minimization of power loss

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 car-

ried out by OKE, Michael Olufemi with matriculation number 01/68EV002

in the Department of Mathematics, Faculty of Science, University of Ilorin,

Ilorin, Nigeria.

........................................

Professor O.M. Bamigbola

(Supervisor)

........................................

Professor M.O. Ibrahim

(Head of Department)

........................................

(External Examiner)

ii

.......................................

Date .......................................

Date .......................................

Date

DEDICATION

This work is dedicated to my late father: Pa David Eniola Oke.

iii

ACKNOWLEDGEMENTS

To God be the glory for the great and marvellous things he has done in

my life. I will forever be grateful to God almighty, the King of Kings, the

Lion of Judah, my messiah and everlasting Father, for giving me the grace

to complete this research work. His protection over me throughout my so-

journ in this university and the manifestation of his invisible hands made

the whole work a success.

I am very grateful for the unrivalled support I enjoyed from my amiable

and indefatigable supervisor, Prof. O.M. Bamigbola. His guidance, en-

couragement and constructive criticisms of the research work at every stage

made it a success.

I will like to thank Engr. (Prof.) I.E. Owolabi, Engr. (Prof.) S.B.

Adeyemo, Engr. (Prof.) J.O. Aribisala, Prof. O. Olaofe, Engr. (Dr.) E.A.

Okunade and Engr. A.A. Adegbemile for their fatherly advice and encour-

agement.

I will like to appreciate Engr. (Prof.) O.S. Onohaebi for the data on

empirical modelling, Engr. D.L. Atandare for the materials on electrical

power systems and some engineers of the Power Holding Company of Nige-

ria who have contributed in one way or the other to the success of this

research work. They include Engr. P.O. Falana, Engr. G.O. Ajayi, Engr.

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 sta of the Department of Math-

ematics, University of Ilorin, particularly Professors M.O. Ibrahim, J.A.

Gbadeyan, T.M. Adeniran, T.O. Opoola and J.S. Sadiku, Drs. O.A. Taiwo,

R.B. Adeniyi, J.O. Omolehin, S.O. Makanjuola, M.S. Dada, A.S. Idowu,

E.O. Titiloye , K. Rauf and K.O. Babalola as well as Dr (Mrs) O.A. Fadipe-

Joseph and Dr (Mrs) C.N. Ejieji.

I cannot but mention the support and encouragement I enjoyed from

Dr (Mrs) Y.O. Aderinto. I will also like to mention the encouragements

from my friends and colleagues who are still on the Ph.D. programme, their

camaraderie made the tension bearable.

I am also grateful to my parents, Late Pa D.E. Oke and Mrs. E.O. Oke,

for the basic education they gave me which qualies me for the postgradu-

ate work. I thank the authority of Ekiti State University, Ado-Ekiti for the

study leave which they gave me to undertake the programme.

Finally, I thank my wife, Olubunmi, and my children, Victor and Peace,

for their understanding and cooperation throughout the period of this re-

search work.

v

TABLE OF CONTENT

page

TITLE PAGE

CERTIFICATION

DEDICATION

ACKNOWLEDGEMENTS

TABLE OF CONTENT

LIST OF TABLES

LIST OF FIGURES

ABSTRACT

CHAPTER ONE : GENERAL INTRODUCTION

1.1 BACKGROUND TO THE STUDY

1.2 GOAL AND OBJECTIVES OF THE STUDY

1.3 SIGNIFICANCE OF THE STUDY

1.4 ORGANIZATION OF THE THESIS

1.5 NOTATIONS

1.6 DEFINITION OF SOME BASIC TERMS

i

ii

iii

iv

vi

x

xi

xiii

1

4

5

5

6

7

CHAPTER TWO : ELECTRIC POWER TRANSMISSION SYS-

TEMS

2.1 ELECTRIC POWER SYSTEMS

2.1.1 Historical Developments

2.1.2 Importance of Electric Power System

2.1.3 Electric Power Systems in Nigeria

vi

11 11 12 13

2.2 ELECTRIC SUPPLY SYSTEMS

2.2.1 Alternating Current and Direct Current Transmission Systems

2.2.2 Overhead and Underground Systems

2.3 MECHANICAL REQUIREMENTS FOR OVERHEAD LINES

2.4 MAIN COMPONENTS OF OVERHEAD LINES

2.4.1 Conductors

2.4.2 Line Supports

2.4.3 Insulators

2.4.4 Cross-arms

2.4.5 Stays

2.4.6 Miscellaneous Components of Overhead Lines

2.5 TRANSMISSION LINE CONSTANTS

2.5.1 Line Resistance

2.5.2 Line Inductance

2.5.3 Line Capacitance

2.5.4 Shunt Conductance

2.6 SKIN EFFECT

2.7 ECONOMICS OF POWER TRANSMISSION

2.7.1 Economic Choice of Conductor Size

2.7.2 Economic Choice of Transmission Voltage

2.8 CORONA PHENOMENON

2.8.1 Factors Aecting Corona

2.8.2 Advantages and Disadvantages of Corona

2.8.3 Methods of Reducing Corona

vii

19 20 21 23 23 24 25 26 26 27 27 28 28 28 29 29 29 30 31 31 31 32 33 33

CHAPTER THREE : MATHEMATICAL MODELS FOR POWER

FLOW OVER TRANSMISSION LINES

3.1 MATHEMATICAL PRELIMINARIES

3.1.1 Modelling

3.1.2 Dierential Equations

3.1.3 Laplace Transformation

3.2 KIRCHOFFS CIRCUIT LAWS

3.2.1 Kircho s Current Law

3.2.2 Kircho s Voltage Law

34 34 35 36 37 37 37

3.3 MATHEMATICAL MODEL FOR ELECTRIC POWER FLOW ALONG

LOSSY TRANSMISSION LINES

3.3.1 Model Formulation

3.3.2 Model Solution

38 38 40

3.4 MATHEMATICAL MODEL ALONG TRANSMISSION LINES WHEN

LEAKAGE TO GROUND IS SMALL

3.4.1 Model Formulation

3.4.2 Model Solution

3.5 ANALYSIS OF MATHEMATICAL MODELS

43 43 44 46

CHAPTER FOUR : MINIMIZATION OF POWER LOSSES OVER

TRANSMISSION LINES

4.1 OHMIC AND CORONA LOSSES

4.1.1 Ohmic Loss

4.1.2 Corona Loss

4.2 MATHEMATICAL MODELS FOR POWER LOSSES

viii

47 47 48 48

4.2.1 Model Based on Ohmic and Corona Losses

4.2.2 Empirical Models as Functions of Distance

48 50

4.3 MULTIVARIABLE OPTIMIZATION WITHOUT CONSTRAINTS 71

4.3.1 Properties of Hessian Matrix

71

4.3.2 Necessary and Sucient Conditions for the Existence of Extremal

Points

4.4 MINIMIZATION OF POWER LOSSES

4.5 DISCUSSION ON RESULTS

CHAPTER FIVE : GENERAL CONCLUSION

5.1 SUMMARY OF THESIS

5.2 SUMMARY OF RESULTS

5.3 CONCLUSION

5.4 RECOMMENDATION

REFERENCES

ix

72 78 79 80 80 81 82 83

LIST OF TABLES

Table 2.1: Per Capital Consumption of Electricity in some Countries

15

Table 4.1: Simulated Results of Power Losses on 330 KV Single Circuit of

the Nigerian Transmission Network

51

Table 4.2: Simulated Results of Power Losses on 330 KV Double Circuit of

the Nigerian Transmission Network

Table 4.3: Summations for a Load of 100 MW on Single Circuit



Table 4.6: Summations for a Load of 100 MW on Double Circuit



x

52 55 57 61 65 67 70

LIST OF FIGURES

Figure 2.1: Pictorial view of 330 KV double circuit transmission line tower

of the Nigerian transmission network. 17

Figure 2.2: Pictorial view of 330 KV single circuit transmission line tower

of the Nigerian transmission network.

Figure 3.1: Equivalent Circuit of a Transmission Line

18 38

Figure 4.1: Scatter Diagram for Power Losses in MW for a load of 100

MW on Single Circuit

53

Figure 4.2: Graph of Power Losses in MW for a load of 100 MW on Single

Circuit

53



56


Circuit

56



59


Circuit

59


MW on Double Circuit

Figure 4.8: Graph of Power Losses in MW for a load of 100 MW on

Double Circuit

63 63


xi



Double Circuit

66 66




Double Circuit

xii

69 69

ABSTRACT

Availability of electric power has been the most powerful vehicle for fa-

cilitating economic, industrial and social developments of any nation. Elec-

tric power is transmitted by means of transmission lines which deliver bulk

power from generating stations to load centres and consumers. For electric

power to get to the nal consumers in proper form and quality, losses along

the lines must be reduced to the barest minimum. A lot of research has been

carried out on analysis and computation of losses on transmission lines us-

ing reliability indices, but hardly any on the minimization of losses using

analytical methods. In another vein, a large body of literature exists for the

solution of optimal power ow problems using evolutionary methods, but

none of them has employed the versatile tool of mathematical modelling.

Thus, the goal of this work is to use the classical optimization approach

coupled with the mathematical modelling technique to minimize the trans-

mission power losses. Specically, the objectives of the study were to:

(i.) develop mathematical models for power ow and power losses along

electric power transmission lines and solve the mathematical models

for electric power ow along transmission lines using an analytical

method;

(ii.) develop empirical models of power losses as functions of distance; and

(iii.) minimize the power losses using the classical optimization technique.

In the research, I employed Kircho s circuit laws and a combination

xiii

of corona and ohmic losses in obtaining the mathematical models for the

power ow and power losses respectively. Empirical models of the power

losses were developed using regression analysis.

The ndings of this study were:

(i.) the models for power ow along transmission lines evolved as homo-

geneous second-order partial dierential equations which were solved

analytically using the method of Laplace transform;

(ii.) the model for power losses over the transmission lines was obtained

as the sum of the ohmic and corona losses;

(iii.) the empirical models developed are monotonic increasing functions of

distance. Thus, establishing that power losses increases with distance;

(iv.) power losses are minimized when the operating transmission voltage

is equal to the critical disruptive voltage.

With the above results, a workable strategy can be formulated to reduce

to the barest minimum electric power losses along transmission lines so as

to ensure availability of electric power, in proper form and quality, to con-

sumers. Hence, this research work has addressed the problem of minimizing

electric power losses during transmission.

Xiv

MINIMIZATION OF POWER LOSSES OVER ELECTRIC POWER TRANSMISSION LINES

Abstract

Chapter 1

GENERAL INTRODUCTION

1.1

BACKGROUND TO THE STUDY

Energy is a basic necessity for the economic development of a nation. There are dierent

forms of energy, but the most important form is the electrical energy, Gupta (2008) and

Mehta and Mehta (2008). A modern and civilized society is so much dependent on the use

of electrical energy. Activities relating to the generation, transmission and distribution of

electrical energy have to be given the highest priority in the national planning process of any

nation because of the importance of electrical energy to the economic and social development

of the society. In fact, the greater the per capital consumption of electrical energy in a

country, the higher the standard of living of its people. Therefore, the advancement of

a country is measured in terms of its per capital consumption of electrical energy, Gupta

(2008) and Mehta and Mehta (2008).

Power plants planning in a way to meet the power network load demand is one of

the most important and essential issues in power systems. Since transmission lines connect

generating plants and substations in power network, the analysis, computation and reduction

of transmission losses in these power networks are of great concern to scientists and engineers.

A lot of research works have been carried out on the above listed aspects. Zakariya

(2010) made a comparison between the corona power loss associated with HVDC trans-

mission lines and the ohmic power loss. The corona power loss and ohmic power loss were

measured and computed for dierent transmission line congurations and under fair weather

and rainy conditions. It was pointed out in the work that the general trend of neglecting

the corona power loss is not always valid. It was found from the comparison that, when

1

the transmission line is moderately or lightly loaded, the percentage of corona power loss to

ohmic power loss could reach up to one hundred percent especially if the transmission line

is operating at a voltage well above the corona onset value. This percentage is also found to

increase substantially under rainy conditions. Finally, it was also discovered that, the ratio

of corona to ohmic power loss, decreases with increasing number of bundles. Numphetch

et al. (2011) worked on loss minimization using optimal power ow based on swarm in-

telligences. Thabendra et al. (2009) considered multi-objective optimization methods for

power loss minimization and voltage stability while Abdullah et al. (2010) looked at trans-

mission loss minimization and power installation cost using evolutionary computation for

improvement of voltage stability. Bagriyanik et al. (2003) used a fuzzy multi-objective

optimization and genetic algorithm-based method to nd optimum power system operating

conditions. In addition to active power losses, series reactive power losses of transmission

system were also considered as one of the multiple objectives. Onohaebi and Odiase (2010)

considered the relationship between distance and loadings on power losses using the exist-

ing 330 KV Nigerian transmission network as a case study in his empirical modelling of

power losses as a function of line loadings and lengths in the Nigeria 330 KV transmission

lines while Moghadam and Berahmandpour (2010) developed a new method for calculating

transmission power losses based on exact modelling of ohmic loss. Ramesh et al. (2009)

looked at minimization of power loss in distribution networks by using feeder restructuring,

implementation of distributed generation and capacitor placement method. Lo and Gers

(2006) considered feeder reconguration for losses reduction in distribution systems. Others

who researched into power losses include Rugthaicharoencheep and Sirisumrannukul (2009),

Crombie (2006), Marwan and Imad (2002), Ayman (2004), Sarajcev et al. (2003) and Daniel

(2005), to mention a few.

Various researchers have also worked on the ow of power on electrical networks. Pandya

and Joshi (2008) presents a comprehensive survey of various optimization methods for solving

optimal power ow problems. The methods considered in the work include linear program-

ming, Newton-Raphson, quadratic programming, nonlinear programming, interior point and

articial intelligence. Under the articial intelligence method, the following were also con-

sidered articial neural network method, fuzzy logic method, genetic algorithm method,

evolutionary programming method, ant colony optimization method and particle swarm

optimization method. It was found in the paper that the classical methods have a lot of

2

limitations. In most cases, mathematical formulations have to be simplied to get the solu-

tions because of the extremely limited capability to solve real-world large-scale power system

problems. The classical methods are weak in handling qualitative constraints and they have

very poor convergence. The methods are also very slow and computationally expensive in

handling large-scale optimal power ow problems. It was also discovered in the paper that

the articial intelligence methods are relatively versatile for handling various qualitative

constraints and that the methods can nd multiple optimal solutions in a single simulation.

They are therefore suitable in solving multi-objective optimization problems. William and

Jose (2002) looked at alternative optimal power ow formulations while Claudio et al. (2001)

worked on comparison of voltage security constraint using optimal power ow techniques.

Roya et al. (2008) considered power ow modelling for power systems with dynamic ow

controller. Other researchers who also worked on power ow include Bouktir et al. (2004),

Swarup (2006), Tarjei (2006), Bouktir and Slimani (2005), Burchett et al. (1982), Dommel

and Tinney (1968), Heinkenschloss and Vicente (1994) and Taiyou and Robert (2006).

In addition, several researchers have also worked on electric power systems. Aderinto

(2011) worked on an optimal control model of the electric power generating system. In

the research work, she developed a mathematical model for the electric power generating

system using the optimal control approach and characterized the mathematical model by

prescribing the conditions for the optimality of the electric power generating system and the

analytic requirements for the existence and uniqueness of the solution to the system. The

optimality condition for the model was determined and the model was solved analytically

and numerically. In the study, two control variables were identied, the rst for load shed-

ding among the generators in the system and the second for restriction on the capacity of

the generators. The problem was formulated based on the second control variable since the

rst control variable can only be on or o as the case may be. The optimality conditions

for the system were imposed implicitly on the controls and the mathematical model repre-

sents a stable loss-free generating system. From the work, it was shown that the generation

loss can be controlled and stabilized. Oke et al. (2007) considered the perspectives on

electricity supply and demand in Nigeria while Ibe and Okedu (2007) looked at optimized

electricity generation in Nigeria. Bamigbola and Aderinto (2009) characterized an optimal

control model of electric power generating system. Karamitsos and Orfanidis (2006) con-

sidered an analysis of blackout for electric power transmission systems while Aderinto et

3

al. (2010) looked at optimal control of air pollution with application to power generating

system model. Others whose researches touched on electric power systems include Savenkov

(2008), Youssef and Hackum (1989), Williams and John (2006), Anderson (2008), Bansal

(2005), Nanda et al. (1989), Aribia and Abdallah (2007), Vaisakh and Rao (2008), Kamin-

skyi (2009), Billinton (1994), Schenk and Ahsan (1985), Jocic et al. (1983), Doraiswami et

al (1995), Caprio (1984), Dandeno (1982), Miroslav et al. (2001), Bockarjova et al. (2003)

Okafor and Adebanji (2009), Dmytro et al. (2007), Grigsby (1998), Komolafe et al. (2009),

Kundur (1994), Kusko (1968), Lee et al. (1986), Rajput (2003), Shahildehpour and Labudda

(2005), Thomas and Martin (2002), Wayne (2001), Youssef and Hackum (1989), Authur and

Connie (1988), Branimir and Radivo (1993), Hicks (1966), Joe et al. (2004), Baskaran and

Palanisamy (2005), Ayodele et al. (2008) and Lee et al. (1988), to list a few. As such, much

emphasis has been on proper design of electrical power systems and reduction of losses using

feeder reconguration and evolutionary techniques.

Loss minimization is a critical component for ecient electric power supply systems.

Losses in an electric power system should be around 3 percent to 6 percent, Ramesh et al.

(2009). In developed countries, it is not greater than 10 percent. However, in developing

countries it is still over 20 percent, Ramesh et al. (2009). Therefore stakeholders in the power

sector are currently interested in reducing the losses on electric power lines to a desired and

economic level. The purpose of this research work, therefore, is to develop mathematical

models for power losses along transmission lines and to minimize the losses using classical

optimization techniques.

1.2

GOAL AND OBJECTIVES OF THE STUDY

Power losses result in lower power availability to the consumers, leading to inadequate

power to operate their appliances. High eciency of power system is determined by its

low power losses. The goal of this research work therefore is to use classical optimization

techniques to minimize the transmission power losses on transmission lines. The objectives

of the research work are to:

(i.) Develop mathematical models for electric power ow and power losses along electric

power transmission lines;

4

(ii.) Solve the mathematical models for electric power ow along transmission lines analyt-

ically;

(iii.) Develop empirical models of power losses as functions of distance; and

(iv.) Minimize power losses using the classical optimization technique.

1.3

SIGNIFICANCE OF THE STUDY

The mathematical representation of power ow along transmission lines provides a bet-

ter understanding of the ow of electric power on transmission lines and the evolution of

voltage and current along the lines. The mathematical representation of power losses along

transmission lines gives an insight into the major problems on electric power transmission.

The minimization of losses on electric power transmission lines using classical optimization

technique provides a solution, in a compact form, to the major problem encontered in power

transmission.

1.4

ORGANIZATION OF THE THESIS

The remaining part of this thesis are organised as follows:

Various notations used in the thesis are listed in section 1.5 while section 1.6 gives the

denition of some basic terms used in the thesis. Chapter two focuses on electric power

transmission systems detailing on requirements for transmitivity. Chapter three is devoted

to the development of mathematical models for power ow over transmission lines. Mathe-

matical preliminaries were considered in section 3.1. In section 3.2, we formulated and solved

the model for electric power ow along lossy transmission lines, while in section 3.3, we de-

rived and solved the model for electric power ow along transmission lines when leakage to

ground along the line is small. We then analysed the models in section 3.4.

In chapter four, we treated minimization of power losses over transmission lines. Specif-

ically, secion 4.1 is on preamble where we detailed the requirements for the existence of

an extemum of a function of several variables. In this section, we also discussed ohmic

and corona losses which we now used in subsection 4.2.1 for the development of a model for

power losses along transmission lines and in subsection 4.2.2, we developed empirical models

5

of power losses as functions of distance. In Section 4.3, we considered the power loss func-

tion as a multivariable optimization without constraints and minimized it using the classical

optimization technique while in section 4.4, we looked at the minimization of power losses

using dierential calculus. Discussion on results is what we have in section 4.5. The thesis

is rounded up in chapter ve with general conclusion. Section 5.1 treated a summary of the

work reported in the thesis and summarized the results obtained in section 5.2. Section 5.3

is on conclusion while section 5.4 suggests outstanding issues for further research work.

1.5

NOTATIONS

We made use of the following notations in this thesis:

(Ik) represents current along the kth branch.

(Vk) represents voltage along the kth branch.

represents summation.

L represents Laplace transform.

L1 represents inverse Laplace transform.

Isc(x) represents complementary function.

Isp(x) represents particular solution.

I represents current along the conductor.

R represents resistance of the conductor.

f represents frequency of transmission.

represents air density factor.

r represents radius of conductors.

d represents space between the transmission lines.

q represents charge on the transmission line.

v represents potential dierence between the conductors.

V represents operating voltage.

V0 represents distruptive voltage.

represents resistivity of the conductor.

represents ux leakage.

L represents length of the conductor.

A represents cross-sectional area of the conductor.

6

represents conductivity of the conductor.

TLoss represents total loss on transmission lines.

LOhmic represents ohmic loss.

LCorona represents corona loss.

1.6

DEFINITION OF SOME BASIC TERMS

In this section, we give the denition of some basic terms used in the thesis.

1. Optimization

Optimization is the act of getting the best result under given circumstances, Rao

(1998). It can therefore be dened as the process of obtaining the optimal (best)

solution to certain mathematical problems, which are often models of physical reality,

Minoux (1986). Many problems in engineering, management and planning lead to

mathematical models requiring the idea of optimization for solution, Craven (1995).

2. Classical Optimization

The classical optimization techniques are methods used in nding the optimum of

continuous and dierentiable functions. It is an analytical method that makes use of

dierential calculus techniques in nding the optimum points. The classical optimiza-

tion method forms the basis for the development of most of the numerical optimization

techniques.

3. Hessian Matrix

An Hessian matrix is a square matrix of second order partial derivatives of a function

of several variables. It was developed in the 19th century by a German mathematician

called Ludwig Otto Hesse.

4. Degenerate and Non-degenerate Critical Point

If the derivative of a function f is equal to zero at some point x, then f has a critical

or stationary value at x. The determinant of the Hessian matrix at x is called the

discriminant. If this discriminant is equal to zero then, the point x is called a degener-

7

ate or non-morse critical point of f. Otherwise it is a non-degenerate or morse critical

point of f.

5. Positive Denite Matrix

A matrix A of order n is said to be positive denite if all its eigenvalues are positive.

That is, if all values of which satises the determinant equation

|A I | = 0

are positive, Rao (1998).

Another test of the positive deniteness of a matrix A of order n is the evaluation of

its determinants:

A1 =

a11

A2 =

a11 a12

a21 a22

a11 a12 a13

A3 = a21 a22 a23

a31 a32 a33

....

a11 a12 a13.....a1n

a21 a22 a23.....a2n

An = a31 a32 a33.....a3n

an1 an2 an3.....ann

A matrix A of order n will therefore be positive denite if and only if all values of A1,

A2, A3, ....., An are positive.

6. Negative Denite Matrix

A matrix A of order n is said to be negative denite if and only if the signs of Ai in

(5) above is (1)i for i = 1,2,3,4,.....,n.

8

dx

dx

7. Positive Semidenite Matrix

A matrix A of order n is said to be positive semidenite if and only if some of the Ais

in (5) above are positive and the remaining ones are zero.

8. Eigenvalues

Eigenvalues of a matrix A are all values of which satises the determinantal equation

det (A I ) = |A I | = 0

(1.1)

where I is an identity matrix of the same order as A

9. Initial Value Problem

An initial value problem (IVP) is a dierential equation in which the solution y(x)

satises prescribed side conditions imposed on the unknown y(x) or its derivatives at

an initial point x0 , Dennis and Michael (2005) and Eagleeld (1989). An initial value

problem is of the form

Solve

subject to

dny n

= f (x, y, y , y , ....., y(n1)) (1.2)

y(x0) = y0, y (x0) = y1, y (x0) = y2, ....., y(n1)(x0) = yn1

(1.3)

where y0, y1, y2, ..., yn1. are arbitrarily specied real constants.

The values of y(x) and its rst (n - 1) derivatives at a single point x0 , that is y(x0) =

y0, y (x0) = y1, y (x0) = y2, ....., y(n1)(x0) = yn1 are called the initial conditions.

10. Boundary Value Problem

A boundary value problem (BVP) is a dierential equation in which the solution y(x)

satises prescribed conditions imposed on the unknown y(x) or its derivatives at more

than one point. A dierential equation of the form:

Solve

a2(x)

d2y 2

+ a1(x)

dy dx

+ a0(x)y = g(x).

(1.4)

subject to

y(a) = ya, y(b) = yb,

9

(1.5)

dx dx

dx dx

is called a boundary value problem. The prescibed values y(a) = ya, y(b) = yb are

called boundary conditions, Dennis and Michael (2005), Etgen (1999) and Kreyszig,

(1987).

11. Homogeneous and Nonhomogeneous Dierential Equations

An nth-order linear dierential equation of the form in (1.6) below is said to be non-

homogeneous if g(x) is not identically zero, Dennis and Michael (2005).

an(x)

dny

n

+ a(n1)(x)

d(n1)y

(n1)

+ ... + a1

dy dx

+ a0(x)y = g(x).

(1.6)

If g(x) is equal to zero, then the nth-order dierential equation is called homogeneous

and we have

an(x)

dny

n

+ a(n1)(x)

d(n1)y

(n1)

+ ... + a1

dy dx

+ a0(x)y = 0.

(1.7)

This explanation also holds for partial dierential equations.

12. Critical Disruptive Voltage

The critical disruptive voltage (V0) is the minimum voltage at which corona occurs.

13. Node or Junction

This is a point where two or more branches meet.

14. Ohmic Loss

Ohmic loss is a loss of power on transmission lines which occurs as a result of the

resistance of conductors against the ow of current.

15. Corona Loss

Corona loss is a loss of power on transmission lines which normally occurs as a result

of the ionization of thin layer of air around the line. This ionization of air is experi-

enced when the applied voltage exceeds the critical disruptive voltage in high voltage

transmission lines.

10

Chapter 2

ELECTRIC POWER

TRANSMISSION SYSTEMS

2.1

2.1.1

ELECTRIC POWER SYSTEMS Historical Developments

Before 1800, researches on electrical and magnetic phenomena were only carried out by

very few scientists. As at that time, no real applications were known. People illuminated

their homes with candles , whale oil lamps and kerosine lamps, Atandare (2007) and Duncan

and Muluktla (1986). Between 1800 and 1810, commercial illuminating gas companies were

formed. It was rst formed in Europe and later in the United States of America. Scientic

research increase in the area of electrical and magnetic phenomena throughout the 19th

century. Two independent researchers Michael Faraday and Joseph Henry Ampere had

already observed that magnetic elds were created by electric currents but no one had

discovered how electrical currents could be produced from magnetic elds. Faraday worked

on such problems between 1821 and 1831 and nally succeeded in formulating a law on

it that bears his name. He subsequently built a machine that generated voltage based

on the principle of magnetic induction. Between 1840 and 1877 several people including

Charles Wheatstone, Carl Siemens and Gramme, applied the principle of induction for the

construction of primitive electrical generators, Atandare (2007), Charles (1986) and Duncan

and Muluktla (1986).

11

In 1878, a 29-year old inventor named Thomas Edison worked on a number of projects

including the development of an incandescent electric lamp. In October 1879, after several

unsuccessful trials and experiments, an enclosed evacuated bulb was energised. In 1882 the

rst system installed to sell electrical energy for incandescent lighting in the United States of

America began operations. The system was DC, three wire, 220/110 volts. The early days

electrical companies referred to themselves as illuminating companies because lighting

was their only service. In 1890, the newly formed Westinghouse Company (WC) developed

another form of electricity name Alternating Current. With this, most of the problems

associated with DC generators were eliminated, Atandare (2007), Olle (1987) and Duncan

and Muluktla (1986).

2.1.2

Importance of Electric Power System

It is no doubt that the civilization of mankind are closely interwoven with energy. Electri-

cal energy occupies a top position in the energy hierarchy because of its usefulness at home,

industry, agriculture and even in the transportation sector. Electrical energy can be gener-

ated centrally in bulk and transmitted economically over long distance. The advancement

in science and technology has made it possible to convert electrical energy into any desired

form like heat, light, motive power etc. This has given electrical energy a place of pride

in the modern world. The social structures and the industrial development of any country

depends primarily upon low cost and uninterrupted supply of electrical energy, Mehta and

Mehta (2008). Availability of electricity has been the most powerful vehicle of introducing

economic development and social change throughout the world. The process of moderni-

sation, increase in productivity, agriculture and industry basically depend upon adequate

supply of electrical energy. The annual per capital consumption of electrical energy is a very

important yardstick for measuring the development of a nation, Gupta (2008).

Generation of electrical energy is the conversion of energy available in dierent forms

in nature to electrical energy. The ever increasing use of electrical energy for industrial,

domestic and commercial purposes necessitated the bulk production of electrical energy.

This bulk production is achieved with the help of suitable power production stations which

are generally referred to as electric power generating stations or electric power plants. A

generating station usually employs a prime mover coupled with an alternator to produce

electric power.

12

Electrical energy is generated at power stations which are usually situated far away

from load centres. Hence an extensive network of conductors between the power stations

and the consumers is required. This network of conductors may be divided into two main

components, called the transmission system and the distribution system. The transmission

system is to deliver bulk power from power stations to load centres and large indusrial

consumers while the distribution system is to deliver power from substations to various

consumers.

Electrical energy produced must be transmitted and distributed to the point of use as

soon as it is needed. Transmission lines and other materials are needed to achieve this pur-

pose. Transmission lines are materials or media that are used to transmit electric energy and

signals from one point to another, specically from a source to a load. They can be regarded

as a set of conductors being run from one place to another and supported on transmission

towers. This involves connections between an electric generating plant and a substation

which is several hundred kilometers away. The transmission and distribution stages are

very important to electric power system, because without these stages the generated power

cannot get to the load centres not to talk of getting to the nal consumers. Power losses

along these stages should be reduced to the bearest minimum so that the nal consumer

will get the normal power to operate their appliances, Mehta and Mehta (2008), Wadhwa

(2009) and Atandare (2007).

Power plants planning in a way to meet the power network load demand is one of

the most important and essential issues in power systems. Since transmission lines con-

nect generating plants and substations in power network, the analysis and computation of

transmission losses of these power networks are of great concern to scientists and engineers.

Another issue of great importance to scientists and engineers is nding methods to reduce

the losses on electric power lines to a desired and economic level.

2.1.3

Electric Power Systems in Nigeria

Source of electric power was rst known in Nigeria in 1896 when a 30 KW, 80 Hz, single

phase locomotive generator was installed in Ijora, Lagos, the then seat of British colony. The

operation, maintenance and distribution of this generator was solely the responsibility of the

Power Works Department (PWD). In 1924, with the increasing population, a three phase,

50 Hz system of power system became known and electric power were been distributed in

13

few cities of the country by some isolated generating stations like Cameroons Development

Corporation (CDC), African Timber and Polywood Company (ATPC) and Nigeria Electrical

Supply Corporation (NESCO), Atandare (2007).

In 1946, the management of electrical power supply in the country was taken over by the

Nigeria Government Electricity Undertaking (NGEU). This new organ of government took

care of electricty distribution and expansion in the country. In 1952, Electricity Corpora-

tion of Nigeria (ECN) was establised and this gave birth to the Ijora Power Station which

had 10 MW coal-red turbo-generators, Atandare (2007). Some investigations for possible

siting of hydro electric power stations ware carried out in 1953 by Netherlands Engineering

Consultants on behalf of Electricity Corporation of Nigeria. This now resulted in the con-

struction of Kainji Dam and the associated hydro-generators for power production. With

the construction of Kainji Dam, Niger Dam Authority (NDA) was established in 1964 with

the responsibility of further constructing the dam, power station and the associated 330

KV transmission lines between Kainji and the national control centre at Osogbo, Atandare

(2007), Manafa (1978).

In Nigeria, there cannot be any successful survey on generation, transmission and distri-

bution of electricity without reference to National Electric Power Authority (NEPA) which

was established by Decree 24 of 1st April, 1972, with the almalgamation of Electricity Cor-

poration of Nigeria (ECN) and Niger Dams Authority (NDA). The decree gave NEPA the

mandate to maintain and co-ordinate an ecient electricity supply to all parts of the coun-

try. NEPA was also empowered to manage and maintain electrical power undertakings,

establish new electric power undertakings, generate, transmit and distribute electric power

to every part of the country, Power Sector Reforms (2005) and Atandare (2007).

However, in March, 2006 NEPA was renamed Power Holding Company of Nigeria (PHCN)

with eighteen business units. NEPA (now PHCN) has eight major generating stations lo-

cated nationwide. These stations are connected by transmission substations to form the

National Grid System with the control centre at Osogbo, Osun State. These stations in-

clude three hydropower stations and ve thermal stations. The total installed capacity of

the existing government-owned generating stations in Nigeria is 6200MW. Although the

stations produced below the actual installed capacity of 6200MW, Power Sector Reforms

(2005). In order to improve the power generation in the country, the Federal government has

seven new on-going thermal power projects in the Niger Delta Area. The total generating

14

Country Per Capital Consumption (in KW)

United State of America 3.2

Cuba 0.38

United Kingdom 1.33

Ukraine 1.33

Iraq 0.42

South Korea 1.09

Nigeria 0.03

Egypt 0.27

capacity of these on-going thermal projects is 2250MW, Popopla et al. (2008). There are

some existing independent power producers in the country with total generating capacity of

2552MW. These independent power producers also have on-going projects with a generating

capacity of 378MW. If all the existing and on-going power generating stations are producing

at optimum level, Nigeria will be generating a total of 11380MW, Atandare (2007).

The per capital consumption of electricity in a country is one of the strongest and most

reliable indices for measuring the degree of development of that nation. The per capital

consumption of electricity in Nigeria is 0.03 KW. This is very low compared to the per

capital consumption of electricity in other countries. We can see this in Table 2.1 which

gives the per capital consumption of electricity in some selected countries as given by the

International Energy Institutes comparative analysis of the per capital consumption of

electricity worldwide, Atandare (2007).

Table 2.1: Per Capital Consumption of Electricity in some Countries, Atandare

(2007).

Improvement in the quality and quantity of infrastructural services, especially electricity,

is fundamental to rapid and sustainable economic growth in any country. But inadequate

quantity, quality and access to electricity services have been a regular feature in the Nigerian

power sector, Iwayemi (2008), Adeniyi (2008) and Adeyemo (2008). The Transmission

15

Company of Nigeria PLC (TCN) manages Nigerians power grid. TCN ensures that power

is transmitted eciently over the national grid and delivered to the distribution companies

in their designated franchise areas, TCN Reports (2006). The Transmission Company of

Nigeria (TCN) is subdivided into ve zones for management and operational purposes. It

is managed from a national control centre at Osogbo, Osun State and a secondary control

centre at Shiroro, Niger State. It has six regional oces and several satellite work centres,

TCN Reports (2006), Atandare (2007),Fasina (2008) and Onohaebi and Odiase (2010)

The Nigerian 330KV transmission network employed 350mm2 aluminium conductor steel

re-inforced (ACSR). Single and double circuits are used in the trasmission network. The

double circuit has the advantage that it ensures continuity of power supply. In case there is

breakdown of one circuit, the continuity of supply can be maintained by the other circuit.

The supporting structures are made of steel towers and are spanned at an average distance

of 500m apart. The towers have heights of 75 metres for double circuits and 54 metres for

single circuits, Onohaebi and Odiase (2010). Figures 2.1 and 2.2 show the 330 KV double

circuit and single circuit transmission line towers respectively.

16

The Nigerian transmission network comprises of over 11000km of transmission lines. (i.e

over 5000km of 330KV transmission lines and 6000km of 132KV transmission lines). It

also has about 24000km of 33KV subtransmission lines and 19000km of 11KV distribution

lines together with 22500 substations all over the country, Atandare (2007) and Onohaebi

and Odiase (2010). The National Electric Power Authority (NEPA), now Power Holding

Company of Nigeria (PHCN), had built twenty three 330KV and ninety 132KV transmission

substations as at 1992 and all these trasmission lines and substations are put into operation

nationwide, TCN Reports, (2006).

With all these in place, there are still a lot of problems with the transmission of electricity

in Nigeria. Loss of power on transmission lines is a global problem and this is a major

problem we have with the transmission of electricity in Nigeria. The Nigerian 330 KV

transmission grid is characterized by high power losses. Most of these power losses are

due to very long transmission lines. Some of these lines include, Benin to Ikeja West (280

km), Osogbo to Benin (251 km), Osogbo to Jebba (249 km), Jebba to Shiroro (244 km),

Birnin Kebbi to Kainji (301 km), Jos to Gombe (265 km) and Kaduna to Kano (230 km),

Onohaebi and Odiase (2010). Distance is not the only factor responsible for loss of power on

transmission lines. Other factors include, the type and size of the conductor, enviromental

factors such as temperature, air density factor etc.

The power loss in Nigerian transmission system was estimated at 337.5 GWH in 2005.

High power losses in an electrical system imply high nancial losses to the nation. The nan-

cial loss associated with the loss in power in 2005 was estimated at 2.6 billion Naira, Kuale

and Onohaebi (2007). In order to maintain a good electric power system, the power losses

on transmission lines must be minimal. Minimal losses will help to ensure that generators,

transformers, lines, etc are subjected to less stresses, Onohaebi and Odiase (2010). Power

generation in a system and the cost involved in the generation will be reduced if the total

losses in transmission are minimal. This is because power generation must meet with load

demands as well as losses, Mehta and Mehta (2008), Wadhwa (2009) and Atandare (2007).

2.2

ELECTRIC SUPPLY SYSTEMS

The convayance of electric power from a power station to consumwers premises is known

as electric supply system. Therefore an electric supply system consists of three main compo-

19

nents which include the power stations, the transmission system and the distribution system.

Electric power is produced at power stations which are usually located far away from con-

sumers. It is then stepped up and transmitted over long distances from the power stations

to load centres by means of conductors known as transmission lines. We have primary and

secondary (or sub-) transmission stages. Finally, power is distributed to a large number

of consumers through a distribution network. We also have primary and secondary (sub-)

distribution stages. The electric supply system can be broadly classied into:

i. Alternating Current and Direct Current Systems

ii. Overhead and Underground Systems.

2.2.1

Alternating Current and Direct Current Transmission Sys-

tems

Electrical power can be transmitted and distributed by either alternating current (AC) or

direct current (DC) systems but in practice 3-phase, 3-wire AC system is generally used

for transmission of large blocks of power and 3-phase, 4-wire AC system is used for the

distribution of electric power. The main advantage of AC transmission system is that voltage

can be stepped up at generating end by means of step up transformers to a desired value for

transmission purposes and then stepped down at the distributing end by means of step down

transformers for distribution purposes. This permits the transmission of electric power at

high voltage. Apart from this, the maintenance of AC sub-stations is easier and cheaper.

Also in AC transmission system, electric power can be generated at high voltages easily,

Gupta (2008), Mehta and Mehta (2008). The AC system also has its own disadvantages

which include the following:

i. An AC line requires more copper than a DC line

ii. In overhead transmission lines, spacing between the conductors is always kept more in

order to provide adequate insulation and avoid corona loss.

iii. The construction of an AC transmission line is more complicated than the one for a

DC transmission line.

iv. The eective resistance of the transmission line is increased because of skin eect in

AC line.

20

v. AC transmission line has capacitance. Therefore there is a continuous loss of power

due to charging current even when the line is open

Transmission of electric power by high voltage DC system is superior to that of AC system

because of the following reasions.

i. There is no skin eect in a DC system. This enables the entire cross-section of the

conductor to be utilized.

ii. It requires only two conductors for transmission as against three for the AC system.

iii. There is less corona loss in a DC line. Therefore there is less interference with com-

munication circuits.

iv. For the same operating voltage, the stress on the insulation is less in a DC line than

in an AC line. This implies that a DC system requires less insulation.

v. There is no inductance, capacitance and surge problems in a DC transmission.

A major disadvantage of a DC system is that the DC voltage cannot be stepped up for

transmission of power at high voltages. Another disadvantage is that electric power cannot

be generated at high DC voltage.

It is clear from the above explanations that high voltage DC transmission is better than

high voltage AC transmission even though transmission of electricity is being done at present

in most countries by AC system. Therefore there is an increasing interest by engineers in

DC high voltage transmission of electricity. The introduction of mercury arc rectiers and

thyratrons have made it possible to convert AC to DC and vice versa. This arangement

will now enable generation and distribution of electricity to be done by AC system and high

voltage transmission of electricity to be done by DC system.

2.2.2

Overhead and Underground Systems

Electric power can be transmitted or distributted either by means of overhead lines or by

underground cables. The underground cables are rarely used for power transmission because

of the following reasons. In the rst place, power is generally transmitted over long distances

to load centres so the installation costs for underground transmission will be very high. The

initial installation costs of underground system is almost double that of overhead system.

21

Secondly electric power has to be transmitted at high voltages for economic reasons. It

will therefore be very dicult to provide proper insulation for the cables to withstand the

high pressures. The underground system cannot be operated above 66 KV because of the

insulation problem whereas overhead transmission system can be designed to operate at 400

KV or above, Gupta (2008). With the continuous rise in voltage level as a result of increase

in power demand, power transmission by overhead transmission lines is now the order of

the day. Another advantage of overhead transmission system over underground system is

that overhead system is more exible than underground system. In overhead system, new

conductors can be laid along with the existing ones for load expansion. In underground

transmission systems such new conductors needed for load expansion will be laid in new

channels. Though there are very rare chances of faults occuring in undergroung systems, if

it occurs it is always very dicult to locate and more expensive to repair than in overhead

systems. The underground system also has its own advantage over the overhead system

which include the following:

i. The underground system is safer than the overhead system.

ii. The maintenance cost of underground system is very low compared to that of overhead

system.

iii. In underground systems there is no interference to communication circuits.

iv. Because of less spacing between conductors in underground systems, the inductance

on the line is very low and therefore voltage drop is low in underground cables than

overhead cables.

v. Underground transmission and distribution systems are neater because no wire is vis-

ible outside.

vi. There are very few chances of faults in underground system.

vii. Underground system is free from interruption of services on account of thunder storm,

lightning or objects falling across the wires.

22

2.3

MECHANICAL REQUIREMENTS FOR OVER-

HEAD LINES

Transmission line is a very important link between generating stations and major load

centres because power from generating stations is transmitted at high voltage over long

distances to these load centres. It has now become imperative that transmission of power

is carried out with minimum loss and disturbance because of the increase in the demand

for power as a result of industrial growth. To achive this goal, the transmission line should

be designed and constructed in such a way that the current carring capacity would be high

so as to transmit the required power over a given distance without much voltage drop and

overheating. The losses on the line should be small and the insulation of the line should

be enough to cope with the high voltage in the system. An overhead transmission line is

subjected to uncertain weather conditions and other external interference. This now calls for

the use of proper mechanical factors to give the transmission system sucient mechanical

strength so that it will be technically sound, reliable and ecient. In general, the strength

of the line should be such as to cope with the worst probable weather conditions and provide

satisfactory service over a long period of time without too much maintenance.

2.4

MAIN COMPONENTS OF OVERHEAD LINES

The main components of overhead lines are:

i. Conductors

ii. Line supports

iii. Insulators

iv. Cross-arms

v. Guys and Stays

vi. Miscellaneous Components of Overhead Lines which include: lightning arrestors, fuses

and isolating switches, barbed wires, danger plates, continuous earth wires, vee-guards,

guard wires and bird guards.

23

2.4.1

Conductors

The conductor is one of the most important items in the transmission of electric power.

Therefore proper choice of material and size of the conductor is of considerable importance.

The conductor materials used in the transmission of electricity should have the following

properties:

i. high electrical conductivity;

ii. high tensile strength (in order to withstand mechanical stresses);

iii. low specic gravity (so that weigth per unit volume is small); and

iv. low cost (so that it can be used for long distances).

All the above properties are not found in a single material. Therefore, while selecting the

conductor material for a particular transmission purpose, a compromise is made between

the cost and the required mechanical and electrical properties.

All conductors used for overhead transmission lines are preferably stranded in order to

increase its exibility. Solid wires are only used as conductors when the cross-sectional area

needed is small and the conductor is for a short distance. If solid wires are used for larger

cross-section and very long distances, continuous vibrations and swinging would produce

mechanical fatigue and the wire would fracture at the point of support, Mehta and Mehta

(2008). In stranded conductors, there is generally one central wire and round this wire we

have successive layers of wires containing 6, 12, 18, 24, 30, ....... wires.

Copper is an ideal material for the transmission of electric power because of its high

electrical conductivity, lower electrical resistivity, high current density and greater tensile

strength. However, because of its high cost and non-availability, it is rarely used for the

purpose.

Aluminium is cheap, light and has a lower electrical conductivity, higher electrical re-

sistivity, lower current density and tensile strength as compared to copper. Aluminium is

also available for use in abundance. The smaller conductivity of aluminium implies that, for

any particular transmission eciency, the cross-sectional area of conductor must be greater

in aluminium than in copper. In fact, the diameter of aluminium conductor will be about

1.26 times the diameter of copper conductor, Mehta and Mehta (2008). The specic gravity

24

of aluminium (2.71 gm/cc) is less than that of copper (8.9 gm/cc). The increased cross-

sectional area of aluminium exposes a greater surface of it to wind pressure and its lightness

made it liable to greater swings and hence larger cross-arms are required. Due to lower

tensile strength and higher co-ecient of linear expansion of aluminium, the sag is greater

in aluminium conductors than copper.

Considering the combined properties of cost, resistivity, conductivity, availability, ten-

sile strength, weight etc., aluminium has an edge over copper. Therefore, aluminium is

widely used as a conductor material for transmission purposes. But due to its low tensile

strength, aluminium conductors generally produce greater sag. In order to increase the

tensile strength, aluminium conductors are normally reinforced with a core of galvanised

steel wires. The composite conductor that is formed with this reinforcement is known as

Aluminium Conductor Steel Reinforced, (ACSR). It will now comprise of central core of

galvanised steel wires surrounded by a number of aluminium strands. For better tensile

strength, the diameters of both steel and aluminium wires are the same and the cross

section of the two metals are generally in the ratio of between 1:6 and 1:4. With this ar-

rangement, the steel core takes greater percentage of mechanical strength while aluminium

strands carries the bulk of current. The Nigerian 330 KV transmission network employed

350mm2 aluminium conductor steel reinforced (ACSR).

2.4.2

Line Supports

The main function of line support is to assist the conductors in a way to keep them at an

appropriate level above the ground. Line support must be capable of carrying insulator and

conductors load as well as loads due to wind. The line support for long distance transmission

at higher voltage is usually steel towers. This is because of its high mechanical strength and

longer life span than any other line supports. Also, it can withstand most of the severe

climatic conditions and it permits the use of longer spans. Therefore the risk of interrupted

service due to broken insulation is drastically reduced because of the longer span. The heigth

of steel towers depends on line voltage and the length of span. In Nigeria, double circuit

and single circuit steel towers are used with heights of 75 metres and 54 metres respectively,

Onohaebi and Odiase (2010). Reinforced Concrete (RCC) poles, steel poles and wooden

poles are used as supports for distribution of low voltage of up to 11 KV.

25

2.4.3

Insulators

The current along the conductors in the overhead transmission lines should not be allowed

to ow to the earth through the line supports. This implies that the conductors should be

properly insulated from the line supports. The insulators provides appropriate insulation

between the conductors and the line support. It therefore prevents any leakage of current

from the conductors to the earth. Air is a general insulator for overhead lines. The most

commonly used material for the insulation of overhead lines is porcelain. Glass and steatite

are occassionally used as insulator materials. Porcelain is stronger mechanically than glass

and steatite. It is also less aected by temperature changes. To be able to function eectively,

a very good insulator should have the following properties:

i. An insulator should have high electrical resistance in order to prevent leakages of

current to the earth.

ii. It should have high mechanical strength in order to withstand wind load and conductor

load.

iii. It should have high relative permittivity so that the dielectric strength will be high.

iv. The insulator materials should be non-porous in order not to lower the permittivity.

2.4.4

Cross-arms

The function of cross-arms is to keep the conductors at a safe distance from each other and

also from the poles. It is a cross-piece tted to the end portion of the top of the pole by means

of brackets. These brackets are known as pole brackets and are general used for supporting

insulators. Steel cross-arms are generally used for steel poles because they are stronger than

any other cross-arms. There are various other types of cross-arms like MS channel, angle

iron or angle wooden which are used for 11 KV and 33 KV lines. Cross-arms are also of

various shapes which include U-shape, V-shape, straigth or zig-zag shape. The length of

cross-arms should be suitable enough for the spacing of the conductors. The cross-arms

should also be strong enough to withstand the resultant forces caused by insulators.

26

2.4.5

Stays

These are braces or cables that are fastened to the pole, at the terminal end, at a very good

angle to resist forces. This becomes essential in order to enable the overhead line supports

to stay at a very good position to withstand the pull by conductors and other lateral forces.

The theoretical angle between the stay and the pole should be 450. But in general practice,

it is not always possible to achive this, so stay designs are based on a minimum angle of 300

between the pole and the stay.

2.4.6

Miscellaneous Components of Overhead Lines

Other components of overhead lines which include lightning arrestors, fuses and isolating

switches, barbed wires, danger plates, continuous earth wires and guard wires, are discussed

below.

i. Lightning Arrestors - This is a device to discharge excessive voltages due to lightning

built upon the line to the earth.

ii. Fuses and Isolating Switches - These are to isolate dierent parts of the overhead

system

iii. Barbed Wires - Barbed wires are wrapped on poles at a height of about 2.5 metres

from the ground. This will prevent climbing of the poles by unauthorised people.

iv. Danger Plates - It is provided on poles as a warning measure to indicate the working

voltage of the line together with the word danger. It is posted at a heigth of about

2.5 metres above the ground.

v. Continuous Earth Wire - Countinuous earth wire is generally run on top of the towers

to protect the transmission line against lightning discharges.

vi. Guard Wires - Guard wires, which are solidly connected to the earth, are provided

above and below power lines while crossing telephone or telegraph lines.

27

2.5

TRANSMISSION LINE CONSTANTS

Transmission lines are basically electrical circuits having distributed constants (or parame-

ters). These constants includes:

i. Line Resistance

ii. Line Inductance

iii. Line Capacitance

iv Shunt Conductance

The performance of a transmission line depends upon these constants to a considerable

extent.

2.5.1

Line Resistance

Every electric conductor oers opposition to the ow of current and this opposition is called

the resistance (R) of the conductor. The resistance is distributed uniformly along the whole

length of the line. The resistance of transmission line conductors, against current ow, is

the most important cause of power loss in transmission line and this aects the transmission

eciency of the line, Mehta and Mehta (2008) and Wadhwa (2009). The resistance of a line

conductor having resisitivity (), length (L) and cross-sectional area (A) is given by

L R = [ ]

A

(2.1)

2.5.2

Line Inductance

Series inductance (L) mainly governs the power transmission capacity of the line. When

an alternating current ows through a conductor, a charging ux is set up which links the

conductor, Mehta and Mehta (2008). The conductors therefore posses inductance due to

these ux leakages. The inductance is also uniformly distributed along the whole length of

the transmission line. Inductance oers opposition to the ow of varying current in a circuit,

Mehta and Mehta (2008). This is dierent from resistance which oers opposition to the

ow of both steady (direct) and varying (alternating) current. The opposition to the ow

28

of varying current, as a result of inductance, is called voltage drop. Inductance is generally

dened as ux per unit current. That is

where represents Flux leakage

and

I represents Current

2.5.3 Line Capacitance

L =

I

(2.2)

Shunt capacitance (C) causes a charging current to ow in the transmission line. Any two

conductors separated by an insulating medium constitute a capacitor or a condenser, Mehta

and Mehta (2008) and Wadhwa (2009). As we know, any two conductors of an overhead

transmission line are separated by air which acts as insulation, therefore, capacitance exists

between any two overhead line conductors. The capacitance is uniformly distributed over

the total length of the transmission line. It may therefore be regarded as a uniform series

of condensers that are connected between the conductors. Capacitance is generally dened

as charge per unit potential dierence. That is,

C = q v

(2.3)

where

q represents charge on the transmission line

and

v represents Potential dierence between the conductors

2.5.4

Shunt Conductance

The shunt conductance (G) is mostly due to leakages over the insulator and is always very

small, Mehta and Mehta (2008). Just like any other transmission parameters, it is also

uniformly distributed over the total length of the transmission line.

2.6

SKIN EFFECT

Current is uniformly distributed over the whole cross-section of the conductor when a

29

conductor is carrying steady direct current (DC). But in alternating current (AC) the ow

of current is not unformly distributed. In fact, in an AC system, no current ows through

the core of the conductor as most current concentrates near the surface of the conductor as

frequency of transmission increases. This is as a result of the fact that a solid conductor

usually consists of a large number of strands each carrying a small part of the current.

Normally, the inductance of each strand will vary with its position. Therefore, the strand

near the centre is surrounded by greater magnetic ux than the one at the surface. Hence

the strand at the centre has greater inductance than the one at the surface. The high

reactance of the inner strands causes the alternating current to ow near the surface of the

conductor particularlly when the transmission frequency is high, Mehta and Mehta (2008),

Gupta (2008).

When an electromagnetic wave interacts with a conductive material, mobile charges

within the material are made to oscillate. The movement of these mobile charges (which are

usually electrons) constitute an alternating electric current. As the frequency of the current

increases, current density tends to decrease in the central axis of the conductor and increase

near the surface of the conductor. That is, the electric current tends to ow at the skin of

the conductor at an average depth called the skin depth. The skin depth is a measure of the

distace over which the current falls to 1 e (about 0.37) of its original value. This phenomenon

is known as skin eect. Skin eect will cause a decrease in the eective cross-sectional

area of the conductor and hence increase the resistance of the conductor. An increase in

the resistance of the conductor will consequently increase the ohmic or line losses of the

transmission line.

2.7

ECONOMICS OF POWER TRANSMISSION

The commercial aspect of the design of power transmission is very essential to an electrical

engineer. He must design the various aspect of the transmission scheme in a way to achieve

maximum economy. Two fundamental economic principles which inuences the electrical

design of a transmission line are:

i. Economic choice of conductor size

ii. Economic choice of transmission voltage

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 Kelvins law, Mehta and Mehta

(2008). The total annual cost of transmission line is a function of the annual charge on

capital outlay and annual cost of energy wasted in the conductor.

2.7.2

Economic Choice of Transmission Voltage

We all know that if transmission voltage is increased, the volume of conductor material

required is reduced and this will denitely decrease the expenditure on the conductor ma-

terial. It should also be noted that, an increase in the transmission voltage will lead to

a rise in the cost of transformers, switchgear, insulation materials for the conductor and

other terminal apparatus of the line. Therefore, there is an optimum transmission voltage

for every transmission line beyond which there is nothing to gain in terms of economy. The

transmission voltage where the costs of conductors, insulators, switchgear, transformer and

other terminal apparatus is minimum is called Economical Transmission Voltage (ETV).

2.8

CORONA PHENOMENON

When an alternating potential dierence is applied across two conductors whose spacing is

large as compared to their diameters, then the atmospheric air surrounding the conductor

is subjected to electro-static stresses. At low voltage there is no apparent change in the

condition of the atmospheric air around the conductors. However, when the applied voltage

is gradually increased and it exceeds a certain value called the critical disruptive voltage then

the conductors are surrounded by a faint violet glow. This phenomenon is called corona and

is accompanied by the production of ozone, hissing sound, power loss and radio interference.

The higher the voltage is raised, the higher and larger the luminous envelops become and

the greater the hissing noise, the power loss and the radio interference. The production of

ozone is readily detected because of its characteristic odour. The glow is due to the fact that

the atmospheric air around the conductor becomes conducting due to electro-static stresses.

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 Aecting Corona

Since corona occurs as a result of the ionization of the air surrounding the line conductors,

it is aected by the physical state of the atmosphere as well as by the condition of the line.

The following are the factors upon which corona depends

2.8.1.1 Atmosphere

Since corona is caused by the bombardment of molecules with subseqent dislodging

of electrons by ionised particles, it will denitely be aected by the physical state of

the atmosphere. The voltage gradient for the breakdown of the air is proportional

to its density. In the stormy weather, the number of ions will be more than normal,

therefore corona may occur at much less voltage than in fair weather.

2.8.1.2 Conductors Size, Shape and Condition

The corona is greatly aected by the size, shape and surface condition of the conductor.

An irregular or rough surface will give rise to more corona. Therefore a stranded

conductor will have more corona eects than a solid conductor because of its irregular

surface. The corona decreases with increasing diameter of conductor.

2.8.1.3 Spacing between Conductors

An increase in the spacing between conductors reduces the electro-static stresses. This

therefore reduces the corona eect. If the spacing between the conductors is made very

large as compared to their diameter, there may not be any corona eect.

2.8.1.4 Line Voltage

The line voltage considerably aects corona. If it is low, there is no change in the

condition of air surrounding the conductors and hence no corona is formed. But when

the line voltage is increased to such a value that electro-static stresses developed at the

conductor surfaces, then corona will occur because the atmospheric air surrounding

the conductor will start conducting.

32

2.8.2

Advantages and Disadvantages of Corona

Corona eect has advantages and disadvantages. An electrical engineer has to strike a

balance between the advantages and the disadvantages in order to design a very good high

voltage transsmission line. The advantages include

i. Corona usually reduces the eects of transients produced by surges.

ii. As a result of corona formation, the air surrounding the conductor becomes conducting

and hence the diameter of the conductor is increased. This increase in diameter reduces

electro-static stresses between the conductors.

Corona eect also has the following disadvantages

i. Ozone is produced by corona and this may cause corrosion of the conductor due to

chemical action.

ii. Corona is accompanied by a loss of energy and this greatly aects the transmission

eciency of the line.

2.8.3

Methods of Reducing Corona

Intense corona eects are observed at an operating voltage of 33 KV and above. Therefore

careful design should be made to avoid corona on the sub-station rated for 33 KV and higher

voltages. The following methods can be used to reduce corona

i. By increasing conductors size so that the voltage at which corona occurs is raised.

This will reduce the eect of corona

ii. By increasing the spacing between conductors, the voltage at which corona occurs is

also raised to reduce corona eects. It is to be noted that there is a limit to which we

can increase the spacing between conductor as this may cause an increase in the cost

of supporting structures considerably.

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 simplied form.

A mathematical model is a model developed using mathematical concepts like equations,

variables, operators, etc, Dilwyn and Hamson (1993), Ruhul and Charles (2008). It is often

desirable to describe the behavior of some real life phenomenon or system, whether physical,

sociological, ecological, scientical, technological or even economical, in mathematical terms.

The mathematical desciption of a system or phenomenon is called a mathematical model and

is constructed with certain goals in mind, Ruhul and Charles (2008), Dennis and Michael

(2005). Thus, mathematical modelling is the art of translating real life problems from an

application area into tractable mathematical formulations whose theoretical and numerical

analysis provides insight, answers and guidance useful for the originating application, Arnold

(2003). Hence, mathematical modelling serves as a bridge between the study of mathematics

and the applications of mathematics to various elds of human endeavous, and is an essential

part of the process of solving real life problem optimally, Ruhul and Charles (2008). An

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 t and

simplicity of mathematical representation, Dilwyn and Hamson (1993). Empirical models

are not based on physical laws or principles neither are they derived from assumptions

concerning the variables, Dilwyn and Hamson (1993).

In this chapter, we developed mathematical models of electric power ow along trans-

mission lines. We developed a mathematical model for power losses along tranmission lines

in chapter four. Also in chapter four, we developed empirical models of power losses for

dierent loads along transmission lines as functions of distance.

3.1.2

Dierential Equations

A Dierential Equation (DE) is an equation containing the derivatives of one or more depen-

dent variables, with respect to one or more independent variables. Dierential equations are

of fundamental importance in engineering because many physical laws and relations appear

mathematically in the form of dierential equations, Kreyszig (1987), Khorasani and Adibi

(2003). The order of a DE is the order of the highest dierential coecient contained in it.

The power to which the highest derivative is raised is called the degree of the DE.

An Ordinary Dierential Equation(ODE) is an equation containing derivatives of one

or more dependent variables with respect to a single independent variable. An equation

involving partial derivatives of one or more dependent variables with respect to one or

more independent variables is called a Partial Dierential Equation(PDE). The independent

variables can be anything such as time, velocity, distance, etc. In most of the applications

of control systems engineering, the independent variable is time, Matilde, Jose and Sanchez

(2009), Otarod and Khodakarim (2008).

An nth-order ordinary dierential equation given by

F (x, y, y , y , ..., yn) = 0

is said to be linear if F is linear in y, y , y , ..., yn. This implies that the dependent variable y

and all its derivatives are of the rst degree. Also for linearity of the dierential equation, the

coecients of the dierential equation must depend at most on the independent variable. A

non linear ordinary dierential equation is just an ordinary dierential equation that is not

35

linear. In this case, non linear functions of the dependent variable or its derivatives can occur

in the equation and the coecients can be functions of both dependent and independent

variables.

An nth-order ODE is said to be nonhomogeneous if

F (x, y, y , y , ..., yn) = g(x).

. If g(x) = 0 then the dierential equation is said to be homogeneous. The models of

the electric power ow along a transmission line are in form of homogeneous second order

partial dierential equations, which are then transformed into a non-homogeneous ordinary

dierential equation by making use of Laplace transformation.

3.1.3

Laplace Transformation

A function F(s) dened by the integral

F (s) = f (t)estdt 0

is called the Laplace transform of the function f(t) and is usually denoted by

F (s) = L[f (t)].

The Laplace transform of f(t) is said to exist if

f (t)estdt 0

converges for some values of s. f(t) is called the inverse Laplace transform of F(s) and is

usually denoted by

f (t) = L1[F (s)].

The Laplace transformation is a method for solving dierential equations and corresponding

initial and boundary value problems. It will transform initial and boundary value ordinary

dierential equations into algebraic equations, Gupta (2009), Stroud and Dexter (2003),

Kreyszig (1987) and Binoy (2009). It will also transform initial and boundary value partial

dierential equations into ordinary dierential equations, Kreyszig (1987), Murray (1967)

and Luke (1982). The Laplace transform method is widely used in engineering. We applied

it to solve the model for electric power ow along transmission lines.

36

3.2

KIRCHOFFS CIRCUIT LAWS

In 1845, a German physicist, Gustav Kircho, rst described two laws that became central

to electrical engineering. The laws were generalized from the work of George Ohm. The

laws can also be derived from Maxwells equations, but were developed prior to Maxwells

work. The Kircho s circuit laws, or simply Kircho s rules, deal with the conservation of

charge and energy in electrical circuits. The two laws are the Kircho s current law and

Kircho s voltage law which are described below.

In this chapter, we applied these two Kircho laws to the equivalent circuit of transmis-

sion lines and then we formulated the model for power ow along transmission lines.

3.2.1

Kircho s Current Law

Kircho s current law (KCL), also known as Kircho s Junction Law, Kircho s Point

Rule, Kircho s Nodal Law or Kircho s First Law, denes the way that electrical current

is distributed when it crosses through a junction. Specically, the law states that: The

algebraic sum of currents in a network of conductors meeting at a junction is zero. That is,

n

(Ik) = 0 k=0

where n is the total number of branches in which current is owing. Since current is the ow

of electrons through a conductor, it cannot build up at a jun

minimization of power loss

Documents

university of ilorin

research work

ilorin matric

department of mathematics

falana of ilorin business

dr mrs

ibrahim head of department

electrical power systems