
0.26
3600
15.6
0.0676
80
0.34
6400
27.2
0.1156
100
0.41
10000
41
0.1681
120
0.49
14400
58.8
0.2401
140
0.56
19600
78.4
0.3136
160
0.63
25600
100.8
0.3969
180
0.70
32400
126
0.49
200
0.76
40000
152
0.5776
220
0.83
48400
182.6
0.6889
240
0.90
57600
216
0.81
260
0.97
67600
252.2
0.9409
280
1.00
78400
280
1.0
300
1.10
90000
330
1.21
320
1.17
102400
374.4
1.3689
340
1.32
115600
448.8
1.7424
Xi =
3070
n
i=1
Yi =
11.76
n
i=1
Xi2 =
714100
n
i=1
XiYi =
2693.3
55
n
i=1
Yi2 =
10.1736
Single
Circuit.
.
56
From the scatter diagram and the graph, we can easily see again that the function can
be represented approximately by a straight line. Table 4.4 gives all the summations
for the determination of the values of a and b for this problem.
Table 4.4:
Summations
for
Distance(Xi) Losses(Yi)
n
i=1
a
Load
of
200
MW
XiYi
Xi2
on
Yi2
10
0.18
100
1.8
0.0324
20
0.37
400
7.4
0.1369
40
0.73
1600
29.2
0.5329
60
1.1
3600
66
1.21
80
1.46
6400
116.8
2.1316
100
1.85
10000
185
3.4225
120
2.22
14400
266.4
4.9284
140
2.52
19600
352.8
6.3504
160
2.98
25600
476.8
8.8804
180
3.37
32400
606.6
11.3569
200
3.77
40000
754
14.2129
220
4.17
48400
917.4
17.3889
240
4.57
57600
1096.8
20.8849
260
5.06
67600
1315.6
25.6036
280
5.40
78400
1512
29.16
300
5.85
90000
1755
34.2225
320
6.30
102400
2016
39.69
340
7.10
115600
2414
50.41
Xi =
3070
n
i=1
Yi =
59
n
i=1
Xi2 =
714100
n
i=1
XiYi =
13889.6
57
n
i=1
Yi2 =
270.5552
Single
Circuit.
Substituting the summations from Table 4.4 in equations (4.10) and (4.11) we have
a=
(59)(714100) − (3070)(13889.6)
18(714100) − (3070)2
.
−509.172
3428900
=
.
= −0.148494269
. and
b=
18(13889.6) − (3070)(59)
18(714100) − (3070)2
=
68882.8
3428900
= 0.020088891
The estimated regression equation is therefore given by
Y = −0.148494269 + 0.020088891X.
(4.13)
4.2.2.3 Empirical Model for Losses on Single Circuit 330 KV Nigerian Transmission Network
with a Load of 300 MW.
The scatter diagram of losses against distance for a load of 300 MW on a single circuit
network is as shown in Figure 4.5 and the graph is in Figure 4.6.
58
.
59
From the scatter diagram and the graph, we can easily see that the function cannot
be perfectly represented approximately by a straight line. We now take the logarithms
of the data for losses and perform regression analysis on the values for distance and
the logarithmic values of losses so as to get an exponential regression model. The
estimated regression equation for the curve will therefore be given by
logY = a + bX.
(4.14)
where
a=
(
n
k=1
and
b=
n nn
2
k=1(Xi) ) − (
n
n( k=1(Xi) 2)
logYi)(
n(
n
k=1
−(
n
n
nk=1(Xi)(logYi)) − (
n
n( k=1 (Xi)2)
−
Xi)(
k=1 Xi)
k=1
n
(k=1
k=1(Xi)(logYi))
2
Xi)(
k=1
logYi)
Xi)2
.
.
(4.15)
(4.16)
Table 4.5 gives all the summations for the determination of the values of a and b for
this problem.
Substituting the summations from Table 4.5 in equations (4.15) and (4.16) we have
a=
(27.9599)(496100) − (2410)(5729.664)
16(496100) − (2410)2
.
=
62416.15
2129500
.
= 0.029310237
. and
b=
16(5729.664) − (2410)(27.9599)
16(496100) − (2410)2
=
24291.265
2129500
= 0.011407027
The estimated regression equation is therefore given by
logY = 0.029310237 + 0.011407027X
(4.17)
Y = 1.02974401.e0.011407027X
(4.18)
This now gives
60
Table 4.5:
Distance(Xi)
Summations
for
Losses(Yi)
a
Load
of
logYi
X
300
2
i
MW
on
Single
Circuit.
XilogYi
logY i
2
10
0.43
-0.8440
100
-8.44
0.7123
20
0.87
-0.1393
400
-2.786
0.0194
40
1.75
0.5596
1600
22.384
0.3132
60
2.84
1.0438
3600
62.628
1.0895
80
3.64
1.2920
6400
103.36
1.6693
100
4.66
1.5390
10000
153.9
2.3685
120
5.71
1.7422
14400
209.064
3.0353
140
6.84
1.9228
19600
269.192
3.6972
160
8.07
2.0882
25600
334.112
4.3606
180
9.39
2.2396
32400
403.128
5.0158
200
10.86
2.3851
40000
477.02
5.6887
220
12.47
2.5233
48400
555.126
6.3670
240
14.4
2.6672
57600
640.128
7.1140
260
16.74
2.8178
67600
732.628
7.9400
280
18.69
2.9280
78400
819.84
8.5732
300
24.4
3.1946
90000
958.38
10.2055
n
i=1
Xi =
2410
n
i=1
Yi =
141.76
n
i=1
n
logYi =
i=1
27.9599
2
Xi =
496100
61
n
i=1
XilogYi =
5729.664
n
i=1
2
logY
i =
68.1695
4.2.2.4 Empirical Model for Losses on Double Circuit 330 KV Nigerian Transmission Network
with a Load of 100 MW.
The scatter diagram of losses against distance for a load of 100 MW on a double circuit
network is as shown in Figure 4.7 and the graph is in Figure 4.8.
62
.
63
Again from the scatter diagram and the graph, we can easily see that the function
cannot be perfectly represented approximately by a straight line. We now take the
logarithms of the data for losses and perform regression analysis on the values for
distance and the logarithmic values of losses so as to get an exponential regression
model. The estimated regression equation for the curve will therefore be given by
equation (4.14) above where a and b are as given in equations (4.15) and (4.16) above.
Table 4.6 gives all the summations for the determination of the values of a and b for
this problem.
Substituting the summations from Table 4.6 in equations (4.15) and (4.16) we have
a=
(−23.9731)(714100) − (3070)(−2201.08)
18(714100) − (3070)2
.
=
−10361875.11
3428900
.
= −3.021923973
. and
b=
18(−2201.08) − (3070)(−23.9731)
18(714100) − (3070)2
=
33977.977
3428900
= 0.009909293651
. The estimated regression equation is given by
logY = −3.021923973 + 0.009909293651X.
(4.19)
Y = 0.048707416.e0.009909293651X .
(4.20)
This now gives
4.2.2.5 Empirical Model for Losses on Double Circuit 330 KV Nigerian Transmission Network
with a Load of 200 MW.
The scatter diagram of losses against distance for a load of 200 MW on a double circuit
network is as shown in Figure 4.9 and the graph is in Figure 4.10.
64
Table 4.6:
Distance(Xi)
Summations
for
Losses(Yi)
a
Load
of
logYi
X
100
2
i
MW
on
Double
Circuit.
XilogYi
logY i
2
10
0.02
-3.9120
100
-39.12
15.3037
20
0.04
-3.2189
400
-64.378
10.3613
40
0.08
-2.5257
1600
-101.028
6.3792
60
0.12
-2.1203
3600
-127.218
4.4957
80
0.15
-1.8971
6400
-151.768
3.5990
100
0.18
-1.7148
10000
-171.48
2.9405
120
0.22
-1.5141
14400
-181.692
2.2925
140
0.25
-1.3863
19600
-194.082
1.9218
160
0.29
-1.2379
25600
-198.064
1.5324
180
0.34
-1.0788
32400
-194.184
1.1638
200
0.39
-0.9416
40000
-188.32
0.8866
220
0.45
-0.7985
48400
-175.67
0.6376
240
0.52
-0.6539
57600
-156.936
0.4276
260
0.60
-0.5108
67600
-132.808
0.2609
280
0.70
-0.3567
78400
-99.876
0.1272
300
0.77
-0.2614
90000
-78.42
0.0683
320
0.95
-0.0513
102400
-16.416
0.0026
340
1.23
0.2070
115600
70.38
0.0428
n
i=1
Xi =
3070
n
i=1
Yi =
7.3
n
i=1
n
logYi =
i=1
-23.9731
2
Xi =
714100
65
n
i=1
XilogYi =
-2201.08
n
i=1
2
logY
i =
52.4435
.
66
From the scatter diagram and the graph, we can easily see that the function can be
represented approximately by a straight line. Table 4.7 gives all the summations for
the determination of the values of a and b for this problem.
Table 4.7:
Summations
for
Distance(Xi) Losses(Yi)
n
i=1
a
Load
of
200
MW
XiYi
Xi2
on
Yi2
10
0.09
100
0.9
0.0081
20
0.18
400
3.6
0.0324
40
0.35
1600
14
0.1225
60
0.51
3600
30.6
0.2601
80
0.68
6400
54.4
0.4624
100
0.81
10000
81
0.6561
120
0.95
14400
114
0.9025
140
1.09
19600
152.6
1.1881
160
1.23
25600
196.8
1.5129
180
1.37
32400
246.6
1.8769
200
1.50
40000
300
2.25
220
1.63
48400
358.6
2.6569
240
1.77
57600
424.8
3.1329
260
1.90
67600
494
3.61
280
2.04
78400
571.2
4.1616
300
2.19
90000
657
4.7961
320
2.34
102400
748.8
5.4756
340
2.53
115600
860.2
6.4009
Xi =
3070
n
i=1
Yi =
23.16
n
i=1
Xi2 =
714100
n
i=1
XiYi =
5309.1
67
n
i=1
Yi2 =
39.506
Double
Circuit.
Substituting the summations from Table 4.7 in equations (4.10) and (4.11) we have
a=
(23.16)(714100) − (3070)(5309.1)
18(714100) − (3070)2
.
=
239619
3428900
.
= 0.069882177
. and
b=
18(5309.1) − (3070)(23.16)
18(714100) − (3070)2
=
24462.6
3428900
= 0.007134241302
Y = 0.069882177 + 0.007134241302X.
(4.21)
4.2.2.6 Empirical Model for Losses on Double Circuit 330 KV Nigerian Transmission Network
with a Load of 300 MW.
Finally, we consider the model for losses on double circuit 330 KV Nigerian transmission network with a load of 300 MW. The scatter diagram of losses against distance
for a load of 300 MW on a double circuit network is as shown in Figure 4.11 and the
graph is in Figure 4.12.
68
.
69
Again from the scatter diagram and the graph, we can easily see that the function can
be represented approximately by a straight line. Table 4.8 gives all the summations
for the determination of the values of a and b for this problem.
Table 4.8:
Summations
for
Distance(Xi) Losses(Yi)
n
i=1
a
Load
of
300
MW
XiYi
Xi2
on
Yi2
10
0.01
100
0.1
0.0001
20
0.41
400
8.2
0.1681
40
0.81
1600
32.4
0.6561
60
1.2
3600
72
1.44
80
1.61
6400
128.8
2.5921
100
1.96
10000
196
3.8416
120
2.32
14400
278.4
5.3824
140
2.68
19600
375.2
7.1824
160
3.04
25600
486.4
9.2416
180
3.40
32400
612
11.56
200
3.74
40000
748
13.9876
220
4.09
48400
899.8
16.7281
240
4.43
57600
1063.2
19.6249
260
4.76
67600
1237.6
22.6576
280
5.06
78400
1416.8
25.6036
300
5.48
90000
1644
30.0304
320
5.97
102400
1910.4
35.6409
340
5.97
115600
2029.8
35.6409
Xi =
3070
n
i=1
Yi =
56.94
n
i=1
Xi2 =
714100
n
i=1
XiYi =
13139.1
70
n
i=1
Yi2 =
241.9784
Double
Circuit.
Substituting the summations from Table 4.8 in equations (4.10) and (4.11) we have
a=
(56.94)(714100) − (3070)(13139.1)
18(714100) − (3070)2
.
=
323817
3428900
.
= 0.094437574
. and
b=
18(13139.1) − (3070)(56.94)
18(714100) − (3070)2
=
61698
3428900
= 0.017993525
Y = 0.094437574 + 0.017993525X.
(4.22)
4.2.2.7 Remarks on Empirical Models.
From the empirical models, it can be established that power losses increases with
distance. That is power losses is directly proportional to distance. The losses on single
circuit 330 KV transmission network with loads of 100 MW and 200 MW and the
losses on double circuit 330 KV transmission network with loads of 200 MW and 300
MW were represented approximately by straight lines. The power losses on single
circuit 330 KV transmission network with a load of 300 MW and the losses on double
circuit transmission network with a load of 100 MW were represented by exponential
models.
4.3
MULTIVARIABLE OPTIMIZATION WITHOUT
CONSTRAINTS
4.3.1
Properties of Hessian Matrix
i An Hessian matrix is a square matrix
71
ii It describes the local curvature of a function of many variables
iii The following tests can be applied to a non-degenerate critical point of x. If the Hessian
matrix is positive definite at x, then the function attains a local minimum at x. If the
Hessian matrix is negative definite at x, then the function attains a local maximum
at x. If the Hessian matrix is positive semidefinite, then the function is convex. If the
Hessian matrix has both positive and negative eigen-values, then x is a saddle point.
For positive semidefinite and negative semidefinite Hessians, the test is inconclusive.
4.3.2
Necessary and Sufficient Conditions for the Existence of Extremal Points
In this section, we shall state some theorems which gives the necessary and sufficient conditions for the optimum values of a function of several variables and then apply the theorems
in finding the optimum power losses over transmission lines. The power loss function is
considered as a multivariable optimization without constraints.
Theorem 4.1: Rao (1998). If a function of several variables f(x) has a stationary point
at x = x∗ and if the first partial derivatives of f(x) exist at x = x∗, then
∂f
(x∗) =
∂x1
∂f
∂x2
∂f
∂x3
(x∗) =
(x∗) = ............ =
∂f
∂xk
(x∗) = 0
(4.23)
Proof: Suppose that one of the first partial derivatives, say the kth one, does not vanish
at x = x∗. Then by Taylor’s theorem
n
∗
∗
hi
f (x + h) = f (x ) +
i=1
∂f
∂xi
(4.24)
(x∗) + Ri(x∗, h)
This now gives
∂f
1
(x∗) + d2f (x∗ + θh), 0 < θ < 1
2
∂xk
f (x∗ + h) − f (x∗) = hk
(4.25)
Since d2f (x∗ + θh) is of order h2i , the terms of order h will dominate the higher order terms
∗
for small h. Thus the sign of f (x∗ +h)−f (x∗) is decided by the sign of hk ∂f∂x(xk ) . Suppose that
∂f (x∗)
∂xk
is greater than zero. Then the sign of f (x∗ + h) − f (x∗) will be positive for hk greater
than zero and negative for hk less than zero. This therefore means that x∗ cannot be an
extremum point. Now suppose that
∂f (x∗)
∂xk
is less than zero. Then the sign of f (x∗ +h)−f (x∗)
will be positive for hk less than zero and negative for hk greater than zero. This therefore
72
implies again that x∗ cannot be an extremum point. Since this conclusion is in contradiction
∂f
with the original statement that x∗ is an extremum point, we therefore say that
∂xk
= 0 at
point x = x∗.
Theorem 4.2: Rao (1998). If f(x) is a function of several variables, a sufficient condition
for a stationary point at x = x∗ to be an optimum point of the function is that the Hessian
matrix of f(x) evaluated at x = x∗ is
(i) positive definite when x = x∗ is a minimum point and
(ii) negative definite when x = x∗ is a maximum point.
Proof: From Taylor’s theorem, we can write
n
∂f
(x∗) +
hi
∂xi
f (x + h) = f (x ) +
∗
∗
i=1
1
2
n
n
hihj
i=1 j=1
∂ 2f
∂xi∂xj
(4.26)
for
x = (x∗ + θh), 0 < θ < 1
Since x∗ is a stationary point, we have from Theorem 4.1 that
∂f
∂xi
=0
(4.27)
, for i = 1,2,3,.....,n
Thus equation (4.26) reduces to
∗
1
2
∗
f (x + h) − f (x ) =
n
n
hihj
i=1 j=1
for
x = (x∗ + θh), 0 < θ < 1
The sign of
f (x∗ + h) − f (x∗)
will be the same as that of
n
n
hihj
i=1 j=1
∂ 2f
∂xi∂xj
for
x = (x∗ + θh), 0 < θ < 1
73
∂ 2f
∂xi∂xj
(4.28)
Since the second partial derivative
∂ 2f
∂xi∂xj
∂ 2f
∂xi∂xj
is continuous in the neighbourhood of x∗ then,
∂ 2f
∂xi∂xj
at x = (x∗ + θh) will have the same sign as
at x = x∗ for all sufficiently small
values of h.
Thus f (x∗ + h) − f (x∗) will be positive, and hence x∗ will be a relative minimum, if
n
n
Q=
hihj
i=1 j=1
∂ 2f
∂xi∂xj
(4.29)
at x = x∗ is positive. This quantity Q is a quadratic form and can be written in matrix
form as
Q = hT Jh, atx = x∗
where
∂ 2f
J=
(4.30)
(4.31)
∂xi∂xj
at x = x∗ is a matrix of second partial derivatives called the Hessian matrix of f(x).
The quadratic form in (4.29) or (4.30) will be positive for all h if and only if J is positive
definite at x = x∗. This means that a sufficient condition for the stationary point x∗ to
be a relative minimum is that the Hessian matrix evaluated at the same point be positive
definite. This completes the proof for minimization case.
For the case of maximization, f (x∗ + h) − f (x∗) will be negative, and hence x∗ will be a
relative maximum, if
n
n
Q=
hihj
i=1
j=1
∂ 2f
∂xi∂xj
(4.32)
at x = x∗ is negative. This quantity Q is a quadratic form and can be written in matrix
form as
Q = hT Jh, atx = x∗
where
J=
∂ 2f
∂xi∂xj
(4.33)
(4.34)
at x = x∗ is a matrix of second partial derivatives called the Hessian matrix of f(x).
The quadratic form in (4.32) or (4.33) will be negative for all h if and only if J is negative
definite at x = x∗. This means that a sufficient condition for the stationary point x∗ to be
a relative maximum is that the Hessian matrix evaluated at the same point be negative
definite.
Let consider the minimization of the total power losses on transmission lines with respect
to the operating voltage (V) and current (I).
74
From equation (4.6) we have
TLoss = I 2
ρL
A
f + 25 4 A
(V − V0)2.(10)−5KW/Km/P hase.
.
η
πd2
+ 242
Differentiating equation (4.6) partially with respect to I, we have
2I ρL
∂TL
=
A
∂I
.
(4.35)
and differentiating equation (4.6) partially with respect to V, we have
∂TL
∂V
(f + 25) 4 A
.
(V − V0).(10)−5.
η
πd 2
= 484
(4.36)
For stationary values, we have
∂TL
∂I
∂TL
.
∂V
=0=
(4.37)
Therefore
2I ρL
A
and
484
(f + 25)
η
4
.
. =0
A
(4.38)
(V − V0).(10)−5. = 0
πd 2
(4.39)
From equations (4.38) and (4.39), we have I = 0, and V = V0
Second derivative of equation (4.6) with respect to I will give
2ρL
∂ 2T L
=
2
A
∂I
.
(4.40)
and second derivative of equation (4.6) with respect to V will give
∂ 2T L
∂V
2
= 484
(f + 25) 4 A
.(10)−5.
.
η
πd2
(4.41)
Differentiating equation (4.35) with respect to V we have
∂ 2TL
∂V ∂I
= 0.
(4.42)
and differentiating eqution (4.36) with respect to I we have
∂ 2TL
∂I ∂V
= 0.
The Hessian matrix of total power losses on transmission lines is therefore given by
∂ 2T L
∂I 2
∂ 2T L
∂I ∂V
∂ 2T L
∂V ∂I
∂ 2T L
∂V 2
75
(4.43)
From which we now have the determinants as
|A1| =
∂ 2T L
∂I 2
and
|A2| =
Substituting the values of
∂ 2T L
∂I 2
∂ 2T L
∂I 2
∂ 2T L
∂I ∂V
∂ 2T L
∂V ∂I
∂ 2 TL
∂V 2
2
∂ 2T
, ∂∂VT2L , ∂V ∂IL and
∂ 2T L
∂I ∂V
in the determinants of the Hessian
matrix, we have
2ρL
A
|A1| =
and
2ρL
A
|A2| =
0
0
484 (f +25)
.4
η
A
πd2
.(10)−5
We can easily see that
2ρL
A
|A1| =
is greater than zero and
|A2| = [
2ρL
A
][484
(f + 25) 4 A
.(10)−5]
.
η
πd2
is also greater than zero. Therefore, the Hessian matrix of power losses over transmission
lines is positive definite. Hence the power loss is minimum at I = 0 and V = V0
Let us now consider the minimization of total power losses with respect to distance and
current along the transmission line. In this case
|A1| =
∂ 2T L
∂I 2
and
|A2| =
∂ 2T L
∂I 2
∂ 2T L
∂I ∂d
∂ 2T L
∂d∂I
∂ 2T L
∂d2
where |A1| is the first determinant of the matrix of second order partial derivatives and |A2|
is the second determinant of the matrix of second order patial derivatives.
Differentiating equation (4.6) partially with respect to d, we have
∂TL
∂d
= −121
3
(f + 25) 4 A
.
(V − V0)2.d− 2 (10)−5.
η
π
76
(4.44)
From equation (4.35), we have
2I ρL
∂TL
=
A
∂I
.
For stationary values, we have
∂TL
∂I
∂TL
.
∂d
=0=
(4.45)
which gives
2I ρL
A
and
−121
(f + 25)
4
.
A
. =0
(4.46)
3
(V − V0)2.d− 2 (10)−5. = 0
η
π
From equations (4.46) and (4.47), we have I = 0, and d −→ ∞
(4.47)
From equation (4.40), we have
2ρL
∂ 2T L
=
.
2
A
∂I
Second derivative of equation (4.6) with respect to d will give
5
363 (f + 25) 4 A
∂ 2T L
=
.
(V − V0)2.d− 2 (10)−5.
2
∂d
2
η
π
(4.48)
Differentiating equation (4.35) with respect to d we have
∂ 2TL
= 0.
∂d∂I
(4.49)
and differentiating eqution (4.45) with respect to I we have
∂ 2TL
= 0.
∂I ∂d
Substituting the values of
∂ 2T L
∂I 2
2
(4.50)
∂ 2T L
∂I ∂d
2T
, ∂∂dT2L , ∂∂d∂IL and
in the Hessian matrices, we have
2ρL
A
|A1| =
and
|A2| =
2ρL
A
0
0
363 (f +25)
2
η
.4
A
π
(V
5
− V0)2.d− 2 (10)−5
We can easily see that
2ρL
A
|A1| =
is greater than zero and
|A2| = [
2ρL 363 (f + 25) 4 A
][
.
A 2
η
π
5
(V − V0)2.d− 2 (10)−5]
is also greater than zero. Therefore, the Hessian matrix of power losses over transmission
lines is positive definite. Hence the power loss is minimum at I = 0 and d −→ ∞.
77
4.4
MINIMIZATION OF POWER LOSSES
In this section, we shall apply the method of finding the extremum of functions of several
variables in finding the optimum power losses over transmission lines.
A function of several variables is said to have a maximum or minimum at the stationary
values if
[
This can only be so if
(2002). If
∂ 2T L
∂V 2
∂ 2T L
∂I 2
and
∂ 2T L
∂I 2
and
∂ 2T L
∂V 2
∂ 2TL
∂I 2
∂ 2TL
]−[
∂V 2
]. [
∂ 2T L 2
] > 0.
∂I ∂V
(4.51)
have the same sign, Kreyszig (1987) and Riley et al.
∂ 2T L
∂V 2
∂ 2T L
∂I 2
are both positive, then TL(I, V ) is a minimum. But if
and
are both negative, then TL(I, V ) is a maximum.
I
f
∂TL
∂I
∂TL
∂V
=0=
∂ 2T L 2
and
∂ 2TL
[
∂I 2
]. [
∂ 2TL
∂V 2
]−[
]<0
∂I ∂V
Then, we have a saddle point at TL(I, V ).
Now, since
[
and
∂ 2T L
∂I 2
=
∂ 2TL 2
∂∂I
2TL
2ρL
A
2
]. [
∂∂V
2TL
2
]−[
∂I ∂V
] =[
2ρL
A
= 484 (f +25) . 4
∂ 2T L
∂V 2
together with
(f + 25) 4 A
.(10)−5] > 0
.
η
πd2
].[484
A
πd2
η
.(10)−5
are both greater than zero, we
can say that TL(I, V ) is minimum at I = 0 and V = V0
Let us now look at the minimization of total power losses with respect to distance and
current along the transmission line.
A function of several variables is said to have a maximum or minimum at the stationary
∂ 2T L 2
values if
∂ 2TL
[
∂I 2
This can only be so if
If
∂ 2T L
∂I 2
and
∂ 2T L
∂d2
∂ 2T L
∂I 2
and
∂ 2T L
∂d2
]. [
∂ 2TL
∂d2
]−[
∂I ∂d
] > 0.
(4.52)
have the same sign, Kreyszig (1987), Richmond (1972).
are both positive, then TL(I, d) is a minimum. But if
both negative, then TL(I, d) is a maximum.
If
∂TL
=0=
∂I
78
∂TL
∂d
∂ 2T L
∂I 2
and
∂ 2T L
∂d2
are
and
∂ 2TL
[
∂I 2
]. [
∂ 2TL
∂d2
]−[
∂ 2T L 2
]<0
∂I ∂d
Then, we have a saddle point at TL(I, d).
Now, since
∂ 2TL
∂ 2TL
∂ 2TL 2
]−[
[ 2 ]. [
] =[
2
∂I ∂d
∂I
∂d
∂ 2T L
∂I 2
and
=
2ρL
A
together with
∂ 2T L
∂d2
2ρL
A
=
]. [
363 (f + 25) 4 A
.
2
η
π
363 (f +25)
2
η
.
4
A
π
5
(V − V0)2.d− 2 (10)−5] > 0
5
(V − V0)2.d− 2 (10)−5 are both greater
than zero, we can say that TL(I, d) is minimum at I = 0 and d −→ ∞.
4.5
DISCUSSION ON RESULTS
The Hessian matrix of power losses over transmission lines is positive definite at the
stationary values. That is, the power loss is minimum at I = 0, V = V0 and d −→ ∞. Technically, this implies that the total power losses on transmission lines will only be minimum
if
(i.) power is transmitted at a very low current along transmission lines. This will reduce
the ohmic or line loss on the conductors to the barest minimum. This conforms with
the principle of electric power transmission;
(ii.) the operating voltage is equal to the critical disruptive voltage. When this happens,
there is no ionisation of air around the conductor and hence no corona is formed.
Therefore, there will be no corona loss; and
(iii.) the spacing between the conductors on the transmission line should be large. This
is because, an increase in the spacing between conductors reduces the electro-static
stresses. This therefore reduces the corona effect. If the spacing between the conductors is made very large as compared to their diameter, there may not be any corona
effect or losses on the line.
79
Chapter 5
GENERAL CONCLUSION
In this chapter, we present the general conclusion for the whole thesis.
5.1
SUMMARY OF THESIS
The aim of this thesis is to minimize the losses on electric power transmission lines to a
desired and economic level by applying the classical optimization technique to the mathematical model of power losses on transmissin lines.
Mathematical models were proposed for power flow and power losses along transmission
lines using the Kirchoff ’s circuit laws and the combination of corona and ohmic losses.
The mathematical models for power flow along transmission lines were solved analytically
using the Laplace transform method and the mathematical model for power losses along
transmission lines was minimized using the classical optimization technique.
Empirical models of power losses as functions of distance were also developed. With
these models, we can determine the losses on electric power transmission lines if given the
distance, type of circuit and the load on the line.
5.2
SUMMARY OF RESULTS
The mathematical models for power flow along transmission lines were solved analytically
using the Laplace transform method. From the results obtained, it was observed that the
model gave an actual representation of power flow along transmission lines because of the
80
exponential form of the solution. The solution it(x) gives the value of the current on transmission lines at any point x and time t.
With the empirical models developed, we can determine the power losses along transmission lines if given the distance, type of circuit and the load on the line.
The mathematical model for power losses based on ohmic and corona losses was minimized using the classical optimization technique. From the results obtained, it was discovered that the Hessian matrix of power losses over transmission lines is positive definite at the
stationary values. That is, the power loss is minimum at I = 0, V = V0 and d −→ ∞. Technically, this implies that the total power losses on tranmission lines will only be minimum
if
(i.) Power is tranmitted at a very low current along transmission lines. This will reduce
the ohmic or line loss on the conductors to the barest minimum. This conforms with
the principle of electric power transmission.
(ii.) The operating voltage is equal to the critical disruptive voltage. When this happens,
there is no ionisation of air around the conductor and hence no corona is formed.
Therefore, there will be no corona loss and
(iii.) The spacing between the conductors on the transmission line should be large. This
is because, an increase in the spacing between conductors reduces the electro-static
stresses. This therefore reduces the corona effect. If the spacing between the conductors is made very large as compared to their diameter, there may not be any corona
effect or losses on the line.
5.3
CONCLUSION
Accurate loss minimization on high voltage transmission line is a very important factor for
the efficiency of electric power network.
In this thesis, mathematical modelling has been applied to study the flow of power on
electric power transmission lines. We also utilized the notion of modelling to study power
losses on electric power transmission lines.
The classical optimization technique was then used to minimize the power losses on
transmission lines thereby proferring solutions to the power loss problem. The application
81
of the classical optimization technique to the mathematical model of power losses on electric
power transmission lines provides a better understanding of the problem of power losses on
high voltage transmission lines.
5.4
RECOMMENDATION
In the course of this research work, we observed that the power loss function is a multivariable function. We treated it as a multivariable optimization problem without constraint.
However, the following are recommended for further investigation and research work
i There is need to impose constraints on the power loss function to enable precise determination of the optimal value of space between conductors on the transmission
lines.
ii Empirical models of power losses on transmission lines should be developed as a multivariable function of load, circuit type and distance.
iii Models should be formulated for power losses that arises during the cause of generation
of electricity. The minimization of this kind of losses should be studied.
iv Instead of minimizing power losses along the distribution lines by using feeder reconfiguration and other methods which are capital intensive, similar research should
be conducted on distribution systems. Mathematical models should be developed for
losses in distribution systems and this kind of models should be minimized using any
of the optimization techniques.
82
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Figure 2.1: Pictorial view of 330 KV double circuit transmission line tower of the Nigerian
transmission network.
Figure 2.2: Pictorial view of 330 KV single circuit transmission line tower of the Nigerian
transmission network.
i
½RΔx
½LΔx
+
Ig
v
GΔx
.
KCL
½LΔx
½RΔx
i + Δi
+
Ic
v + Δv
CΔx
KVL
Figure 3.1: Equivalent Circuit of a Transmission Line
-
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
300
350
400
Figure 4.1: Scatter Diagram for Power Losses in MW for a load of 100 MW on Single
Circuit
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
300
350
Figure 4.2: Graph of Power Losses in MW for a load of 100 MW on Single Circuit
400
8
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
400
Figure 4.3: Scatter Diagram for Power Losses in MW for a load of 200 MW on Single
Circuit
8
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
Figure 4.4: Graph of Power Losses in MW for a load of 200 MW on Single Circuit
400
30
25
20
15
10
5
0
0
50
100
150
200
250
300
350
Figure 4.5: Scatter Diagram for Power Losses in MW for a load of 300 MW on Single
Circuit
30
25
20
15
10
5
0
0
50
100
150
200
250
300
Figure 4.6: Graph of Power Losses in MW for a load of 300 MW on Single Circuit
350
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
300
350
400
Figure 4.7: Scatter Diagram for Power Losses in MW for a load of 100 MW on Double
Circuit
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
300
350
Figure 4.8: Graph of Power Losses in MW for a load of 100 MW on Double Circuit
400
3
2.5
2
1.5
1
0.5
0
0
50
100
150
200
250
300
350
400
Figure 4.9: Scatter Diagram for Power Losses in MW for a load of 200 MW on Double
Circuit
3
2.5
2
1.5
1
0.5
0
0
50
100
150
200
250
300
350
Figure 4.10: Graph of Power Losses in MW for a load of 200 MW on Double Circuit
400
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
400
Figure 4.11: Scatter Diagram for Power Losses in MW for a load of 300 MW on Double
Circuit
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
400
Figure 4.12: Graph of Power Losses in MW for a load of 300 MW on Double Circuit