1

CHAPTER ONE

UNIVERSITY O F NIGERIA, NSUKKA

DEPARTMENT OF ELECTRICAL ENGINEERING

A CRITICAL ANALYSIS OF TRANSIENT

STABILITY OF ELECTRICAL POWER SYSTEM

A CASE STUDY OF NIGERIAN 330KV POWER

SYSTEM:

A PROJECT WORK SUBMITTED IN PARTIAL

FULFILLMENT OF THE REQUIREMENT FOR THE

AWARD OF MASTER OF ENGINEERING DEGREE

(M. ENG) IN ELECTRICAL ENGINEERING.

BY:

IKELI, HYGINUS NDUBUISI

PG/M.ENG/O6/40613

NOVEMBER, 2009

UNIVERSITY O F NIGERIA, NSUKKA

2

DEPARTMENT OF ELECTRICAL ENGINEERING

A CRITICAL ANALYSIS OF TRANSIENT STABILITY OF

ELECTRICAL POWER SYSTEM:

A CASE STUDY OF NIGERIAN 330KV POWER SYSTEM .

A PROJECT WORK SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENT FOR THE AWARD OF MASTER OF

ENGINEERING DEGREE (M. ENG) IN ELECTRICAL

ENGINEERING.

BY:

IKELI, HYGINUS NDUBUISI

PG/M.ENG/O6/40613

NOVEMBER, 2009

AUTHOR: ………………………………………………

IKELI, HYGINUS NDUBUSISI

SUPERVISOR: ………………………………………………

VEN. ENGR. (PROF) T.C. MADUEME

HEAD OF DEPARTMENT: ………………………………………………

ENGR .DR. L.U. ANIH

EXTERNAL EXAMINER: ………………………………………………

ENGR. PROF. J.C. EKEH

TITLE PAGE

3

A CRITICAL ANALYSIS OF TRANSIENT STABILITY OF

ELECTRICAL POWER SYSTEM

A CASE STUDY OF NIGERIAN 330KV POWER SYSTEM.

4

DECLARATION PAGE

I, IKELI, HYGINUS NDUBUISI, a postugraduate student in

the Department of Electrical Engineering with Registration

Number, PG/M.ENG/06/40613, hereby declare that the work

embodied in this dissertation is Original and has not been

submitted in part or full for any other Diploma or Degree of this

University or other Institution to the best of my knowledge.

CERTIFICATION

5

IKELI, HYGINUS NDUBUISI, a postgraduate student in the Department of

Electrical Engineering with Registration Number, PG/M.ENG/06/40613,

has satisfactorily completed the requirement for course and research work

for the Degree of Master of Engineering(M.Eng) in the Department of

Electrical Engineering, University of Nigeria, Nsukka.

The work embodied in this dissertation is original and has not been

submitted in part or full for any other Diploma or Degree of this University

or other institution to the best of our knowledge.

…………………………….. ………………………………

Ven. Engr. (Prof) T.C. Madueme Engr. (Dr.) L.U. Anih

Supervisor Head of Department

DEDICATION

6

This work is dedicated to the Almighty God, the Giver of life, the author and

finisher of faith.

ACKNOWLEDGEMENT

7

I wish to express my profound gratitude to Almighty God, the Giver of life,

for his continuous guidance, I owe everything to his love and Grace upon my

life.

My sincere and humble thanks go to my able and wonderful

supervisor, Ven. Engr. Prof T.C. Madueme for his fatherly guidance, sincere

advice, encouragement and un-alloyed interest to seeing this work to

completion. I am highly indebted to him.

My special appreciation goes to Engr. Dr L.U. Anih, the Head of

Department Electrical Engineering for all his contributions and invaluable

support in my life.

I wish to also specially appreciate the efforts of Engr. Prof M.U. Agu,

Engr. Dr E.S. Obe, Engr. Prof O.I. Okoro and Engr.Dr. B.O. Anyaka for

their constant encouragement and enormous sacrifices they made in shaping

me to what I am today.

This work would have suffered serious setback if not for the

contributions of my very good friends and colleagues who assisted me at all

times with relevant information. They include Engr. (Prof) F.N. Okafor

UNILAG), Engr. A.J. Onah (PHCN, ENUGU), Engr. Emma Okonkwo

(PHCN, Delta power station), Engr. Iloma Davis, Engr. Chindo (both of

National Centre Oshogbo PHCN)

My special acknowledgement goes to my parents, Mr and Mrs

Hyacinth – Choma Ikeli for laying a good foundation for me in life. Long

life and prosperity shall be your portion in Jesus Name Amen. My special

thanks goes to my Uncle, Rev. Dr Godfrey Anyaka for his constant

encouragement, invaluable support and advice. I am indeed very grateful to

you.

8

I also remain grateful to the following as lecturers in this Department

for their encouragement, Engr. B.O. Nnadi, Engr. C. Nwosu, Engr. C. Odeh

and Engr. S.O. Oti.

Also not left out in my train of appreciation were my school mates,

Mbadiwe, Cosmas, Umuoh, Douglas, Nelson, Chinedu, Emeka, Samuel and

Benjamin for our cooperation cannot be forgotten in a hurry.

………………………………...

Ikeli, Hyginus Ndubuisi

November, 2009

U.N.N.

ABSTRACT

9

This work involves the investigation and analysis of critical clearing angle

and time of Protection System of Nigerian 330KV power system Network.

In the recent times, the importance of transient stability assessment

has been increasing since the electric power systems are being operated

closer to their stability limits. This poses a variety of challenging problems

at the planning and design stages as well as during the system operation.

In this work, the composition of the Nigerian 330KV Electric power

system (the national grid) is looked into and the critical clearing time and

angle evaluated and determined using fourth (4th) order Runge Kutta method

after obtaining power flow solution results with Ifnewton power flow

program(Newton-Raphson Method) and other programs such as Lfybus,

Busout, Trstab, Afpek, Dfpek, Ybusaf, Ybusbf and Ybusdf in Matlab software

package environment such that the probability of total system collapse is

reduced to the barest minimum.

The performance of protective system during transient period is

evaluated and the system critical clearing time and angle obtained.

Heavy Egbin – Ikeja West 330KV line was faulted and removed and

the system critical clearing time and angle determined such that the

Nigerian power system is transiently stable thereby averting widespread

black-out. The results so obtained is used to predict the transient stability of

the entire power system since Egbin – Ikeja West 330KV line is one of the

heaviest feeders in the National grid.

10

TABLE OF CONTENT

COVER PAGE- - - - - - - - - - - - - - - - - - - --------------------------------i

2ND

COVER PAGE- - - - - - - - - - - - - - - - - - ----------------------------ii

TITLE PAGE --- --- --- --- --- --- --- --- --- iii

DECLARATION ----- --- --- ---- ----- ---- ----- ---- ---- ---- -- iv

CERTIFICATION --- --- --- --- --- --- --- --- v

DEDICATION --- --- --- --- --- --- --- --- vi

ACKNOWLEDGEMENT --- --- --- --- --- --- --- vii-viii

ABSTRACT --- --- --- --- --- --- --- --- --- ix

TABLE OF CONTENT --- --- --- --- --- --- --- x-xii

LIST OF SYMBOLS AND ABBREVIATIONS --- --- --- --- xiii - xviii

LIST OF FIGURES AND DIAGRAMS --- --- --- --- xix-xx

LIST OF TABLES --- --- --- --- --- --- --- --- xxi

CHAPTER ONE: INTRODUCTION --- --- --- --- --- 1

1.1 PREAMBLE --- --- --- --- --- --- --- --- 1-2

1.2 OBJECTIVE OF THE STUDY --- --- --- --- --- 3

1.3 SCOPE OF STUDY --- --- --- --- --- --- --- 3

1.4 Methodology --- --- --- --- --- ---- ----- 3-4

1.5 Existing 330Kv National grid - - - - - - - - - 5

1.6 NEED FOR STABILITY STUDIES --- --- --- --- --- 6

CHARPTER TWO --- --- --- --- --- --- --- --- 7

2.1 CONCEPT OF ENERGY FUNCTION MODEL

IN TRANSIENT STABLITY ANALYSIS --- --- --- --- 7

2.2 MODELLING ISSUES --- --- --- --- --- --- 8

2.3 NUMBERICAL METHODS --- --- --- --- --- --- 9

2.3.1 STEADY STATE ANALYSIS --- --- --- --- --- ---- 9

2.3.2 DYNAMIC (TRANSIENT) STATE ANALYSIS --- --- --- 9-10

2.3.3 RUNGE-KUTTA METHOD --- --- --- --- --- --- 10-12

2.4 NIGERIAN NATIONAL GRID--- --- --- --- ---- ---- ---- ---- --- 12-13

2.5 BASIC CONCEPTS OF STABLITY STUDIES --- --- -- 13-14

11

2.6 ESSENTIAL FACTORS IN THE STABILITY PROBLEM --- --- 14

2.5.1 FACTORS AFFECTING STABILITY --- --- --- --- ---- 14-15

2.5.2 POWER CIRCLE DIAGRAM --- --- ---- ---- ---- ---- ---- --- 16-19

POWER ANGLE DIAGRAM --- --- --- --- --- ---- 17-19

2.5.3 FACTORS AFFECTING TRANSIENT STABILITY --- --- --- 19-20

2.5.4 STEADY-STATE STABILITY LIMIT --- --- --- --- ---- 20-21

2.5.5 TRANSIENT STABILITY LIMIT --- --- --- --- --- 22-24

2.6 THE SWING EQUATION --- --- --- --- --- --- 25-29

2.6.1 THE POWER-ANGLE EQUATION --- --- --- --- --- 30

2.6.2 TRANSIENT AND SUBTRANSIENT EFFECTS --- --- --- 30-31

2.7 EQUAL AREA CRITERION --- --- --- --- --- --- 31-39

2.8 MULTIMACHINE DYNAMIC MODEL --- --- --- --- 40

2.8.1 INTERCONNECTION OF SYNCHRONOUS

MACHINE DYNAMIC CIRCUIT AND THE

REST OF THE NETWORK --- --- --- --- --- --- 41

2.8.2 NETWORK EQUATIONS --- --- --- --- --- --- 42

2.9.0 POWER FLOW SOLUTION ---- ---- ----- ---- ---- ---- --- 43

2.9.1 POWER FLOW EQUATION --- ---- ---- ----- ---- ----- 44-45

CHAPTER THREE -- --- --- --- --- --- --- --- 46

3.1 PROJECT DESIGN --- --- --- --- --- --- --- 46

3.2 DATA SOURCES --- --- --- --- --- --- --- 47

3.3 DATA ANALYSIS TECHNIQUE --- --- --- --- --- 48

3.4 MULTIMACHINE TRANSIENT STABILITY

WITH MATLAB SOFTWARE PACKAGE --- --- --- --- 48-50

3.6 ASSUMPTION MADE FOR THIS WORK --- --- --- --- 51-52

3.7 PERFORMANCE OF PROTECTIVE RELAYING --- --- --- 51-52

3.7.1 FAULT CLEARING TIMES --- --- --- --- --- --- 52

3.7.2 FACTORS INFLUENCING TRANSIENT STABILITY --- --- 53

3.8 DATA FOR THE WORK --- --- --- --- --- --- 54-60

CHAPTER FOUR --- --- --- --- --- --- --- --- 61

12

4.1 EXISTING 330KV NATIONAL GRID DIAGRAM --- --- 61

4.2 SIMULATION AND DISCUSSION

OF RESULTS --- --- --- --- --- --- --- --- 62- 74

CHAPTER FIVE ---- ---- ----- ----- ------ ----- ------ ----- ----- ------ 75

5.0 CONCLUSION AND RECOMMENDATION --- --- --- --- 75- 76

REFERENCES 77 - 81

LIST OF APPENDICES

13

APPENDIX 1 82 -84

APPENDIX 2 85

APPENDIX 3 86

APPENDIX 4 87-88

APPENDIX 5 89

APPENDIX 6 90

APPENDIX 7 91

APPENDIX 8 92

APPENDIX 9 93

APPENDIX 10 94

APPENDIX 11 95

LIST OF SYMBOLS AND ABBREVIATIONS

O Code for the load buses

1 Code for the slack buses

14

2 Code for the voltage – controlled buses

Ra Generator’s armature resistances in per unit expressed on a

100 MVA base

Xa Transient reactance in per unit, expressed on a 100 MVA base.

( )

H Inertia constants in seconds expressed on a 100MVA base.

B Susceptance in per unit expressed on a 100mva base.

ω Angular speed rad/s

VA Phase voltage quantity, V

VB Voltage quantity, V

VC Phase voltage quantity, V

TDQOS Transformed Operator

ωS Synchronously speed rad/s

HIW Oshogbo T.S – Ikeja West T.S 330kv line

H7B Oshogbo T.S – Benin T.S 330kv line

O1W Olorunshogbo G.S – Ikeja West T.S 330kv line

NWIBS Sakete T.S – Ikeja West T.S 330kv line

W3L Ikeja West T.S – Akangba T.S 330kv line 1

W4L Ikeja West T.S – Akangba T.S 330kv line 2

E11 Ikeja West T.S – Egbin G.S 330kv line 4

E21 Ikeja West T.S – Egbin G.S 330kv line 2

N3J Aja T.S – Egbin G.S 330KV line 1

N4J Aja T.S – Egbin G.S 330KV line 2

E1A Egbin/Ikeja West line – AES G.S 330kv line 1

E2A Egbin/Ikeja West line – AES G.S 330kv line 2

A4J AES G.S line – AJA T.S 330KV line

M7W Ikeja West line – Omotosho G.S 330KV line

15

B5M Benin T.S – Omtosho G.S 330KV line

B6W Ikeja West line – Benin T.S 330KV line

B11J Benin T.S – Ajaokuta T.S 330KV line 1

B12J Benin T.S – Ajaokuta G.S 330KV line 2

AIS Ajaokuta T.S – Ascon G.S 330KV line 1

A2S Ajaokuta T.S – Ascon G.S 330KV line 2

AIG Ajaokuta T.S – Geregu G.S 330KV line 1

A2G Ajaokuta T.S – Geregu G.S 330KV line 2

BIT Benin T.S – Onitsha T.S 330KV line

T3H Onitsha T.S – (Enugu) New Heaven 330KV line

K3R Kainji – Benin Kebbi 330KV line

KIJ Kainji – Jebba T.S 330KVline 1

K2J Kainji – Jebba T.S 330KV line 2

RIM Shiroro T.S – Kaduna 330KV line

M6N Kaduna T.S – Kano 330KV line

R2M Shiroro T.S – Kaduna 330KV line

M2S Kaduna T.S – Jos 330KV line

SIE Jos T.S – Gombe 330KV line

B8J Jebba G.S – Jebba T.S 330KV line 1

B9J Jebba G.S – Jebba T.S 330KV line 2

J3R Jebba T.S – Shiroro T.S 330KV line 1

J7R Jebba T.S – Shiroro T.S 330KV line 2

R4B Shiroro T.S – Abuja T.S 330KV line 1

R5B Shiroro T.S – Abuja T.S 330KV line 2

P4A Shiroro T.S – Shiroro T.S 330KV line

JIH Jebba T.S – Oshogbo G.S 330KV line 1

J2H Jebba T.S – Oshogbo T.S 330KV line 2

J3H Jebba T.S – Oshogbo T.S 330KV line 3

16

H2A Ayede T.S – Oshogbo T.S 330KV line 1

W2A Ayede T.S – Ikeja-West T.S 330KV line 2

G3B Benin T.S – Delta G.S 330KV line

GIW Delta G.S – Aladja T.S 330KV line

S4W Sapele G.S – Benin T.S 330KV line

S3B Sapele G.S – Benin T.S 330KV line

KIT Onitsha T.S – Okpai G.S 330KV line 1

K2T Onitsha T.S – Okpai G.S 330KV line 2

T4A Alaoji T.S – Alaoji T.S 330KV line

FIA Alaoji T.S – Afam G.S 330KV line 1

F2A Alaoji T.S – Afam G.S 330KV line 2

MM6 Afam G.S line – Omoku G.S 330KV line

BUS 1 Kainji G.S (Slack bus)

BUS 2 Bernin Kebbi

BUS 3 Jebba G.S

BUS 4 Jebba T.S

BUS 5 Shiroro T.S

BUS 6 Abuja (katampe)

BUS 7 Shiroro G.S

BUS 8 Ayede T.S

BUS 9 Oshogbo T.S

BUS 10 Kaduna T.S

BUS 11 Olorunsongo G.S

BUS 12 Sakete T.S

BUS 13 Kano T.S

BUS 14 Jos T.S

BUS 15 Ikeja West T.S

BUS 16 Benin T.S

17

BUS 17 Gombe T.S

BUS 18 Delta G.S

BUS 19 Ajaokuta T.S

BUS 20 Akangba T.S

BUS 21 Omotosho T.S

BUS 22 Egbin G.S

BUS 23 Onitsha T.S

BUS 24 Sapele G.S

BUS 25 Aladja T.S

BUS 26 Geregu G.S

BUS 27 Ascon G.S

BUS 28 AES G.S

BUS 29 Aja T.S

BUS 30 New Heaven T.S

BUS 31 Okpai G.S

BUS 32 Alaoji T.S

BUS 33 Afam G.S

BUS 34 Omoku G.S

G.S Generating station

T.S Transmitting Station

Yij Transfer Admittance

Yii Self Admittance

X1d Transient Reactance

X11

d Sbtransient Reactance

E1 Voltage behind Transient Reactance

KG Kainji Generator

JG Jebba Generator

SG Shiroro Generator

18

OG Olorunsongo Generator

DG Delta Generator

OMG Omotosho Generator

EG Egbin Generator

SAG Sapele Generator

GG Geregu Generator

AG Ascon Generator

AEG AES Generator

OKG2 Okpai Generator

AFG Afam Generator

OMKG Omoku Generator

K1 is the slope @ the beginning of time step,

K2 is the first approximation to the slope @ midstep

K3 is the second approximation to the slope @ midstep

K4 is the slope @ the end step

Yn+1 is the incremental value of Y given by the weighted average of

estimates based on slopes @ the beginning, midstep, and end of time

step

Vi = voltage @ i bus

Vk= voltage @ k bus

Idi= current @ direct axis of i bus

Iqi = current @ quadrature axis of i bus

Pli= Real power @ i bus

Qli= Reactive power @ i bus

Yik = Admittance of k bus with reference to i bus

θi = Theta @ i bus

θk = Theta @ k bus

αik = phase angle difference between bus i and bus k

19

y = Rotor angle in radii

t = Time in seconds

LIST OF FIGURES AND DIAGRAMS

Figure 1.1 Existing 330KV National grid Network Diagram

Figure 2.1 Various elements of power system Network.

Figure 2.2 Power circle Diagram.

Figure 2.3 Power Angle Curve.

Figure 2.4 Single machine infinite bus system.

Figure 2.5 Generator prime mover and motor Dynamics.

20

Figure 2.6 Circuit diagram of a synchronous machine for transient

stability studies.

Figure 2.7 Plot of power against δ.

Figure 2.8 Plot of δ versus time for stable and unstable systems.

Figure 2.9 Synchronous machine and the rest of the network.

Figure 2.10 A typical bus of the power system.

Figure 4.1 Existing 330KV National grid Network Diagram.

Figure 4.2 Plot of rotor angle [radii] against time [s] for Nigerian

generators swing during fault for 0.025 seconds.

Figure 4.3 Plot of rotor angle [radii] against time [s] for system

fault cleared @ 0.025seconds.

Figure 4.4 Plot of rotor angle [radii] against time [s] for system

fault cleared @ 0.030 seconds.

Figure 4.5 Plot of rotor angle [radii] against time [s] for system

fault cleared @ 0.035 seconds.

Figure 4.6 Plot of rotor angle [radii] against time [s] for fault

cleared @ 0.040 seconds.

Figure 4.7 Plot of rotor angle [radii] against time [s] for system

fault cleared @ 0.045 seconds.

Figure 4.8 Plot of rotor angle [radii] against time [s] for system

fault cleared @ 0.050 seconds.

Figure 4.9 Plot of rotor angle [radii] against time [s] for system

fault cleared @ 0.055 seconds.

Figure 4.10 Plot of rotor angle [radii] against time [s] for system

fault cleared @ 0.060 seconds.

Figure 4.11 Plot of rotor angle [radii] against time [s] for system

Generators swinging in synchronism after the faulted

21

elements have been removed from the system @ 0.065

seconds.

LIST OF TABLES

22

Table 3.1: Load Data ------------------------------------------------54-55

Table 3.2: Generation schedule------------------------------------56-57

Table 3.3 Machines and system vars-----------------------------57-58

Table 3.4: Line Data-----------------------------------------------58-59

Table 3.5: Machine Data -----------------------------------------59-60

23

INTRODUCTION

1.1 PREAMBLE

Electrical energy is an essential ingredient for the industrial and

all-round development of any country. The quality of life in any

country is highly dependent on a reliable electricity supply. The

frequent system collapses in the Nigerian power sector have severally

thrown or plunged the nation into darkness due to system instability in

the Nigeria electric power system. The epileptic nature of the supply

has led to low economic growth and dissatisfaction among the

citizenry. To assist in overcoming the instability problems, analysis of

the Nigerian electric power system transient stability is carried out

employing Newton-Raphson method of load flow solution and fourth

order Runge Kutta method in mat-lab software package environment

[1, 2, 3].

The objective of any electrical power system is to generate

electric energy in sufficient quantities at most suitable locality, transmit

it in bulk quantities to the load centre, which is then distributed to the

individual consumers. In carrying out the desired objectives, the electric

power system is faced with unforeseen circumstances such as faults.

Under this condition, the system voltage collapses resulting in a

dangerous high current. This causes instability within the system which

can result in system breakdown if adequate care is not taken [4, 5].

In order to avoid these undesirable situations, it becomes

necessary to, before hand; predict with a very good accuracy, the extent

of voltage, current and power distribution within the system at anytime

so as to know the protective devices to be incorporated to handle the

abnormal conditions. Hence under these dangerous situations, the

24

transformers, lines, generators, cables, bus-bars etc. need to be

protected [2, 6, and 7].

The transient stability analysis which is the main concern of this

work, deals with the state of the synchronous machine during a fault in

the system. It gives the state and position of the load. The digital

computer is an indispensable tool for power system analysis,

computational algorithms for various system studies such as load flow,

fault-level analysis, stability studies etc. It gives an acceptable working

accuracy to the ever widening complex power system of modern times

[8, 9].

1.2 THE OBJECTIVES OF THE STUDY.

These include the following:

i To determine the critical clearing angle and time of the Nigerian

330KV protection system.

ii To determine the behavior of the Nigerian power system during

large scale disturbance and make necessary recommendations.

iii To specify the circuit break speeds in the system

iv To determine the available transfer capability (ATC) of the

Nigerian 330KV grid system during fault and make

recommendation for improvement such that the system is

transiently stable.

25

1.3 SCOPE OF STUDY

The dissertation work covers the Nigerian 330KV power system. All

the thirty-four (34) buses in the network are critically examined and

analyzed with regard to transient stability .

1.4 Methodology

In this work, the composition of the Nigerian 330KV Electric power

system (the national grid) is looked into and the critical clearing time

and angle evaluated and determined using fourth (4th) order Runge

Kutta method after obtaining power flow solution results with

Ifnewton power flow program(Newton-Raphson Method) and other

programs such as Lfybus, Busout, Trstab, Afpek, Dfpek, Ybusaf,

Ybusbf and Ybusdf in Matlab software package environment such that

the probability of total system collapse is reduced to the barest

minimum.

Power holding company Plc. 330KV electrical network

single Line diagram is used for this study. The generators,

transmission lines and transformer parameters are taken from the most

up-to-date data from National control centre, Oshogbo System

Planning unit and system operations department. The subtransient

reactance X11

of the synchronous machines is used to give maximum

fault levels at the instant of fault.

The MATLAB function ode 23 will be employed to solve the 2m

first order swing equation to give the desired result.

The performance of protective system during transient period will be

Evaluated. Heavy Egbin – Ikeja West 330KV line will be faulted

26

And be removed and the system critical clearing time and angle

determined such that the Nigerian power system is transiently stable

thereby averting widespread black-out.

27

EXISITING 330KV NATIONAL GRID NETWORK

1

K3R

KIJ K2J

10

RIM R2M

M2S

M6N

SIE

13

17

14

7

P4A

R5B R4B J7R

J3R 5

6

B8J

B9J

4

JIH J2H J3H

9 H2A

W2A

01W

NWIBS

W3L

W4L

11

12

20

N3J

N4J

29

E21

22

E1A 15

E11

E2A

28

M7W

21

B5M

B6W

H7B

16

ABUJA

B11J

B12J AIG A2G

19

AIS

A2S

27

26

T3H

23 30

BIT

G3B

18

G1W

25

S4w

24

S3B

S4B KIT

K2T T4A

32

33

F1A F2A MM6

8

A4J

31

Figure 1.1 Existing 330kv National grid Network Diagram.

3

2

28

1.5 NEED FOR STABILITY STUDIES

In general, stability studies are very important primarily from

the stand point of determining the maximum amount of power that

can be transmitted without instability being incurred under steady

state conditions or as a result of load changes or faults. Potential

stability problems are still the most critical impediments to

maximizing power transfers across interconnected power systems like

the Nigerian national grid. Occurrence of transient instability

problems may result to large excursions of the system machines rotor

angle, and if corrective action fails, loss of synchronism among

generators may result in total system collapse. Recall that in the

summer of 1996, two major transient disturbances occurred in the

Western system co-ordinating council in United States of America

which resulted in partial black-outs that cost the power utilities and

their customers several Millions Dollars[13].

In Nigeria, two system collapses within a three-day interval in

March 2000, plunged the entire nation into darkness. The nation was

without electricity for up to 72 hours in some areas with serious

social, economic and security implications. This incident led to the

sacking of the power utility [NEPA] board and the appointment of a

Technical Board to oversee the day to day activities of the Authority

[13, 18].

The transient stability of a power system is normally taken as

very important and a major determinant of the stability of the power

system because of its non-linear character, its fast evolution and its

disastrous practical implications.

29

CHAPTER TWO

2.1 CONCEPT OF ENERGY FUNCTION MODEL IN TRANSIENT

STABILITY ANALYSIS

In 1892, A.M Lyapunov, in his famous ph.D. dissertation,

proposed that stability of the equilibrium of a non-linear dynamic

system of dimension n can be ascertained without numerical

integration.

According to this model, the critical clearing time of a circuit

breaker can be interpreted in terms of meaningful quantities such as

Maximum power transfer in the pre-fault state.

x = f (x), f (0) = 0 ………………………………………... (2.1)

He said that if there exists a scalar function V(x) for equation (1) that is

position definite, for V(x) > O around the equilibrium point “O” and

the derivative V(x) < O, then the equilibrium is asymptotically stable.

V(x) is obtained as Σi=1 ∂v xi =

∂xi

Σi=1 ∂v fi (x) = ------------(2.2) Where n is the order of the system in (2.1)

Furthermore, in 1948, the application of the energy function model to

power system stability actually began with the early work of

Magusson and Aylett, followed by a formal application of the more

general Lyapunov’s model by EL-Abad and Nagappan[19]

.

.

.

.

30

2.2 MODELLING ISSUES

A Power system undergoing a disturbance can be described by

a set of three differential equations:-

x (t) = f1 (x(t)) - ∞ < t < 0………………………(2.3)

x (t) = ff(x(t) ) O < t < tcl ……………………………..(2.4)

x (t) = f (x(t) ) tcl < t < ∞ ………………………….(2.5)

x (t) is the vector of state variable of the system at time t.

At t = 0, a fault occurs in the system and the dynamics change from fi

to ff .

During 0<t < tcl, called the faulted period, the system is

governed by the fault – on dynamics if actually, before the fault is

cleared at t = tcl, we may have several switching in the network, each

giving rise to a different ff.

For simplicity, we have taken a single ff, indicating that there

are no structural changes between t = 0 and t = tcl. When the fault is

cleared at t = tcl, we have the post fault system with its dynamics fi

(x(t)).

However, the energy function methods have proved to be

reliable after many decades of research for single machine system but

for multi-machine system with complex network like that of Nigerian

330KV power system network, the value of the tcl is not as reliable as

that got with numerical integration method with digital computer.

More research is still going on, on multi-machine system on this

concept. [19]

. .

.

31

2.3 NUMERICAL METHODS

Differential equations’ solution is highly restricted to the

employment of numerical methods since accuracy is needed. In

stability studies, non-linear highly-dimensional mathematical

problems are encountered hence engineers resort to using numerical

methods in their analysis. Numerical methods for the analysis of the

steady state and dynamic (transient) behaviour of power system

network are listed below:

2.3.1 STEADY STATE ANALYSIS.

This is done by four methods viz:

1. Gauss – Elimination Method.

2. Grammar’s rule.

3. Gauss – Jordan method.

4. Newton’s method.

The first three methods are suitable for linear expressions while the

last two methods are suitable for linear and non-linear expressions.

2.3.2 DYNAMIC (TRANSIENT) STATE ANALYSIS

Nine methods are available for this analysis namely:

1. Euler method.

2. Improved Euler method.

3. Euler-Cauchy.

4. Adams-Bash forth fourth-order method.

5. Adams-Moulton fourth-order.

32

6. Gear’s method.

7. Finite Difference method.

8. Fourth-order Runge-Kutta method.

9. Crank-Nicolson method.

Note: Crank-Nicolson and finite difference methods are mainly used

for the analysis involving partial differential equations, while the rest

seven methods are very suitable for analysis involving-ordinary

differential equations. [20]

2.3.3 RUNGE-KUTTA METHOD

One of the popular and most accurate, numerical procedures or

methods used in cracking a system of differential equation is the

fourth-order Runge-Kutta method. There are different orders of this

Runge-Kutta method but the interesting thing is that they are derived

using the Taylor’s series-expansion with remainder of function y(x).

The first order Runge-Kutta method is normally and basically the

same with basic Euler method. The second order Runge-Kutta method

is the same with improved Euler method. The fourth order Runge-

kutta solution to a system equation is as follows:

Y(x) = y(a) + y(a)1 x – a + y(a)

11 (x – a)

2 + … Y(c) (k+1) (x-a) k+1 …. (2.6)

1! 2! (k+1)!

Where c is some number between a and x boils down to:

Yn+1 = yn + ak1 + bk2 + ck3+ dk4 ………………… (2.7)

Where :

K1 = hf (Xn, Yn)

K2 = hf (Xn + 0.5h, Yn + 0.5K1)

K3 = hf (Xn + 0.5h, Yn + 0.5K2)

K4 = hf (Xn +h, Yn + K3)

33

This agrees with a Taylor Polynomial of degree 4.

The final solution formula for using in system analysis gives:

Yn+1 = Yn +1/6 [K1 + 2k2 + 2K3 + k4] ----------------- (2.8)

Careful look at the analysis of equation (6) shows that K2 depends on

K1, K3 depends on K2 and K4 depends on K3. This method amongst all

other methods allows usage of variable step size and this improves

accuracy of result. k1 is the slope @ the beginning of time step,

k2 is the first approximation to the slope @ midstep, k3 is the

second approximation to slope @ midstep While k4 is the slope

@ the end step. Yn+1 is the incremental value of Y given by the

weighted average of estimates based on slopes @ the beginning,

midstep, and end of the time step.

In this work, Runge-Kutta has been chosen for the stability

assessment of the Nigerian-Electric Power System based on the

following advantages it gives;

a. The method is inherently self-stating and this quality directly ceases

the handling of discontinuities and the adjustment of step length

where required (“easy for all automatic error control”)

b. It has high accuracy when the step can be made small.

Moreover, since this is a stability study which needs knowledge

of steady-state initial-condition values before starting transient-

stability integration, Runge-Kutta method was chosen.

c. Experiments have shown that several applications of a corrector say

three applications have to be made to match accuracy obtained using

Runge-Kutta method. This exhibits it’s time saving quality.

34

d. Easy to program for s digital computer and are always designed to

give greater accuracy [21], [23].

2.4 NIGERIAN NATIONAL GRID

The Nigerian National Grid is being run and controlled by

Power Holding Company of Nigerian PLC (PHCN), formally known

as National Electric Power Authority (NEPA). The Nigerian National

Grid System has a total of fourteen (14) Generating Power stations:

Namely:- Kanji, Jebba, Shiroro, Ascon, Egbin, AES, Delta, Okpai,

Afam, Sapele, Omotosho, Olorunsongo, Omoku and Geregu [10, 12,

13]. The national grid is made up of interconnected network of

5000km of 330KV transmission lines.

The control of the grid is affected by eight (8) Regional control

centers (RCC) located at Lagos, Oshogbo, Benin, Enugu,

PortHarcourt, Bauchi, Kaduna and Shiroro. Shiroro is normally taken

as sub-National control center (SNCC). The operations in these

regional control centers (RCC) are co-ordinated, directed and

supervised by the National control center at Oshogbo [14,15].

The radial nature of most stations in the National gid has made

some of the transmission lines very important and critical to the

integrity of the whole grid. These lines are Egbin – Ikeja West line,

Benin – Ikeja West line, Benin – Onitsha line, Jebba – Oshogbo line

and lastly the Oshogbo – Benin line. This could be clearly seen from

fig. 1.1. above. Records show that Lagos area consumes about 55%

of the total generated power [16]. These critical lines once disturbed,

reflect heavily on the whole power system most at times lead to a total

system outage. This situation critically affects the stability of the

system. The size of disturbance or fault distinguishes between

35

small disturbance stability and large disturbance stability. Small

disturbance stability is usually handled or solved by the linearization

of dynamic equations of motion while the large disturbance stability

requires non-linear approaches [17].

2.5 BASIC CONCEPTS OF STABILITY STUDIES

On commercial power systems, the large machines are of the

synchronous type; these include substantially all of the generators and

condensers, and a considerable part of motors

On such systems it is necessary to maintain synchronism

between the synchronous machines under steady-load conditions.

Also, in the event of transient disturbance, it is necessary to maintain

synchronism; otherwise a standard of service satisfactory to the user

will not be obtained.

These transient disturbances can be produced by load changes,

switching operations and particularly faults and loss of excitation.

Thus, maintenance of synchronism during steady-state conditions and

regaining of synchronism or equilibrium after a disturbance are of

prime-importance to the electrical utilities. Electrical manufacturers

are likewise concerned because stability considerations determine

many features of apparatus and under many conditions affect their

cost and performance. The characteristics of virtually every element

of the system have an effect on stability. It introduces important

problems in the co-ordination of electrical-apparatus and lines in order

to provide, at lowest cost, a system capable of carrying the desired-

36

loads and of maintaining a satisfactory-standard of service, both for

steady-state conditions and at times of disturbances [22], [23].

2.6 ESSENTIAL FACTORS IN THE STABILITY PROBLEM

The essential factors in the stability problem are illustrated in

connection with the two machine system shown schematically below in

figure 2.21.

The various elements of the system; prime mover synchronous

generator, reactance line, synchronous motor, and the shaft load, are

indicated:-

Figure 2.1: Various elements of Power System Network

PM - Prime Mover

G - Synchronous Generator

X - Reactance of the Line

M - Synchronous Motor

SL - Shaft Load

2.6.1 FACTORS AFFECTING STABILITY

These factors are the mechanical and electrical factors.

The essential mechanical factors include:

i. Prime Mover:- Turbine

Input

Torque

PM G X

M

SL

Output

Torque Inertia

37

ii. Inertial:- Mover of Turbine (crank shaft of the motor)

iii. Shaft:- Load out-put torque

The essential Electrical factors include:

i. Internal voltage of Synchronous generator

ii. Reactance of the system Viz: Generator, line and motor.

iii. Internal voltage of Synchronous motor [22]

38

θ11

θ1

P1

P11

Radius

Es Er

X

Receiver Circle

Centre

E2

X

θθ

P1

θ1

θ11

P11

Pmax

Es Er

X

Supply Circle

Centre

E2

O

O

X

Leading Reactive Power -

+ Lagging Reactive Power

2.6.2 POWER CIRCLE DIAGRAM AND POWER ANGLE CURVE.

Figure 2.2: Power Circle Diagram

39

Pmax

δ0 δ

O

Pe

pe

Pm

π/2

Figure 2.3: Power Angle Curve

The performance characteristics of the simple two-machine power

transmission system in figure 2.21 are shown by power-circle diagram and

power angle Curve as given in figure 2.22 and 2.23 respectively.

40

When a synchronous machine is in parallel with infinite bus bars, the

power generated equals the power received by the bus bars for a generator

in the absence of losses the system depends mainly on four factors namely:-

Eg, Em, θ and X.

The equation relating power transfer to the four factors in a three-phase

system is given by:-

P = Eg Em Sin θ ……………………………………………….……… (2.9)

X

P = three-phase power transferred in Watts

Eg = internal voltage of generator (line-to line in volts)

Em = internal voltage of motors (line-to-line in volts)

X = reactance between generator and motor internal voltages in OHMS

per phase.

θ = angle by which the internal voltage of generator leads the internal

voltage of the motor.

The maximum power is given by:-

Pmax = Eg Em ………………………………………………… (2.10)

X

(For θ = 900)

The radius of the circle is Pmax. The angle θ is varied between O and

1800. The relationship of the angle and the power is indicated in figure

2.23. As θ varies between O and 900, the power increases thus

reaching a maximum value at 900 and falling gradually from the

maximum to zero as θ increases from 900

to 1800.

The term “Stability” and “Maintenance of Synchronous” are

quite frequently used interchangeably. A system consisting of a

41

synchronous generator, a reactance line, and an induction motor may

become unstable but cannot loose synchronous. System stability is

ordinarily of importance only when it deals with the conditions of

stable operation between synchronous machines. The problem is of

importance, primarily from the stand-point of the maximum amount

of power that can be transmitted without instability being incurred

under steady-state conditions or as a result of circuit changes or faults.

Stability when used with reference to a power system is the attribute

of the system, or part of the system, which enables it to develop

restoring forces between the elements there of equal to or greater than

the disturbing forces so as to restore a state of equilibrium between the

elements. Stability applies to both steady state and transient conditions

in a power system [23],[24], [ 25].

2.5.3 FACTORS AFFECTING TRANSIENT STABILITY

Two factors which indicate the relative stability of a generating unit

are:

i. The angular swing of the machine during and following fault

condition.

ii. The critical clearing time.

H constant and the transient reactance X1d of the generating unit

have a direct effect on both of those factors.

Any development which lower the H constant and increase transient

reactance X1d of the machine causes the critical clearing time to

decrease and lessen the probability of maintaining stability under

transient conditions.

As power systems continually increase in size, there may be a

corresponding need for higher rated generating units. These larger

42

units have advanced cooling system which allows higher-rated

capacities without comparable increases in rotor size. As a result H

constants continue to decrease with potential adverse impact on

generating unit stability. Stability control techniques and transmission

system designs have also been evolving to increase overall system

stability in the modern times. The control schemes are

a. Excitation systems

b. Turbine valve control

c. Single-pole operation of circuit breakers

d. Faster fault clearing times

2.5.4 STEADY-STATE STABILITY LIMIT

A stability limit is the maximum power flow possible through

some point in the system when the entire system or part of the system

to which the stability limit refers is operating with stability. For the

two simple two-machine-transmission systems illustrated in figure

2.24 above, the steady-state stability is given by the maximum-power

obtained from either the power circle diagram or the power-angle

diagram of figures 2.22 and 2.23. The steady state stability of a

system without loss occurs at the angle of 90 degrees between the

sending and receiving ends as shown by these diagrams or reading

obtained from equation 8.

The steady-state limit for a three-phase system is given by:-

Pmax = Eg Em ………………………………………..(2.11)

X

Which gives the maximum power in watts when the voltages

are expressed as line-to-line voltage and reactance as ohms per phase.

43

If the criterion of stability is applied which states that the steady

limit for a three-phase system is given by the maximum power, then

the following exist.

i. For all load conditions with the power and angle less than those

corresponding to the 90 degrees limit, the system will be inherently

stable.

ii. For all loads at angle greater than 90 degrees the system will be

unstable. The 90 degrees-load point for a system without loss is the

critical load or the maximum value for all steady state operating

points that are inherently stable.

2.5.5 TRANSIENT STABILITY LIMIT

Transient stability refers to the amount of power that can be

transmitted with stability when the system is subjected to an

“aperiodic disturbance”. The three principal types of transient

disturbances that receive consideration in stability studies, in order of

increasing importance are:-

A. Load increases

B. Switching operation

C. Faults with subsequent circuit isolation.

44

A. LOAD INCREASE

Load increases can result in transient disturbances that are

important from the stability-standpoint if the following exist.

i. The total load exceeds the steady-state stability limit for specified

voltage and circuit reactance conditions.

ii. If the load increases set up an oscillation that causes the system to

swing beyond the critical point from which recovering would be

impossible.

If a large increment of load is added suddenly instead of

gradual, the synchronous machine may fall out of step even though

the steady-state stability limit has not been exceeded [22], [26], [27].

B. SWITCHING OPERATION

The transient-stability limits for switching- operations can be

investigated by looking at the initial condition and the final condition

after the switching operation has taken place.

The switching operation takes place the electrical out-put is

reduced. This change produces an increment power which is available

for accelerating the generator and decelerating the motor, both

changes tending to increase the angle between the sending and

receiving machines. Thus, the two machines depart from synchronous

speed, accelerating and decelerating forces increases the angle. At this

point, the generator rotor is traveling above the synchronous speed

with the result that both rotors tend to over shoot.

The amount of power transferable without loss of synchronous

depends upon the followings:-

45

i. The steady-state stability limit for the condition after switching

operation takes place.

ii. The difference between the initial and final state operating angles.

C. FAULTS

If short circuit faults of any degree occur within the systems,

the output power of the generator will be affected and become zero

while the mechanical input power to the generator remains constant

due to relating large time constant of the governor.

Consequently the speed of the synchronous machine decreases

thus, the machine will fall out of step.

The procedure of determining the stability of a system upon

occurrence of a disturbance followed by various switching off and on

actions is called a stability study. Steps to be followed in a stability

study are outlined below:-

46

G

Infinite

Busbar

Fault

Figure 2.4: Single machine infinite bus system

The fault is assured to be a transient one which is cleared by the time of first

reclosure of the circuit breaker. The steps listed below also apply to a system

of any size.

i. From the prefault loading, the voltage behind the transient reactance

and the torque angle δo of the machine with reference to the infinite

bus is determined.

ii. For the specified fault, the power transfer Pe (δ) is determined during

faults.

iii. From the swing equation, starting with δo as obtained in step i , δ is

calculated as a function of time using a numerical technique of

solving the non-linear differential equation.

iv. After the clearance of the fault, Pe(δ) is calculated and solved further

for (δ) (t).

47

v. After the switching on the transmission line, Pe((δ) ) is calculated and

likewise (δ)(t).

vi. If (δ)(t) goes through a maximum value and starts to reduce, the

system is regarded as stable. It is unstable if (δ)(t) continues to

increase. Calculation is ceased after a suitable length. [28, 29, 30].

2.6 THE SWING EQUATION

The swing equation describes the dynamics of a generator or motor.

Considering the generator in figure 2.25(a) .

It receives mechanical power Pim at torque at torque Ti and rotor speed

w via shaft from the prime mover. It delivers electrical power Pe to

the power system network via the bus bars. The generator develops

electromechemical torque Te in opposition to Ti. Assuming that

windage and frictional torque is negligible.

Figure 2.5: Generator, Prime Mover Diagram and Motor Dynamics.

Considering that the synchronous generator develops an electromagn etic

torque Te and running at the synchronous speed ωSm. If Tm is the driving

Generator Motor

Ti ω (a)

Pm

ω Te

Ti

(b)

Pe Pe Te

Pm

48

mechanical torque, then under steady-state operation with losses neglected

we have

Tm = Te -------------------------------------------------------------------------- (2.12)

A departure from stesy state due to a disturbance results in an accelerating

(Tm > Te) or decelerating (Tm< Te) torque Ta on the rotor.

Ta = Tm – Te --------------------------------------------------------------(2.13)

If J is the combined moment of inertia of the prime miver and generator,

neglecting the frictional and damping torques, from law’s of relation we

have

J = Ta = Tm – Te------------------------------------------- (2.14)

Where θm is the angular displacement of the rotor with respect to the

stationary reference axis on the stator. Since we are interested in the rotor

speed relative to synchronous speed, the angular reference is chosen relative

to a synchronously rotating reference frame moving with constant angular

velocity ωsm, that is:

θm = ωSmt + δm --------------------------------------------------------- (2.15)

Where δ is the rotor position before disturbance at time t = 0, measured from

the synchronously rotating reference frame. Derivative of (14) gives the

rotor angular velocity.

ωm = dθm = ωmS + dδm --------------------------------------------------- (2.16)

dt dt

and the rotor acceleration is

d2θ m = d

2δm ----------------------------------------------------------- (2.17)

dt2 dt

2

dt2

d2 θm

49

Substituting (16) into (13), we have

J = Tm – Te---------------------------------------------- (2.18)

Multiplying (17) by ωm, results in

Jωm d2δm = ωm Tm – ωm Te-------------------------------------- (2.19)

dt2

Since angular velocity times torque is equal to the power, we write the abo

ve equation in terms of power

Jωm = Pm - Pe ------------------------------------------- (2.20)

The quantity Jωm is called the inertia constant and is denoted by M. It is

related to Kinetic energy of the rotating masses, Wk.

Wk = ½ Jω2

m = ½ Mωm ----------------------------------------------- (2.21)

or

M = 2Wk -----------------------------------------------------------------(2.22)

ωm

Although M is called inertia constant, it is not really constat when the roror

speed deviates from the synchrounous speed. However, since Wm does not

change by a large amount before stability is lost, M is evaluated at the

synchrounous speed and is considered to reamain constant, ie,

M = 2Wk -----------------------------------------------------------------(2.23)

ωsm

The swing equation in terms of the inertia constant becomes:

M d2δm = Pm – Pe ------------------------------------------------------ (2.24)

dt2

dt2

d2δm

dt2

d2δ m

50

It is more convenient to write the swing equation in terms of the electrical

power angle δ . If P is the number of poles of a synchronous generator, the

electrical power angle δ is related to the mechanical power angle δm by:

δ = P/2 δm -------------------------------------------------------------- (2.25)

also, ω = P/2 ωm ------------------------------------------------------------(2.26)

Swing equation in terms of electrical power angle is

2/p M d2 δ = Pm – Pe ------------------------------------------------- (2.27)

dt2

Since power system analysis is done in per unit system, the swing equation

is usually expressed in per unit. Dividing (26) by the base power Sb, and

substituting for M from (22) results in

2/P 2 W k d2 δ = Pm – Pe ---------------------------------------- (2.28)

ω sm sb dt2 Sb Sb

We now define the important quantity known as the H constant or per unit

inertia constant.

H = Kinetic energy in MJ at rated speed = W k ------------------------- (2.29)

Machine rating in MVA Sb

The unit of it is seconds. The val ue of H ranges from I to 10 seconds,

depending on the size and type of machine. Substituting in (2.28), we get

2/p 2H d2 δ = Pm(p.u) – Pe(pu) -------------------------------------- (2.30)

ωsm dt2

Where Pm(pu) and Pe(pu) are the per unit mechanical power and electrical

power, respectively. The electrical angular velocity is related to the

mechanical angular velocity by ωsm = (2/p) ωs. Writing (2.30) in terms of

electrical angular velocity is

2H d2δ = Pm(p.u) - Pe(p.u) ----------------------------------- (2.31)

ωs dt2

51

Exressing the above equation in terms of frequency fo and simplifying the

notation, the subscript p.u is omitted and the powers are understood to be in

per unit.

H d2 δ = Pm - Pe --------------------------------------------- (2.32)

πfo dt2

Where δ is in electrical radian. If δ is expressed in electrical degrees, the

swing equation becomes

H d2 (δ) = Pm - Pe ------------------------------------------ (2.33)

180fo dt2

The above equation is called the swing equation and describes the

dynamics of a generator or a motor. It is strictly a non-linear equation where

accelerated power (Pa) has a non-linear functional dependence on δ, the

internal angle of machine with respect to busbar [31], [32], and [33].

Equation (2.32) is transformed into state variable model as follows:

dδi = Δωi ---------------------------------------------------------------------------------------------- (2.34)

dt

dΔωi = πf0 (pm - pe) ------------------------------------------------ (2.35)

dt Hi

This is the two state equation for each generator, with initial

power angle δ0i, Δω0i = 0.

The MATLAB function ode 23 is employed to solve the above 2m

first order differential equation.

52

2.6.1 THE POWER-ANGLE EQUATION

In the swing equation, for generator, the input-mechanical

power from the prime mover, Pm is assured to be constant.

Considering equation (2.31) Pm is constant and the electrical power

output Pe will be determined whether rotor accelerates, decelerates, or

remains at synchronous speed. When Pe changes from this value, the

rotor deviates from synchronous speed; when Pe equals Pm the

machine operates at steady-state synchronous speed. Changes in Pe are

deter mined by conditions on the transmission and distribution

networks, the loads on the system to which the generator supplies

power. Electrical network disturbance resulting from severe load

changes, network faults, or circuit breaker operations may cause the

generator out-put Pe to change rapidly in which case

electromechanically transients exists. Each synchronous machine is

represented for transient stability studies by its transient internal

voltage E; in senses with the transient reactance X1d, as show in figure

2.26 in which Vt is the terminal voltage.

2.6.2 TRANSIENT AND SUBTRANSIENT EFFECTS

When a fault occurs in a power system networks, the current

flowing is deter mined by the internal e.m.f of the machines in the

network; by their impedances, and by the impedances in the network

between the machines and the fault. The current flowing in a

synchronous machine immediately after the occurrence of a fault

differs from that flowing a few cycles later and from the sustained or

steady-state value of the fault current. This is because of the effect of

53

the fault current in the armature on the flux generating the voltage in

the machine. The current changes relatively slowly from its initial

value to its steady-state value owing to the changes in reactance of the

synchronous machine [31], [32], [33].

Figure 2.6: Circuit diagram of a synchronous machine for stability

studies.

2.7 EQUAL AREA CRITERION

To determine whether a power system is stable after a

disturbance, it is necessary, in general, to plot and to inspect the swing

curves. If these curves show that the angle between any two machines

tends to increase without limit, the system, of course is unstable. If,

on the other hand, after all disturbances, the angle reaches a maximum

and then diminishes, it is probable, although not certain, that the

system is stable.

jX1

d

+

Ei

-

Vt

i

54

Considering the swing equation below

M δ 2 = Pa = Pm - Pe ------------------------------------------ (2.34)

dt2

Where M is the inertia constant of the finite machine and δ is the

angular displacement of this machine with respect to infinite bus.

Multiplying each member of the equation by 2d δ

Mdt

gives:

M d2 δ . 2 d δ = Pa . 2d δ ------------------------------------- (2.35)

dt2 Mdt Mdt

Simplifying (34) gives

2 d2 δ . d δ = 2 . Pa . d δ

dt2 dt M dt

or

d dδ 2 = 2 . Pa . d δ ----------------------------- (2.36)

dt dt 2 M dt

Multiplying both sides of (2.36) by dt we obtain differential instead of

derivatives

d d δ 2 = P a d δ--------------------------------------- (2.37)

dt M

Integrating equation (2.37) gives:

55

d δ 2 = 1/m ∫ Pa . d δ

dt

d δ = ω = 1/m ∫ pa.dδ -------------------- 2.38)

dt

Where δo is the initial rotor angle before it begins to swing due to

disturbance.

When machine comes to rest with respect to the infinite bus, ω

becomes zero (ω = 0)

1/m ∫ Pa . d δ = 0

or

∫ pa.dδ= 0--------------------------- (2.39)

δm

δo

δm

δo

1/2

δm

δo

1/2 δm

δo

56

The integral is the area under a curve of Pa plotted against δ between

limits δo, the initial angle, and δm, the final angle.

Since Pa = Pm –Pe, the integral may be interpreted also as the area between

the curve of Pm versus δ. the curve of Pm versus δ is a horizontal line, since

Pm is assumed to be constant. The curve of Pe versus δ is known as a power

angle curve and it is a sinusoidal if the network is linear and if the machine

is represented by a constant reactance.

57

A2

A1

δ δm

Figure 2.7: plot of power against δ

Pm

Pe

P

0

58

Figure 2.8: plot of δ versus time for stable and unstable systems

The area, to be equal to zero must consist of a positive portion A1, for which

Pm > Pe, and an equal and opposite negative portion A2, for which Pm < Pe.

For a stable system, indication of stability will be given by observation of

the first swing where δ will go to a maximum and start to reduce. The

system is stable if at sometime d δ = 0 and

t

Unstable

dδ = 0

dt

Stable

δ

δmax

59

dt

is Unstable , if d δ

dt

The equal – area criterion cannot be used directly in multimachine

systems because the complexity of the numerical computations increases

with the number of machines considered in a transient stability study. When

a multi-machine system operates under electromechanical transient

conditions, inter-machine Oscillations occur through the medium of the

transmission system connecting the machines.

P1 = E1

2 G11 + E

11 E

12 Y12 Cos (δ 1 – δ2 -θ12) -------------------- (2.40)

Q1 = - E1

2 B11 + E

11 E

12 Y12 Sin (δ 1 – δ 2 -θ12) -------------------- (2.41)

Letting δ = δ 1 – δ 2 ---------------------------------------------------- (2.42)

and

defining a new angle which is temperature dependent such that

= θ12 - π/2 -------------------------------------------------------------- (2.43)

Therefore θ12 = +

Equations (2.40) & (2.41) give or transformed into the following:

P1 = E1

1 2 G11 + E

11 E

12 Y12 Sin (δ- - )- --------------------------- (2.44)

Q1 = - E1

1 2 B11 - E

11 E

12 Y12 Cos (δ- )--------------------------- (2.45)

> 0 (for a Sufficiently longtime, say one second).

60

Equations (43) can be written more simply as:

Pe = Pc + Pmax Sin (δ - ) -------------------------------------------- (2.46)

Where, Pc = E1

1 2 G11 and

Pmax = E1

1 E1

2 Y12

The parameters Pc, Pmax and are constants for a given network

configuration and constant voltage magnitudes E1

1 and E12 . When the

network is considered without resistance, all the elements of Ybus are

susceptance and G11 and are both zero. The power-angle equation which

then applies for the pure reactance network is simply the familiar equation

shown below.

Pe = Pmax Sin δ ----------------------------------------------------------------- (2.47)

Where Pmax = E1

1 E1

2 and

X

X is the transfer reactance between E11 and E

12. The in-phase component of

the admittance is the conductance G, and the quadrant component is the

susceptance [34, 35, 36].

Modern excitation systems employing thyristor controls can respond

rapidly to bus – voltage reduction and can affect from 0.5 to 1.5 cycles gain

in critical clearing times for three phase faults on the high side bus of the

generator step-up transformer. Modern electrohydraulic turbine governing

systems have the ability to close turbine valves to reduce unit acceleration

during severe system faults near the unit. A gain of 1 to 2 cycles in critical

clearing time can be achieved.

61

Reducing the reactance of the system during fault conditions increases Pmax

and decreases the acceleration area. Reducing the reactance of a

transmission line is another way of raising Pmax. Compensating for line

reactance by series capacitors is often an economical means of increasing

stability. Increasing the number of parallel lines between two points is a

common means of reducing reactance. When parallel transmission lines are

used instead of a single line, some power is transferred over the remaining

line even during a three-phase fault on one of the lines unless the fault

occurs at a paralleling bus. For more than two lines in parallel the power

transferred during the fault is even greater. Thus, the more power is

transferred into the system during a fault, the lower the acceleration of the

machine rotor and the greater the degree of stability [34] - [36].

62

2.8 MULTIMACHINE DYNAMIC MODEL

There is probably more literature on synchronous machines

than any other devices in-electrical engineering. Unfortunately, this

vast amount of material often makes the subject complex and

confusing.

In addition, most of the work on reduced order modeling is

based primarily on physical intuition, practical experience, and years

of experimentation. This model considers many synchronous

machines interconnected by transformers and transmission lines.

Here, loads are considered balanced symmetrical R-L elements. For

symbol notation, we have the followings:-

m = number of synchronous machines (if there is an infinite

bus, it is machine number 1).

n = number of system three-phase buses (excluding the datum

or reference bus).

b = total number of machines plus transformers plus lines plus

load (total branches).

63

PLm+1 (Vm+1) +

jQLm+1 (Vm+1) +

-

+

-

PLi (Vi) +

jQLi (Vi)

Network

I= YN V

m+1

PLn (Vn) + jQLn (Vn)

PLm (Vm) +

jQLm Vm

m

i

2.8.1 INTERCONNECTION OF SYNCHRONOUS MACHINE

DYNAMIC CIRCUIT AND THE REST OF THE NETWORK

Jx1

di Rsi

Jx1

dm Rsm

n

Figure: 2.9: Synchronous machine and the rest of the network

(Idi + jIqi) ej(δi -

π/2) = IGie

jγi = IDi + jIQi

64

2.8.2 NETWORK EQUATIONS

The network equations written at n buses are in complex form.

Network equations are represented as:

i GENERATOR BUSES

Vi ejθi

(Idi – jIqi) e-j(δi -

π/2) + PLi(Vi) + jQLi(Vi)

= n

ViVk Yik ej (θi - θk - ik)

k = 1 i = 1, ------------, m

---------------(2.48)

ii LOAD BUSES

PLi(Vi) + jQLi(Vi) = n

ViVk Yik ej (θi - θk - ik)

k = 1 ------------------------------ (2.49)

i = m + 1, ---------, n

In equation (47)

Vi ejθi

(Idi – jIqi) e-j(δi -

π/2)

PGi + jQGi

and it is the complex power “Injected” into bus i due to the generator.

Equations (2.47) and (2.48) represent the real and reactive power

balance equation at the n buses [19],[ 34], [35],[ 36].

65

2.9 POWER FLOW SOLUTION

Power flow studies, commonly known as load flow, form an important

part of power system analysis.

In solving a power flow problem, the system is assumed to be

operating under balanced conditions. Four quantities are associated with

each bus. These are voltage magnitude /V/, phase angle δ, real power P,

and reactive power Q. The system buses are generally classified into

three types, namely:

(i) Slack bus:- This is taken as reference bus where the

magnitude and phase angle of the voltage are specified. This

bus makes up the difference between the scheduled loads and

generated power that are caused by the losses in the network.

The bus is known by other names such as, reference bus, swing

bus, etc.

(ii) Load buses:- At these buses the active and reactive powers

are specified. The magnitude and the phase angle of the bus

voltages are unknown. These buses are called P-Q buses.

(iii ) Regulated buses:- These buses are the generator buses. They

are also known as voltage controlled buses. At these buses, the real

power and voltage magnitude are specified. The phase angles of the

voltages and the reactive power are to be determined. The limits on

the value of the reactive power are also specified. These buses are

called P-V buses [37]- [40].

66

2.9.1 POWER FLOW EQUATION

Consider a typical bus of power system shown in figure 2.10 below.

Vi

yi1 V1

yi2 V2

Ii

yin Vn

yio

Figure : 2.10: A typical bus of the power system.

Ii = yioVi + yi1(Vi - V1) + y12(Vi - V2) + ……+ yin(Vi - Vn)

=(yio + yi1 + yi2 + …….+ yin)Vi - yi1V1 – yi2V2 -------yinVn ----------- (2.50)

Equation (2.51) can be written as

Ii = Vi Σyij - ΣyijVj -------------------------------------------------- (2.51)

The real and reactive power @ bus i is

pi + jQi = ViIi* ---------------------------------------------------------------------------------------------------------- (2.52)

n

J=0

n

J=1

67

or

Ii = pi + jQi ------------------------------------------------------------------------------------------------------------ (2.53)

Vi*

Substituting for Ii in Equation (51) yields

pi + jQi = Vi Σyij - ΣyijVj ----------------------------------- (2.54)

Vi*

n

n

68

CHAPTER THREE

Data Analysis and Simulation

3.1 PROJECT DESIGN

The quality of life in any country is highly dependent on

a reliable electricity supply. In Nigeria, the electricity supply authority

is unable in most cases to meet up with a reliable and efficient power

supply to its consumers. The epileptic nature of the supply has led to

low economic growth and dissatisfaction among the citizenry.

As the size and complexity of electric power system increase

because of pressing economics and population, the desire to predict

system behaviour more accurately will also increase. The digital

computer has given the engineers the ability to predict in situations

where complexity would have been two great before.

For multi-machine system like that of Nigerian power system,

synchronism poses a great problem, when a machine or generator is

out of synchronism within an electric power system. This affects the

quality of supply to the consumers if it does not cause total collapse of

such a system. These undesirable occurrences usually cause a great

loss of revenue to the supplying authority and hardship to consumers

because of the inability of the system to fulfill requirement of its

customers. The undesirable factor of instability can be eliminated by

carrying out the stability studies of the system so that the transient

stability limit of the system loading can be determined.

69

Short circuit studies and load flow studies are carried out. These

results are used in the stability analysis.

3.2 DATA SOURCES

Power holding company Plc. 330KV electrical network single

Line diagram is used for this study. The generators, transmission

lines and transformer parameters are taken from the most up-to-date

data from National control centre, Oshogbo System Planning unit and

system operations department. The subtransient reactance X11

of the

synchronous machines is used to give maximum fault levels at the

instant of fault.

In a station where the number of machines is more than one, the

machines are represented with a single unit by paralleling transient

reactance and adding Inertia.

Line data for the equivalent system are used which contains a

complex tap ratio.

There are two types of bus bars namely:

i) Load bus bar

ii) Generator bus bar

iii) At each bus bar the voltage is assumed to be (1.0 + j0.0)

70

3.3 DATA ANALYSIS TECHNIQUE

The tool used for the data analysis is MAT-LAB software

package. MAT-LAB is a matrix-based software package, with its

extensive numerical resources, it can be used to obtain numerical

solutions that involve various types of vector-matrix-operations. My

choice of mat-lab for this work is as result of it’s high

performance COMPUTATION and VISUALIZATION. The

combination of analysis capabilities, flexibilities, reliability and

powerful graphics makes MAT-LAB the premier software

package for engineers and scientists.

3.4 MULTIMACHINE TRANSIENT STABILITY WITH MAT-LAB

SOFTWARE PACKAGE

The classical transient stability study is based on the application

of a three-phase fault. A solid three-phase fault at bus k in the network

results in Vk = 0. This is simulated by removing admittance matrix.

The new bus admittance matrix is reduced by eliminating all nodes

except the internal generator modes.

The generator excitation voltages during the fault and post fault

modes are assumed to remain constant.

The Electrical power of the ith generator in terms of the new

reduced bus admittance machines are obtained from the equation (43)

The swing equation with damping neglected is given as:

Hi d2δi = Pmi - m Ei

1 Ej

1 Yij Sin (δi - ) ------------ (3.1)

πfo dt2

j = 1

71

Where Yij are the elements of the faulted reduced bus admittance matrix, and

Hi is the inertia constant of machine i expressed on the common MVA base

SB. If HGi is the inertia constant of the machine i expressed on the machine

rated MVA SGi, then:

Hi is given as:

Hi = SGi HGi ------------------------------------------------------------ (3.2)

SB

Representing the electrical power of the ith generator by P

fe and transforming

equation (49) into state variable model yields:

dδi = ωi -------------------------------------------------------- (3.3)

dt

d ωi = fo (Pm - Pfe) --------------------------------------- (3.4)

dt Hi

i = 1-------------m

There are two state equation for each generator, with initial power

angle δo, ωoi = 0.

The mat lab function Ode 23 is employed to solve the above 2 – machine

first order differential equation.

When the fault is cleared, which may involve the removal of the

faulty line, the bus admittance matrix is recomputed to reflect the change in

the network.

Furthermore, the post fault reduced bus admittance matrix is evaluated

and the post fault Electrical Power of the ith generator shown by Pi

pf is

readily determined from equation (43).

72

Using the post fault power Pipf

, the simulation is continued to

determine the system stability, until the plots reveal a definite trend as to

stability or instability. Usually, the slack generator is selected as the

reference, and the phase angle difference of all other generators with respect

to the reference machine are plotted. Usually, the solution is carried out for

two swings to slow that the second is not greater than the first one.

If the angle differences do not increase, the system is stable but if any

of the angle differences increase indefinitely, the system is unstable.

Based on the procedure, a program named trstab is developed for the

transient stability analysis of a multi-machine network subjected to a

balanced three-phase fault. The program trstab must be preceded by the

power flow program. Ifnewton power flow program is used for this work.

In addition to the power flow data, generator data must be specified in

a matrix named gendata. The first column contains the generator bus

number terminal. Column 2 and 3 contain resistance and transient reactance

in per unit on the specified common MVA base. The program trstab

automatically adds additional buses to include the generator impedances in

the power flow line data.

Also, the bus admittance matrix is modified to include the load

admittances y load, returned by the power flow program. The program

prompts the user to enter the faulted bus number, fault clearing time, and the

line number of the removed faulty line.

The program displays the prefault, faulted, and post fault reduced bus

admittances matrices. The machine phase angles are tabulated and a plot of

the swing curves is obtained. The program inquires for other fault clearing

times and fault locations [3], [33], [38].

73

3.6 ASSUMPTIONS MADE FOR THIS WORK

1. Each synchronous machine is represented by a constant voltage source

behind the direct axis transient reactance. This representation neglects the

effect of saliency and assumes constant flux linkages.

2. The governor’s actions are neglected and the input powers are assumed

to remain constant during the entire period of simulation.

3. Using the prefault bus voltages, all loads are converted to equivalent

admittance to the ground and are assumed to remain constant.

4. Damping or asynchronous powers are ignored

5. The mechanical rotor angle of each machine coincides with the angle of

the voltage behind the machine reactance.

6. Machines belonging to the same station swing together and are said to be

coherent. A group of coherent machines is represented by one equivalent

machine.

3.7 PERFORMANCE OF PROTECTIVE RELAYING

Protective relays detect the existence of abnormal system

conditions by monitoring appropriate system quantities, determine

which circuit breakers should be opened, and energize trip circuits of

those breakers. In order to perform their functions satisfactorily,

relays should satisfy three basic requirements: selectivity, speed and

reliability.

Since transient stability is concerned with the ability of the

power system to maintain synchronism when subjected to a severe

disturbance, satisfactory performance of certain protection system is

of paramount importance in ensuring system stability.

Protective relays must be able to distinguish among fault

conditions, stable power swings and out-of-step condition. While the

74

relays should initiate circuit-breaker operations to clear faulted

elements, it is important to ensure that there are no further relaying

operations that cause unnecessary opening of unfaulted elements

during stable power swings. Tripping of unfaulted element would

weaken the system further and could lead to system instability.

One of the important aspects of transient stability analysis is the

evaluation of the performance of protective systems during the

transient period, particularly the performance of relaying used for

protection of transmission lines and generators [38].

3.7.1 FAULT – CLEARING TIMES

The critical clearing time is the maximum elapsed time from the

initiation of fault until its isolation such that the power system is

transiently stable.

The removal of a faulted element requires a protective relay

system to detect that a fault has occurred and to initiate the opening of

circuit breakers which will isolate the faulted element from the

system.. The total fault-clearing time is, therefore, made up of the

relay time and breaker-interrupting time. The relay time is the time

from the initiation of the short-circuit current to the initiation of the

trip signal to the circuit breaker. The interrupting time is the time from

initiation of the trip signal to the interruption of the current through

the breaker.

On High voltage (HV) and Extra-high voltage (EHV)

transmission systems, the normal relay times range from 15 to 30ms

(1 to 2 cycles) and circuit breaker interrupting times range from 30 to

40ms (2 to 2.5 cycles). [34]

75

3.7.2 FACTORS INFLUENCING TRANSIENT STABILITY

Transient stability of the generator is dependent on the

following:

i How heavily the generator is loaded.

ii The generator output during the fault. This depends on the fault

location and type.

iii The fault- clearing time.

iv The post fault transmission system reactance.

v The generator reactance. A lower reactance increases peak power

and reduces initial rotor angle.

vi The generator inertia. The higher the inertia, the slower the rate of

change in angle. This reduces the Kinetic energy gained during

fault ie, area A1 is reduced.

vii The generator internal voltage magnitude ( E ).

This depends on the field excitation.

viii The infinite bus voltage magnitude EB [37], [38]

76

3.8 DATA FOR THE WORK

The data for this work is obtained from the up to date records of the

power Holding company of nigerian's National Control Centre, Oshogbo.

The departments visited inlude:

(1) system planing.

(2) system operations.

(3) SCADA sections.

( 4) Protection division.

Verbal interaction was also carried out with the most senior and principal

Engineers of the departments. These Data are shown below:

TABLE 3.1 LOAD DATA

LOAD DATA

BUS NO LOAD

MW MVar

1 00.00 00.00

2 40.00 - 10.00

3 00.00 00.00

4 140.00 30.00

5 90.00 30.00

6 160.00 70.00

7 00.00 00.00

8 130.00 70.00

9 300.00 90.00

10 210.00 40.00

77

11 00.00 00.00

12 50.00 -20.00

13 100.00 -30.00

14 120.00 60.00

15 500.00 50.00

16 250.00 43.00

17 70.00 38.00

18 00.00 00.00

19 200.00 55.00

20 150.00 35.00

21 00.00 00.00

22 00.00 00.00

23 300.00 45.00

24 00.00 00.00

25 100.00 58.00

26 00.00 00.00

27 00.00 00.00

28 00.00 00.00

29 120.00 80.00

30 130.00 -78.00

31 00.00 00.00

32 200.00 67.00

33 00.00 00.00

34 00.00 00.00

78

TABLE 3.2 GENERATION SCHEDULE

Bus

No

Voltage Magnitude Generation MW MVar Limits

Min Max

1 1.06 00.00 0 0

2 1.0 00.00 0 0

3 1.04 300.00 0 110

4 1.0 00.00 0 0

5 1.0 00.00 0 0

6 1.04 20.00 0 0

7 1.0 400.00 0 140

8 1.0 00.00 0 0

9 1.0 00.00 0 0

10 1.02 00.00 0 0

11 1.0 150.00 0 114

12 1.0 00.00 0 0

13 1.0 00.00 0 0

14 1.0 00.00 0 0

15 1.0 000.00 0 0

16 1.0 00.00 0 0

17 1.0 00.00 0 0

18 1.03 280.00 0 100

19 1.0 00.00 0 0

20 1.0 00.00 0 0

21 1.02 240.00 0 104

22 1.05 700.00 0 108

23 1.0 00.00 0 0

24 1.04 180.00 0 132

79

25 1.0 00.00 0 0

26 1.01 190.00 0 126

27 1.03 150.00 0 100

28 1.02 130.00 0 150

29 1.0 00.00 0 0

30 1.0 00.00 0 0

31 1.03 150.00 0 100

32 1.0 00.00 0 146

33 1.04 200.00 0 140

34 1.02 300.00 0 125

TABLE 3.3: MACHINES AND SYSTEM VARS

Bus No Machine Vars System Injected Vars

1 00.00 0

2 00.00 0

3 40.00 0

4 00.00 0

5 00.00 0

6 00.00 0

7 60.00 0

8 00.00 0

9 00.00 0

10 00.00 0

11 50.00 0

12 00.00 0

13 00.00 0

14 00.00 0

15 00.00 0

80

16 00.00 0

17 00.00 0

18 45.00 0

19 00.00 0

20 00.00 0

21 55.00 0

22 68.00 0

23 00.00 0

24 00.00 0

25 00.00 0

26 -35.00 0

27 51.00 0

28 80.00 0

29 00.00 0

30 00.00 0

31 00.00 0

32 00.00 0

33 59.00 0

34 65.00 0

TABLE 3.4: LINE DATA

Bus No Bus No R (P.U) X (P.U) B( P.U) TRANSFORMER

TAP SETTING

PER UNITY

1 2 0.0121836 0.0916336 1.21 1.0

1 4 0.0015918 0.0119716 0.31 1.0

3 4 0.0001572 0.0094178 0.00 1.0

4 5 0.0047827 0.0360219 0.09 1.0

4 9 0.0020565 0.0154692 0.07 1.0

5 6 0.0018864 0.0141884 0.36 1.0

5 7 0.0003144 0.0188355 0.00 1.0

5 10 0.0018864 0.0141884 0.37 1.0

8 9 0.0053843 0.0404961 0.33 1.0

8 15 0.0053343 0.0405651 0.45 1.0

9 15 0.0065432 0.0426547 0.55 1.0

9 16 0.0098648 0.0741936 0.98 1.0

10 13 0.0090394 0.0679862 0.52 1.0

81

10 14 0.0077425 0.0582316 0.77 1.0

11 15 0.0020643 0.0103951 0.31 1.0

12 15 0.0040534 0.0305160 0.41 1.0

14 17 0.0104150 0.0783319 0.01 1.0

15 16 0.0110045 0.0827653 0.09 1.0

15 20 0.0003527 0.0026574 0.05 1.0

15 21 0.0055023 0.0413829 0.35 1.0

15 22 0.0012184 0.0091634 0.20 1.0

16 18 0.0063843 0.0404961 0.15 1.0

16 19 0.0038336 0.0288242 0.76 1.0

16 21 0.0055023 0.0413829 0.55 1.0

16 23 0.0053843 0.0404961 0.38 1.0

16 24 0.0009826 0.0073898 0.19 1.0

18 25 0.0010218 0.0076553 0.10 1.0

19 26 0.0005109 0.0038427 0.38 1.0

19 27 0.0006105 0.0038427 0.40 1.0

22 28 0.0005109 0.0036458 0.30 1.0

22 29 0.0002749 0.0020654 0.20 1.0

23 30 0.0037730 0.0283768 0.37 1.0

23 31 0.004913 0.0036949 0.09 1.0

23 32 0.00605225 0.0455212 0.02 1.0

24 25 0.0024760 0.0186223 0.24 1.0

28 29 0.0034640 0.0206114 0.30 1.0

32 33 0.0009825 0.0073898 0.09 1.0

33 34 0.0005109 0.0038427 0.30 1.0

TABLE 3.5: MACHINE DATA

GENERATOR

NUMBER

Ra(Ω)) X1d H

1 0.0020 0.0901 9.920

3 0.0080 0.3000 3.390

7 0.0240 0.3000 3.240

11 0.0036 0.2200 4.000

18 0.0020 0.1240 12.400

21 0.0036 0.2200 4.000

22 0.0040 0.3080 3.090

24 0.0030 0.1060 12.690

26 0.0061 0.3400 1.245

27 0.0036 0.3000 1.242

28 0.0051 0.2100 1.249

82

31 0.0061 0.3000 4.000

33 0.0010 0.0610 28.050

34 0.0051 0.1900 1.350

83

CHAPTER FOUR

4.1 EXISTING 330KV NATIONAL GRID NETWORK

1

K3R

KIJ K2J

10

RIM R2M

M2S

M6N

SIE

13

17

14

7

P4A

R5B R4B J7R

J3R 5

6

B8J

B9J

4

JIH J2H J3H

9 H2A

W2A

01W

NWIBS

W3L

W4L

11

12

20

N3J

N4J

29

E21

22

E1A 15

E11 E2A

28

M7W

21

B5M

B6W

H7B

16

ABUJA

B11J

B12J AIG A2G

19

AIS

A2S

27

26

T3H

23 30

BIT

G3B

18

G1W

25

S4w

24

S3B

S4B KIT

K2T T4A

32

33

F1A F2A MM6

8

A4J

31

Figure 4.11 Existing 330kv National grid Network Diagram.

3

2

84

4.2. SIMULATION AND DISCUSSION OF RESULTS

The classical transient stability study is based on the application of a

three-phase fault. A solid three-phase fault at bus k in the network results in

Vk = 0.

Simulations were carried out using raw data on pages 51,52,53,54,

55, & 56 respctively in MATLAB software package environment to

examine the behaviour of Nigeria 330KV power system network during

large scale disturbance and hence, determine the actual critical clearing time

and angle of the system such that the grid is transiently stable.

Egbin – Ikeja West 330KV line was faulted and removed from

the system. This is simulated for a period of 0.025 (25 milliseconds)

and was observed that the phase angle difference of all the fourteen

(14) machine increase with out limit as shown in figure 4.2 until the

fault was cleared at 0.025 seconds as shown in figure 4.3. The

simulation is continued until the Critical Clearing time and angle are

Obtained as shown in figures 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10 and 4.11

respectively. It was observed that the phases angle difference, after

reaching a maximum, start to decrease and the machines regain their

synchronism and start swinging together in unison.

85

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4

5

6

7

Time[s]

Roto

r angle

[radii]]

GENERATORS SWING DURING FAULT FOR 0.025 SECONDS

KG

JG

SG

OG

DG

OMG

EG

SAG

GG

AG

AEG

OKG

AFG

OMKG

Figure 4.2 : Plot of rotor angle against time

86

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1

-0.5

0

0.5

1

1.5

2

Time[s]

Roto

r angle

[radii]]

System fault cleared @ 0.025seconds

KG

JG

SG

OG

DG

OMG

EG

SAG

GG

AG

AEG

OKG

AFG

OMKG

Figure 4.3: Plot of rotor angle against time

87

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -1

-0.5

0

0.5

1

1.5

2

Time[s]

Rotor angle[radii]]

System fault cleared @ 0.030 seconds

KG JG SG OG DG OMG EG SAG GG AG AEG OKG AFG OMKG

Figure 4.4: Plot of rotor angle against time

88

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-1.5

-1

-0.5

0

0.5

1

1.5

2

Time[s]

R

otor angle

[radii]]

System fault cleared @ 0.035 seconds

KG

JG

SG

OG

DG

OMG

EG

SAG

GG

AG

AEG

OKG

AFG

OMKG

Figure 4.5: Plot of rotor angle against time

89

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -1

-0.5

0

0.5

1

1.5

2

Time[s]

Rotor angle [radii]]

System fault cleared @ 0.040seconds

KG JG SG OG DG OMG EG SAG GG AG AEG OKG AFG OMKG

Figure 4.6: Plot of rotor angle against time

90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Time[s]

Rotor angle [radii]]

System generators swinging in synchronism after the fault clearing time of 0.045 seconds

Figure 4.7: Plot of rotor angle against time

KG JG SG OG DG OMG EG SAG GG AG AEG OKG AFG OMKG

91

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -2

0

2

4

6

8

10

12

14

Time[s]

Rotor ange[radii]]

faulted machine 6 removed @ clearing time of 0.050 seconds

Figure 4.8: Plot of angle against time

KG JG SG OG DG OMG EG SAG GG AG AEG OKG AFG OMKG

92

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -3

-2

-1

0

1

2

3

Time[s]

Rotor angle[radii]]

System fault cleared @ 0.055 seconds after faulted machine 6 has been removed from the System

KG JG SG OG DG EG SAG GG AG AEG OKG AFG OMKG

Figure 4.9: Plot of angle against time

93

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -10

0

10

20

30

40

50

Time[s]

Rotor angle [radii]]

Faulted machine 11 removed @ 0.060 seconds which is the critical clearing time of the system

Figure 4.10: Plot of rotor angle against time

KG JG SG OG DG OMG EG SAG GG AG AEG OKG AFG OMKG

94

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -1

-0.5

0

0.5

1

1.5

2

2.5

Time[s]

Rotor angle[radii]]

Healthy System after faults have been completely removed @ 0.065 seconds

Figure 4.11: Plot of rotor angle against time

KG JG SG OG DG EG SAG GG AG AEG AFG OMKG

95

The simulations were repeated for fault clearing times of

0.030s, 0.035s, 0.040s, 0.045s, 0.050s, 0.055s & 0.065s. It was

observed that at 0.050s, faulted machine six (6) was removed from

the system. At the fault clearing time of 0.060s, faulted machine 11

was removed from the rest of the other generators in the system.

Simulation was continued, system generators continue to swing

in synchronism after machines 6 and 11 were removed from the

system.

The critical clearing time of the system is 0.060s and the

corresponding clearing angle is 2.5 radii or 143.20 in degree. This

fault clearing time is used to predict the entire Nigerian 330KV grid

network since the Egbin – Ikeja West 330KV line is the one of the

heaviest feeder in the national grid.

The PHCN protection Engineers need to set their relay and

circuit breaker operating time at 0.060s or less such that the entire

network will be transiently stable after the occurrence of a fault

thereby reducing the probability of total system collapse to the barest

minimum.

According to the annual technical report released by the

National Control Center Oshogbo, PHCN on 5th

March, 2006,

73.33% of all the system disturbances were due to the transmission

faults, mainly from Egbin – Ikeja West 330KV line.

Analysis of the events leading to the system disturbances

generally revealed inadequacy of protection schemes in the national

grid network. Some of the protection problems on the grid are

traceable to the lack of proper coordination of distance relays, failure

96

and spurious relay operations [39, 40, 41, 42, 43, 44]. This work

serves as a guide to the PHCN Protection Engineers to set their

relays and circuit breakers operating time to act at 0.060s (60

millisecond) or less in the event of a three phase fault in the grid, so

that the system regains its synchronising power fast such that the

network is transiently stable and integrity of the national grid network

maintained. Also, customer’s inconvinences are reduced to the barest

minimum.

97

Chapter Five

4.2 CONCLUSION AND RECOMMENDATIONS

In this work, a suitable critical clearing time and angle has been

achieved or obtained for Nigerian Power System so that our

system will be able to survive any severe disturbance.

PHCN protection Engineers should endeavour to set their relay

and circuit breaker operating time to clear fault at 60 milliseconds

and it's corresponding angle of 143.20 or less than this time and angle.

If this is done the incidence of power system collapse will be

minimized to the barest minimum.

The persistent system collapse in the National grid is traceable

to the inappropriate relay and circuit breakers operating time which is

higher than 120 milliseconds [41],[42]. If the system is allowed to

collapse, it takes time for the system to be revived and this leads to the

customer’s frustrations and loss in revenue to PHCN.

RECOMMENDATION: It is very important to point out here that

increasing the number of parallel lines between two points is a common

means of reducing reactance. When a parallel transmission lines are used

instead of a single line, some power is transferred over the remaining line

even during a three – phase fault on one of the lines unless the fault occurs at

a paralleling bus. Thus, the more power is transferred into the system during

a fault, the lower the acceleration of the machine rotor and the greater the

98

degree of stability, hence, a gain in critical clearing time can be achieved.

Benin – Onitsha – Alaoji 330KV line is limited by a single line contingency.

There is an urgent need to construct the second Benin – Onitsha 330KV

circuit which had been under plan for some years past [40], [41], [42]. Also,

construction of the proposed Alaoji – New Heaven – Markurdi – Jos 330KV

circuit configuration should be expedited.

The results obtained from this work are used to predict the transient

stability of the entire Nigerian power system since Egbin – Ikeja West is

one of the heavily loaded 330KV line in the National grid.

99

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103

(39) PHCN National Control Centre Oshogbo “Generation and

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March and 28 March –

10th April, 2001)

104

APPENDIX 1

% Power flow solution by Newton-Raphson

method

ns=0; ng=0; Vm=0; delta=0; yload=0; deltad=0; nbus = length(busdata(:,1)); for k=1:nbus n=busdata(k,1); kb(n)=busdata(k,2); Vm(n)=busdata(k,3); delta(n)=busdata(k, 4); Pd(n)=busdata(k,5); Qd(n)=busdata(k,6); Pg(n)=busdata(k,7); Qg(n) =

busdata(k,8); Qmin(n)=busdata(k, 9); Qmax(n)=busdata(k, 10); Qsh(n)=busdata(k, 11); if Vm(n) <= 0 Vm(n) = 1.0; V(n) = 1 + j*0; else delta(n) = pi/180*delta(n); V(n) = Vm(n)*(cos(delta(n)) + j*sin(delta(n))); P(n)=(Pg(n)-Pd(n))/basemva; Q(n)=(Qg(n)-Qd(n)+ Qsh(n))/basemva; S(n) = P(n) + j*Q(n); end end for k=1:nbus if kb(k) == 1, ns = ns+1; else, end if kb(k) == 2 ng = ng+1; else, end ngs(k) = ng; nss(k) = ns; end Ym=abs(Ybus); t = angle(Ybus); m=2*nbus-ng-2*ns; maxerror = 1; converge=1; iter = 0; % Start of iterations clear A DC J DX while maxerror >= accuracy & iter <= maxiter % Test for max. power

mismatch for i=1:m for k=1:m A(i,k)=0; %Initializing Jacobian matrix end, end iter = iter+1; for n=1:nbus nn=n-nss(n); lm=nbus+n-ngs(n)-nss(n)-ns; J11=0; J22=0; J33=0; J44=0; for i=1:nbr if nl(i) == n | nr(i) == n if nl(i) == n, l = nr(i); end if nr(i) == n, l = nl(i); end J11=J11+ Vm(n)*Vm(l)*Ym(n,l)*sin(t(n,l)- delta(n) + delta(l)); J33=J33+ Vm(n)*Vm(l)*Ym(n,l)*cos(t(n,l)- delta(n) + delta(l)); if kb(n)~=1 J22=J22+ Vm(l)*Ym(n,l)*cos(t(n,l)- delta(n) + delta(l)); J44=J44+ Vm(l)*Ym(n,l)*sin(t(n,l)- delta(n) + delta(l)); else, end

105

if kb(n) ~= 1 & kb(l) ~=1 lk = nbus+l-ngs(l)-nss(l)-ns; ll = l -nss(l); % off diagonalelements of J1 A(nn, ll) =-Vm(n)*Vm(l)*Ym(n,l)*sin(t(n,l)- delta(n) +

delta(l)); if kb(l) == 0 % off diagonal elements of J2 A(nn, lk) =Vm(n)*Ym(n,l)*cos(t(n,l)- delta(n) +

delta(l));end if kb(n) == 0 % off diagonal elements of J3 A(lm, ll) =-Vm(n)*Vm(l)*Ym(n,l)*cos(t(n,l)-

delta(n)+delta(l)); end if kb(n) == 0 & kb(l) == 0 % off diagonal elements of

J4 A(lm, lk) =-Vm(n)*Ym(n,l)*sin(t(n,l)- delta(n) +

delta(l));end else end else , end end Pk = Vm(n)^2*Ym(n,n)*cos(t(n,n))+J33; Qk = -Vm(n)^2*Ym(n,n)*sin(t(n,n))-J11; if kb(n) == 1 P(n)=Pk; Q(n) = Qk; end % Swing bus P if kb(n) == 2 Q(n)=Qk; if Qmax(n) ~= 0 Qgc = Q(n)*basemva + Qd(n) - Qsh(n); if iter <= 7 % Between the 2th & 6th

iterations if iter > 2 % the Mvar of generator buses

are if Qgc < Qmin(n), % tested. If not within limits

Vm(n) Vm(n) = Vm(n) + 0.01; % is changed in steps of 0.01

pu to elseif Qgc > Qmax(n), % bring the generator Mvar

within Vm(n) = Vm(n) - 0.01;end % the specified limits. else, end else,end else,end end if kb(n) ~= 1 A(nn,nn) = J11; %diagonal elements of J1 DC(nn) = P(n)-Pk; end if kb(n) == 0 A(nn,lm) = 2*Vm(n)*Ym(n,n)*cos(t(n,n))+J22; %diagonal elements of

J2 A(lm,nn)= J33; %diagonal elements of J3 A(lm,lm) =-2*Vm(n)*Ym(n,n)*sin(t(n,n))-J44; %diagonal of elements

of J4 DC(lm) = Q(n)-Qk; end end

gx=A; DX=A\DC'; for n=1:nbus nn=n-nss(n); lm=nbus+n-ngs(n)-nss(n)-ns;

106

if kb(n) ~= 1 delta(n) = delta(n)+DX(nn); end if kb(n) == 0 Vm(n)=Vm(n)+DX(lm); end end maxerror=max(abs(DC)); if iter == maxiter & maxerror > accuracy fprintf('\nWARNING: Iterative solution did not converged after ') fprintf('%g', iter), fprintf(' iterations.\n\n') fprintf('Press Enter to terminate the iterations and print the

results \n') converge = 0; pause, else, end

end

if converge ~= 1 tech= (' ITERATIVE SOLUTION DID NOT CONVERGE');

else, tech=(' Newton-Raphson Power Flow Solution'); end V = Vm.*cos(delta)+j*Vm.*sin(delta); deltad=180/pi*delta; i=sqrt(-1); k=0; for n = 1:nbus if kb(n) == 1 k=k+1; S(n)= P(n)+j*Q(n); Pg(n) = P(n)*basemva + Pd(n); Qg(n) = Q(n)*basemva + Qd(n) - Qsh(n); Pgg(k)=Pg(n); Qgg(k)=Qg(n); %june 97 elseif kb(n) ==2 k=k+1; S(n)=P(n)+j*Q(n); Qg(n) = Q(n)*basemva + Qd(n) - Qsh(n); Pgg(k)=Pg(n); Qgg(k)=Qg(n); % June 1997 end yload(n) = (Pd(n)- j*Qd(n)+j*Qsh(n))/(basemva*Vm(n)^2); end busdata(:,3)=Vm'; busdata(:,4)=deltad'; Pgt = sum(Pg); Qgt = sum(Qg); Pdt = sum(Pd); Qdt = sum(Qd); Qsht =

sum(Qsh); inv(A); Vm(4); deltad(4); Vm(2)*YL(1,2) Vm(3)*gx; plot(Vm) grid on %clear A DC DX J11 J22 J33 J44 Qk delta lk ll lm %clear A DC DX J11 J22 J33 Qk delta lk ll lm

107

APPENDIX 2

% This program obtains th Bus Admittance Matrix

for power flow solution

j=sqrt(-1); i = sqrt(-1); nl = linedata(:,1); nr = linedata(:,2); R = linedata(:,3); X = linedata(:,4); Bc = j*linedata(:,5); a = linedata(:, 6); nbr=length(linedata(:,1)); nbus = max(max(nl), max(nr)); Z = R + j*X; y= ones(nbr,1)./Z; %branch admittance for n = 1:nbr if a(n) <= 0 a(n) = 1; else end Ybus=zeros(nbus,nbus); % initialize Ybus to zero % formation of the off diagonal elements for k=1:nbr; Ybus(nl(k),nr(k))=Ybus(nl(k),nr(k))-y(k)/a(k); Ybus(nr(k),nl(k))=Ybus(nl(k),nr(k)); end end % formation of the diagonal elements for n=1:nbus for k=1:nbr if nl(k)==n Ybus(n,n) = Ybus(n,n)+y(k)/(a(k)^2) + Bc(k); elseif nr(k)==n Ybus(n,n) = Ybus(n,n)+y(k) +Bc(k); else, end end end clear Pgg

108

APPENDIX 3

% This program prints the power flow solution in a

tabulated form on the screen.

%clc disp(tech) fprintf(' Maximum Power Mismatch = %g \n',

maxerror) fprintf(' No. of Iterations = %g \n\n',

iter) head =[' Bus Voltage Angle ------Load------ ---Generation---

Injected' ' No. Mag. Degree MW Mvar MW Mvar

Mvar ' '

']; disp(head) for n=1:nbus fprintf(' %5g', n), fprintf(' %7.3f', Vm(n)), fprintf(' %8.3f', deltad(n)), fprintf(' %9.3f', Pd(n)), fprintf(' %9.3f', Qd(n)), fprintf(' %9.3f', Pg(n)), fprintf(' %9.3f ', Qg(n)), fprintf(' %8.3f\n', Qsh(n)) end fprintf(' \n'), fprintf(' Total ') fprintf(' %9.3f', Pdt), fprintf(' %9.3f', Qdt), fprintf(' %9.3f', Pgt), fprintf(' %9.3f', Qgt), fprintf('

%9.3f\n\n', Qsht)

109

APPENDIX 4

% TRANSIENT STABILITY ANALYSIS OF A MULTIMACHINE

POWER SYSTEM NETWORK

f=50; %zdd=gendata(:,2)+j*gendata(:,3); ngr=gendata(:,1); %H=gendata(:,4); ngg=length(gendata(:,1)); %% for k=1:ngg zdd(ngr(k))=gendata(k, 2)+j*gendata(k,3); %H(ngr(k))=gendata(k, 4); H(k)=gendata(k,4); % new end %% for k=1:ngg I=conj(S(ngr(k)))/conj(V(ngr(k))); %Ep(ngr(k)) = V(ngr(k))+zdd(ngr(k))*I; %Pm(ngr(k))=real(S(ngr(k))); Ep(k) = V(ngr(k))+zdd(ngr(k))*I; % new Pm(k)=real(S(ngr(k))); % new

end E=abs(Ep); d0=angle(Ep); for k=1:ngg nl(nbr+k) = nbus+k;

nr(nbr+k) = gendata(k, 1);

%R(nbr+k) = gendata(k, 2); %X(nbr+k) = gendata(k, 3);

R(nbr+k) = real(zdd(ngr(k))); X(nbr+k) = imag(zdd(ngr(k)));

Bc(nbr+k) = 0; a(nbr+k) = 1.0; yload(nbus+k)=0; end nbr1=nbr; nbus1=nbus; nbrt=nbr+ngg; nbust=nbus+ngg; linedata=[nl, nr, R, X, -j*Bc, a]; [Ybus, Ybf]=ybusbf(linedata, yload, nbus1,nbust); fprintf('\nPrefault reduced bus admittance matrix \n') Ybf Y=abs(Ybf); th=angle(Ybf); Pm=zeros(1, ngg); disp([' G(i) E''(i) d0(i) Pm(i)']) for ii = 1:ngg for jj = 1:ngg Pm(ii) = Pm(ii) + E(ii)*E(jj)*Y(ii, jj)*cos(th(ii, jj)-d0(ii)+d0(jj));

110

end, fprintf(' %g', ngr(ii)), fprintf(' %8.4f',E(ii)), fprintf('

%8.4f', 180/pi*d0(ii)) fprintf(' %8.4f \n',Pm(ii)) end respfl='y'; while respfl =='y' | respfl=='Y' nf=input('Enter faulted bus No. -> '); fprintf('\nFaulted reduced bus admittance matrix\n') Ydf=ybusdf(Ybus, nbus1, nbust, nf) %Fault cleared [Yaf]=ybusaf(linedata, yload, nbus1,nbust, nbrt); fprintf('\nPostfault reduced bus admittance matrix\n') Yaf resptc='y'; while resptc =='y' | resptc=='Y' tc=input('Enter clearing time of fault in sec. tc = '); tf=input('Enter final simulation time in sec. tf = ');

111

APPENDIX 5

% This function forms the bus admittance matrix

including load admittances before fault. The

corresponding reduced bus admittance matrix is

obtained for transient stability study.

function [Ybus, Ybf] = ybusbf(linedata, yload, nbus1, nbust) global Pm f H E Y th ngg

lfybus for k=1:nbust Ybus(k,k)=Ybus(k,k)+yload(k); end YLL=Ybus(1:nbus1, 1:nbus1); YGG = Ybus(nbus1+1:nbust, nbus1+1:nbust); YLG = Ybus(1:nbus1, nbus1+1:nbust); Ybf=YGG-YLG.'*inv(YLL)*YLG;

112

APPENDIX 6

% This function forms the bus admittance matrix

including load admittances during fault. The

corresponding reduced bus admittance matrix is

obtained for transient stability study.

function Ypf=ybusdf(Ybus, nbus1, nbust, nf) global Pm f H E Y th ngg nbusf=nbust-1; Ybus(:,nf:nbusf)=Ybus(:,nf+1:nbust); Ybus(nf:nbusf,:)=Ybus(nf+1:nbust,:); YLL=Ybus(1:nbus1-1, 1:nbus1-1); YGG = Ybus(nbus1:nbusf, nbus1:nbusf); YLG = Ybus(1:nbus1-1, nbus1:nbusf); Ypf=YGG-YLG.'*inv(YLL)*YLG;

113

APPENDIX 7

% This function forms the bus admittance matrix

including load admittances after fault. The

corresponding reduced bus admittance matrix is

obtained for transient stability study.

function [Yaf]=ybusaf(linedata, yload, nbus1,nbust, nbrt); global Pm f H E Y th ngg

nl=linedata(:, 1); nr=linedata(:, 2); remove = 0; while remove ~= 1 fprintf('\nFault is cleared by opening a line. The bus to bus nos. of

the\n') fprintf('line to be removed must be entered within brackets, e.g. [5,

7]\n') fline=input('Enter the bus to bus Nos. of line to be removed -> '); nlf=fline(1); nrf=fline(2); for k=1:nbrt if nl(k)==nlf & nr(k)==nrf remove = 1; m=k; else, end end if remove ~= 1 fprintf('\nThe line to be removed does not exist in the line

data. try again.\n\n') end end linedat2(1:m-1,:)= linedata(1:m-1,:); linedat2(m:nbrt-1,:)=linedata(m+1:nbrt,:);

linedat0=linedata; linedata=linedat2; lfybus for k=1:nbust Ybus(k,k)=Ybus(k,k)+yload(k); end YLL=Ybus(1:nbus1, 1:nbus1); YGG = Ybus(nbus1+1:nbust, nbus1+1:nbust); YLG = Ybus(1:nbus1, nbus1+1:nbust); Yaf=YGG-YLG.'*inv(YLL)*YLG; linedata=linedat0;

114

APPENDIX 8

% State variable representation of the multimachine

system during fault. (for use with trstab) function xdot = dfpek(t,x) global Pm f H E Y th ngg Pe=zeros(1, ngg); for ii = 1:ngg for jj = 1:ngg Pe(ii) = Pe(ii) + E(ii)*E(jj)*Y(ii, jj)*cos(th(ii, jj)-x(ii)+x(jj)); end, end for k=1:ngg xdot(k)=x(k+ngg); xdot(k+ngg)=(pi*f)/H(k)*(Pm(k)-Pe(k)); end xdot=xdot'; % use with MATLAB 5 (remove for MATLAB 4)

115

APPENDIX 9

% State variable representation of the multimachine

system

after fault. (for use with trstab)

function xdot = afpek(t,x) global Pm f H E Y th ngg

Pe=zeros(1, ngg); for ii = 1:ngg for jj = 1:ngg Pe(ii) = Pe(ii) + E(ii)*E(jj)*Y(ii, jj)*cos(th(ii, jj)-x(ii)+x(jj)); end, end for k=1:ngg xdot(k)=x(k+ngg); xdot(k+ngg)=(pi*f)/H(k)*(Pm(k)-Pe(k)); end xdot=xdot'; % use with MATLAB 5 (remove for MATLAB 4)

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APPENDIX 10

Plotting Program 1

f=50; a2=0; H2=30.390; Pm2=11.6670; Pc2=0; Pmax2=9.5646; % y2''=((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y2-a2) t0=0 y20=(14/180)*pi dy20=0.1; h=0.005; k=1; ddy20(k)=((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y20-a2)); k1(k)=(1/2)*(h)^2*((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y20-a2)); t01(k)=t0+(h/2); y201(k)=y20+(h/2)*dy20+(k1(k)/4); k2(k)=(1/2)*(h)^2*((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y201(k)-a2)); k3(k)=(1/2)*(h)^2*((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y201(k)-a2)); t011(k)=t0+h; y2011(k)=y20+(h*dy20)+k3(k); k4(k)=(1/2)*(h)^2*((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y2011(k)-a2)); P(k)=(1/3)*(k1(k)+k2(k)+k3(k)); Q(k)=(1/3)*(k1(k)+2*k2(k)+2*k3(k)+k4(k)); t(k)=t0+h; y2(k)=y20+h*dy20+P(k) yy2=(y2(k)/pi)*180; dy2(k)=dy20+(Q(k)/h); ddy2(k)=((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y2(k)-a2));

for k=2:100; ddy20(k)=((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y2(k-1)-a2)); k1(k)=(1/2)*(h)^2*((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y2(k-1)-a2)); t01(k)=t(k-1)+(h/2); y201(k)=y2(k-1)+(h/2)*dy2(k-1)+(k1(k)/4); k2(k)=(1/2)*(h)^2*((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y201(k)-a2)); k3(k)=(1/2)*(h)^2*((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y201(k)-a2)); t011(k)=t(k-1)+h; y2011(k)=y2(k-1)+(h*dy2(k-1))+k3(k); k4(k)=(1/2)*(h)^2*((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y2011(k)-a2)); P(k)=(1/3)*(k1(k)+k2(k)+k3(k)); Q(k)=(1/3)*(k1(k)+2*k2(k)+2*k3(k)+k4(k)); t(k)=t(k-1)+h y2(k)=y2(k-1)+h*dy2(k-1)+P(k) yy2=(y2(k)/pi)*180 dy2(k)=dy2(k-1)+(Q(k)/h); ddy2(k)=((pi*f)/H2)*(Pm2-Pc2-Pmax2*sin(y2(k)-a2)); end figure(1) plot(t,y2,'r') grid on

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APPENDIX 11

Plotting Program 2

f=50; H2=9.920; a2=0; Pmax2=7.2298; P2=0.4618; % y2''=((pi*f)/H2)*(P2-Pmax2*sin(y2-a2) t0=0.060; y20=0.3075; dy20=0.1; h=0.005; k=1; ddy20(k)=((pi*f)/H2)*(P2-Pmax2*sin(y20-a2)); k1(k)=(1/2)*(h)^2*((pi*f)/H2)*(P2-Pmax2*sin(y20-a2)); t01(k)=t0+(h/2); y201(k)=y20+(h/2)*dy20+(k1(k)/4); k2(k)=(1/2)*(h)^2*((pi*f)/H2)*(P2-Pmax2*sin(y201(k)-a2)); k3(k)=(1/2)*(h)^2*((pi*f)/H2)*(P2-Pmax2*sin(y201(k)-a2)); t011(k)=t0+h; y2011(k)=y20+(h*dy20)+k3(k); k4(k)=(1/2)*(h)^2*((pi*f)/H2)*(P2-Pmax2*sin(y2011(k)-a2)); P(k)=(1/3)*(k1(k)+k2(k)+k3(k)); Q(k)=(1/3)*(k1(k)+2*k2(k)+2*k3(k)+k4(k)); t(k)=t0+h y2(k)=y20+h*dy20+P(k) yy2=(y2(k)/pi)*180; dy2(k)=dy20+(Q(k)/h); ddy2(k)=((pi*f)/H2)*(P2-Pmax2*sin(y2(k)-a2));

for k=2:100; ddy20(k)=((pi*f)/H2)*(P2-Pmax2*sin(y2(k-1)-a2)); k1(k)=(1/2)*(h)^2*((pi*f)/H2)*(P2-Pmax2*sin(y2(k-1)-a2)); t01(k)=t(k-1)+(h/2); y201(k)=y2(k-1)+(h/2)*dy2(k-1)+(k1(k)/4); k2(k)=(1/2)*(h)^2*((pi*f)/H2)*(P2-Pmax2*sin(y201(k)-a2)); k3(k)=(1/2)*(h)^2*((pi*f)/H2)*(P2-Pmax2*sin(y201(k)-a2)); t011(k)=t(k-1)+h; y2011(k)=y2(k-1)+(h*dy2(k-1))+k3(k); k4(k)=(1/2)*(h)^2*((pi*f)/H2)*(P2-Pmax2*sin(y2011(k)-a2)); P(k)=(1/3)*(k1(k)+k2(k)+k3(k)); Q(k)=(1/3)*(k1(k)+2*k2(k)+2*k3(k)+k4(k)); t(k)=t(k-1)+h y2(k)=y2(k-1)+h*dy2(k-1)+P(k) yy2=(y2(k)/pi)*180; dy2(k)=dy2(k-1)+(Q(k)/h); ddy2(k)=((pi*f)/H2)*(P2-Pmax2*sin(y2(k)-a2)); end figure(1) plot(t,y2,'r') grid on

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The simulations were repeated for fault clearing times of

0.030s, 0.035s, 0.040s, 0.045s, 0.050s, 0.055s & 0.065s. It was

observed that at 0.050s, faulted machine six (6) was removed from

the system. At the fault clearing time of 0.060s, faulted machine 11

was removed from the rest of the other generators in the system.

Simulation was continued, system generators continue to swing

in synchronism after machines 6 and 11 were removed from the

system.

The critical clearing time of the system is 0.060s and the

corresponding clearing angle is 2.5 radii or 143.20 in degree. This

fault clearing time is used to predict the entire Nigerian 330KV grid

network since the Egbin – Ikeja West 330KV line is the one of the

heaviest feeder in the national grid.

The PHCN protection Engineers need to set their relay and

circuit breaker operating time at 0.060s or less such that the entire

network will be transiently stable after the occurrence of a fault

thereby reducing the probability of total system collapse to the barest

minimum.

According to the annual technical report released by the

National Control Center Oshogbo, PHCN on 5th

March, 2006,

73.33% of all the system disturbances were due to the transmission

faults, mainly from Egbin – Ikeja West 330KV line.

Analysis of the events leading to the system disturbances

generally revealed inadequacy of protection schemes in the national

grid network. Some of the protection problems on the grid are

194

traceable to the lack of proper coordination of distance relays, failure

and spurious relay operations [39, 40, 41, 42, 43, 44]. This work

serves as a guide to the PHCN Protection Engineers to set their

relays and circuit breakers operating time to act at 0.060s (60

millisecond) or less in the event of a three phase fault in the grid, so

that the system regains its synchronising power fast such that the

network is transiently stable and integrity of the national grid network

maintained. Also, customer’s inconvinences are reduced to the barest

minimum.

195

Chapter Five

4.2 CONCLUSION AND RECOMMENDATIONS

In this work, a suitable critical clearing time and angle has been

achieved or obtained for Nigerian Power System so that our

system will be able to survive any severe disturbance.

PHCN protection Engineers should endeavour to set their relay

and circuit breaker operating time to clear fault at 60 milliseconds

and it's corresponding angle of 143.20 or less than this time and angle.

If this is done the incidence of power system collapse will be

minimized to the barest minimum.

The persistent system collapse in the National grid is traceable

to the inappropriate relay and circuit breakers operating time which is

higher than 120 milliseconds [41],[42]. If the system is allowed to

collapse, it takes time for the system to be revived and this leads to the

customer’s frustrations and loss in revenue to PHCN.

RECOMMENDATION: It is very important to point out here that

increasing the number of parallel lines between two points is a common

means of reducing reactance. When a parallel transmission lines are used

instead of a single line, some power is transferred over the remaining line

even during a three – phase fault on one of the lines unless the fault occurs at

a paralleling bus. Thus, the more power is transferred into the system during

a fault, the lower the acceleration of the machine rotor and the greater the

196

degree of stability, hence, a gain in critical clearing time can be achieved.

Benin – Onitsha – Alaoji 330KV line is limited by a single line contingency.

There is an urgent need to construct the second Benin – Onitsha 330KV

circuit which had been under plan for some years past [40], [41], [42]. Also,

construction of the proposed Alaoji – New Heaven – Markurdi – Jos 330KV

circuit configuration should be expedited.

The results obtained from this work are used to predict the transient

stability of the entire Nigerian power system since Egbin – Ikeja West is

one of the heavily loaded 330KV line in the National grid.

197

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