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UNIVERSITY OF NAIROBI
FACULTY OF ENGINEERING
DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINERRING
PROJECT TITLE:
POWER STABILITY WITH RENEWABLE ENERGY
PROJECT INDEX: PRJ (051)
SUBMITTED BY
MC’LIGEYO DIANA ADHIAMBO
F17/37449/2011
SUPERVISOR: PROFESSOR O.N ABUNGU
EXAMINER: MR. PETER MOSES MUSAU
PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT
FOR THE AWARD OF THE DEGREE OF:
BACHELOR OF SCIENCE IN ELECTRICAL AND INFORMATION ENGINEERING OF
THE UNIVERSITY OF NAIROBI
DATE OF SUBMISSION: 16TH May, 2016
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DECLARATION OF ORIGINALITY
1. I understand what plagiarism is and I am aware of the university policy in this regard.
2. I declare that this final year project report is my original work and has not been submitted
elsewhere for examination, award of a degree or publication. Where other people’s work or
my own work has been used, this has properly been acknowledged and referenced in
accordance with the University of Nairobi’s requirements.
3. I have not sought or used the services of any professional agencies to produce this work
4. I have not allowed, and shall not allow anyone to copy my work with the intention of passing
it off as his/her own work.
5. I understand that any false claim in respect of this work shall result in disciplinary action, in
accordance with University anti-plagiarism policy.
Signature:
……………………………………………………………………………………
Date:
…………………………………………………………………………
NAME OF STUDENT: MC’LIGEYO DIANA ADHIAMBO
REGISTRATION NUMBER: F17/37449/2011
COLLEGE: ARCHITERCTURE AND ENGINEERING
FACULTY: ENGINEERING
DEPARTMENT: ELECTRICAL AND INFORMATION ENGINEERING
COURSE NAME: BACHELOR IN SCIENCE IN ELECRICAL AND
ELECTRONIC ENGINEERING
TITLE OF WORK: POWER STABILITY WITH RENEWABLE ENERGY
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CERTIFICATION
This report has been submitted to the Department of Electrical and Information Engineering in
The University of Nairobi with my approval as supervisor.
…………..………………………………
Prof. Nicodemus Abungu Odero
Date:………………………
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DEDICATION
To my family, for their zealous motivation and encouragement throughout my academic journey.
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ACKNOWLEDGEMENTS
First I would like to thank God, for His unending grace throughout my academic life. Without it I
would not have made it this far.
I extend my utmost appreciation to my advisor Mr. Peter Moses Musau for his valuable insights
into my project, criticism and encouragement. I would also like to thank my supervisor, Prof.
Nicodemus Abungu for his priceless motivation and support. I would also like to acknowledge
Mr. Oscar Ondeng for his help in the technical aspects of my project.
I appreciate all my lectures and all non-teaching staff at the Department of Electrical and Electronic
Engineering, University of Nairobi for their contribution towards my degree.
I am indebted to my friends Noreen Moraa, Edwin Oloo, Dennis Njau and Samuel Simiyu
Khaemba for their help in understanding the project.
I am also thankful to my classmates for their moral support as I did the project.
Finally, I would like to thank my family and especially my parents for their support and unmatched
encouragement throughout my academic life.
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ABSTRACT
Renewable Energy such as Photovoltaic (PV) and wind energy power generation play an important
role in energy production. However the output power of PV and wind power generation are
fluctuant due to the uncertainty and intermittence of solar and wind energy. This affects the power
stability of a power system. Hence there is a need to improve power stability when a distributed
generation is incorporated into a power system. This is achieved by carrying out a load flow study
of the power system where a distributed slack bus and STATCOM are implemented. Renewable
Energy is introduced by using the distributed slack bus which is based on participation factors.
The STATCOM is a FACT device used to improve power stability of a power system. In this
project real participation factors are used to distribute system losses among generators while
STATCOM is used provide reactive power compensation. One of the causes of power instability
is lack of sufficient reactive power which leads to voltage instability hence STATCOM provides
reactive power compensation which in turn improves the voltage stability. An algorithm is
developed and implemented using a Newton-Raphson Solver on a MATLAB platform. The IEEE
30 Bus is used as a case study.
When the STATCOM was implemented during the normal operating conditions, it resulted in
overall increase in voltage magnitude in comparison to when it is not included. Real and reactive
power line losses are also reduced in the buses. A disturbance in terms of load change was
implemented leading to decrease in voltage but when a STATCOM was inserted in the system the
voltage magnitudes improved.
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Table of Contents
DECLARATION OF ORIGINALITY ....................................................................................... ii
CERTIFICATION ....................................................................................................................... iii
DEDICATION.............................................................................................................................. iv
ACKNOWLEDGEMENTS ......................................................................................................... v
ABSTRACT .................................................................................................................................. vi
LIST OF FIGURES ...................................................................................................................... x
LIST OF TABLES ......................................................................................................................... xi
LIST OF ABBREVIATIONS ....................................................................................................... xii
CHAPTER 1 ................................................................................................................................... 2
POWER SYSTEM STABILITY WITH RENEWABLE ENERY ............................................. 2
1.1 Introduction ...................................................................................................................... 2
1.1.1. Power System ........................................................................................................ 2
1.1.2. Stability ................................................................................................................. 2
1.1.3. Power System Stability ......................................................................................... 2
1.1.4. Renewable Energy................................................................................................. 2
1.1.5. What is Power Stability with Renewable Energy?................................................ 3
1.2 Problem Statement ........................................................................................................ 3
1.2.1 Project Objectives ................................................................................................. 4
1.2.2 Project Scope ......................................................................................................... 4
1.2.3 Project Questions................................................................................................... 4
1.3 Organization of the Report ........................................................................................... 4
CHAPTER 2 ................................................................................................................................... 6
LITERATURE REVIEW ............................................................................................................ 6
2.1 INTRODUCTION ON POWER STABILITY............................................................. 6
2.1.1 Rotor-Angle stability ............................................................................................. 7
viii
2.1.2 Frequency Stability ............................................................................................. 10
2.1.3 Voltage Stability .................................................................................................. 11
2.2 INTRODUCTION TO FACTS .................................................................................. 12
Types of FACTS [9] .......................................................................................................... 13
2.2.1 Series Controller .................................................................................................. 13
2.2.2 Shunt Controllers................................................................................................. 14
2.2.3 Combined Series-Series Controller ..................................................................... 16
2.2.4 Combined Series-Shunt Controller ..................................................................... 17
2.3 Effects of Renewable Energy on Power Stability ...................................................... 17
CHAPTER THREE ...................................................................................................................... 19
SOLUTION TO POWER STABILITY WITH RENEWABLE ENERGY.............................. 19
3.1 Methods of Solving Power Stability with Renewable Energy Problem ..................... 19
3.2 Load Flow ................................................................................................................... 20
3.2.1 Bus Classifications .............................................................................................. 20
3.2.2 The Load Flow Problem ...................................................................................... 21
3.2.3 Newton Raphson Method .................................................................................... 22
3.2.4 Gauss Seidel Method ........................................................................................... 26
3.2.5 The Fast De-coupled Method .............................................................................. 28
3.3 Power Stability with Renewable Energy Problem Formulation ................................. 30
3.4 Solution of Power Stability with RE using Newton-Raphson Method ...................... 32
3.5 Formulation of The Real Participation factors ........................................................... 33
3.6 Solution Algorithm for Solving Stability with Renewable Energy ............................ 35
3.7 Flow Chart .................................................................................................................. 37
CHAPTER FOUR ......................................................................................................................... 38
RESULTS AND ANALYSIS ................................................................................................... 38
4.1 Case study: IEEE 30-Bus System ............................................................................... 38
4.2 Results and Validation ................................................................................................ 39
4.3 Analysis and Discussion ............................................................................................. 46
CHAPTER FIVE .......................................................................................................................... 51
CONCLUSION AND RECOMMENDATON FOR FURTHER WORK ................................ 51
5.1 Conclusion .................................................................................................................. 51
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5.2 Recommendation ........................................................................................................ 51
REFERENCES ............................................................................................................................. 52
APPENDIX 1: STATCOM Data .............................................................................................. 55
APPENDIX 2: Program for formation of Bus Admittance Matrix ........................................... 56
APPENDIX 3: Newton Raphson Solver ................................................................................... 57
APPENDIX 4: Newton Raphson Solver with STATCOM ....................................................... 61
APPENDIX 5: Program for Calculating Power Outputs, Line Flows and Losses ................... 69
x
LIST OF FIGURES
Figure 1: Classification of Power Stability ..................................................................................... 6
Figure 2: Power Transfer Characteristic ......................................................................................... 8
Figure 3: STATCOM Representation ........................................................................................... 32
Figure 4: Flowchart for Power Stability with Renewable Energy ................................................ 37
Figure 5: IEEE 30 Bus Test Network ........................................................................................... 38
Figure 6: Voltage Profile in Normal Operating Conditions.......................................................... 48
Figure 7: Comparison of Losses under Normal Operating Conditions ........................................ 48
Figure 8: Voltage Profile after Disturbance .................................................................................. 49
Figure 9: Comparison of Total Losses after Disturbance ............................................................. 49
xi
LIST OF TABLES
Table 3.1: Comparison of load Flow Methods ............................................................................. 29
Table 4.2: Line Data ..................................................................................................................... 40
Table 4.3: Comparison between voltages with and without Renewable Energy .......................... 42
Table 4.4: Comparison of Voltages with renewable with and without STATCOM in normal
operating Conditions. .................................................................................................................... 44
Table 4.5: Comparison of voltages with renewable energy without STATCOM and with
renewable energy and STATCOM after disturbance (50% load change). ................................... 46
Table 4.6: Comparison of Total Real and Reactive Power Injected and Generated Without
Renewable Energy, with Renewable Energy and with Renewable Energy and STATCOM. ...... 46
Table 4.7: Comparison of Total Real and Reactive Losses without Renewable Energy, with
Renewable Energy and with Renewable Energy and STATCOM. .............................................. 47
xii
LIST OF ABBREVIATIONS
IEEE - Institute of Electrical and Electronic Engineers
RE - Renewable Energy
NR - Newton Raphson
DG - Distributed Generator
FACT - Flexible Alternating Current Transmission System
STATCOM - Static Synchronous Compensator
SVC - Static VAR Compensator
PV - Photovoltaic
2
CHAPTER 1
POWER SYSTEM STABILITY WITH RENEWABLE ENERY
1.1 Introduction
1.1.1. Power System
A network of components used to generate, transmit and distribute electric power is called a power
system. This is a network consisting of more than one electrical generation units, power
transmission lines and loads including the associated equipment’s connected electrically and
mechanically to the network.
1.1.2. Stability
This is the resistance to change whereby there is equilibrium between opposing forces and a state
of instability will occur when a disturbance leads to a sustained imbalance between opposing
forces [1].
1.1.3. Power System Stability
Power System Stability is defined as that ability of an electric power system to remain in a state
of operating equilibrium under normal operating conditions and after being subjected to a
disturbance to regain an acceptable state of equilibrium [1, 12].
1.1.4. Renewable Energy
Renewable Energy is energy that is obtained from natural resources and thus can be constantly
replenished such as sunlight, wind, rain, tides and geothermal heat. The renewable energy sources
exist over a wide geographical area in contrast to other energy sources which are limited.
Renewable Energy includes bio-mass energy, wind energy, solar energy, hydro-power, geo-
thermal energy just to mention a few. This study is interested in two types of renewable energy
which include solar energy and wind energy.
Solar Energy
Solar energy makes the production of solar electricity possible due to the energy in the form of
solar radiation. Thus solar energy is simply provide by the sun. Electricity can be produced from
photovoltaic cells directly. These cells exhibit photovoltaic due to the materials they are made up
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of. Light excite the electrons in the cell and cause them to flow when sunshine hits the Photovoltaic
cell and hence generating electricity.
Wind Energy
Wind energy is a form of solar energy. The wind energy is harvested by modern wind turbines.
The wind-turbine is used to convert the Kinetic energy in the wind to mechanical power. A
generator is used to convert the mechanical power into electrical power.
1.1.5. What is Power Stability with Renewable Energy?
Stability is a condition of equilibrium between opposing forces. Power stability is generally
divided into three parts [1]. They include:
Voltage Stability
Frequency Stability
Rotor-angle Stability
Frequency stability is the ability of a power system to restore balance between power system
generation and load after a disturbance has occurred. Voltage stability is the ability of the power
system to maintain steady accepted voltages at all buses of the system under normal operation and
after being subjected to a disturbance. While rotor-angle stability is the ability of interconnected
machines of a power system to remain at synchronism after a disturbance [12, 15].
When renewable energy is introduced to a power system it brings a lot of advantages such as, it is
in vast amount and it reduces pollution but it also brings about its own set of challenges in terms
of power stability. The challenges of wind energy and solar energy are brought about due to their
unpredictable nature as well the characteristics of the wind generators and photovoltaic cells.
1.2 Problem Statement
We are endowed with an enormous wind energy source and solar energy resources across the
globe. However the integration of the renewable energy is bound to impact the generation,
transmission and distribution of the power system. Due to the limited predictability of the weather
conditions and the resources of the renewable energy and the characteristics of the wind generators
and photovoltaic cell, the outputs cannot be controlled. Hence there is a need to counter-effect this
consequence when wind energy and solar energy are implemented in a power system.
4
1.2.1 Project Objectives
The objective of the project is to investigate the integration of renewable energy to a power system
and find the effects it has on power stability and explore measures that will deal with the challenges
it brings about.
1.2.2 Project Scope
The project will be limited to the impact of wind energy and solar energy on voltage stability in
normal operating conditions and after a disturbance which was implemented by a 50% load
increase and the method used to mitigate the negative effects.
1.2.3 Project Questions
The project will attempt to answer the following questions:
1) Does renewable energy bring about voltage stability or instability to a power system?
2) If power system instabilities are brought about how can we reduce their effects?
3) If it brings about voltage stability how can we improve the power system further?
1.3 Organization of the Report
The report has been organized into five chapters:
In Chapter 1. An introduction to Power Stability. The problem statement and project objectives
have also been discussed.
In Chapter 2. A literature review on power stability, renewable energy and FACTS Devices have
been discussed independently.
In Chapter 3. Load flow studies. Data to be utilized in the formulation of power stability with
renewable energy is introduced. The Data is obtained from the IEEE 30 bus test network. The
design of power stability with renewable is discussed in detail and a flow chart as well as an
algorithm to be used in the project is developed.
In Chapter 4. The simulated results obtained from the MATLAB simulation in chapter 3 are
analyzed and discussed.
Chapter 5. Concludes this report by giving a review of the study in the preceding chapters and
examining the extent of the achievement of the objectives of the project.
5
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CHAPTER 2
LITERATURE REVIEW
2.1 INTRODUCTION ON POWER STABILITY
Power System Stability is the ability of an electric power system to remain in a state of operating
equilibrium under normal operating conditions and after being subjected to a disturbance to regain
an acceptable state of equilibrium [1, 12]. Power stability is sub-divided into three:
Rotor-Angle Stability
Frequency Stability
Voltage Stability
As power systems are non-linear, their stability depends on both the initial conditions and the size
of a disturbance. Thus rotor-angle and voltage stability can be divided into small and large
disturbance stability.
Figure 1: Classification of Power Stability
Physical quantities such as phase angle and magnitude of the voltage at each bus, the active and
reactive power describe a power- system operating conditions. The system is in steady state if they
are constant in time. When the steady state condition is subjected to a change in the system
quantities, a power system undergoes a disturbance. The disturbance can be a small disturbance or
a large disturbance depending on the origin and the magnitude. A power system is said to be
steady-state stable if after being subjected to a small disturbance, it is able to return to essentially
the same steady state condition. While if a system reaches a new acceptable state different from
7
the original steady-state condition especially under a large disturbance, the system is said to
transiently stable [1, 12, 16].
2.1.1 Rotor-Angle stability
This is the ability of inter-connected synchronous machines of a power system to remain in
synchronism. Instability in the form of increasing angular swings of some generators may result
in loss in synchronism. Rotor angle stability is dependent on the ability to maintain or restore
equilibrium between electromagnetic torque and mechanical torque in each of the machines in the
system [1].
The relationship between interchange power and angular position of the rotors of synchronous
machines is an important characteristic having a bearing on power system stability. It is highly
non-linear. Let us assume machine 1 represents a generator feeding power to a synchronous motor
represented by machine 2. The power transferred from the generator to the motor is given by: [1]
𝑃 =𝐸𝐺𝐸𝑀𝑠𝑖𝑛(δ)
𝑋𝑇
Where EG = emf of the generator
EM = emf of the motor
XT = the total reactance
δ = the angular separation between the rotors of the two machines
As the angle is increased, the power transfer increases up to a certain limit but upon further increase
in the angle, usually 900, it results in a power decrease in the power transferred [16]. The power
angle curve is as shown below [15].
8
Figure 2: Power Transfer Characteristic
The equilibrium of between opposing forces is called stability. The restoring forces is the
mechanism by which interconnected synchronous machines maintain synchronism. The restoring
forces act whenever there are forces acting on one or more machines with respect to other machines
tending to accelerate to decelerate them [1, 16].
Electromechanical oscillations inherent in power systems is the study of the rotor angle stability
problem. Under steady-state, the speed remains constant due to equilibrium between the input
mechanical torque and output electromagnetic torque of each generator. If the equilibrium is upset,
it results in acceleration or deceleration of the rotors of the machines. The angular position relative
to the slower machine, if one machine temporarily runs faster than another, will advance. The
resulting angular difference transfers part of the load from the slow machine to the fast machine
depending on the power-angle relationship. This tends to reduce the speed difference and hence
angular separation [1].
An increase in angular separation beyond a certain limit is accompanied by a decrease in power
transfer. This continued increase in angular separation leads to instability. Hence the stability of a
system depends on whether there is sufficient restoring torques due to deviations in angular
positions of the rotors.
The change in electromagnetic torque can be resolved in two components following a perturbation
in synchronous machines. They include: [1, 16, 18]
a) Synchronizing torque component in phase with rotor angle deviation
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b) Damping torque component in phase with the speed deviation
System stability depends on the existence on both components. Lack of sufficient damping torque
results in oscillatory instability whereas lack of sufficient synchronizing torque results in aperiodic
or non-oscillatory instability. Rotor-angle stability can be sub-divided into:
Transient stability
Small Signal (or small disturbance) Stability
Transient stability
It is concerned with the effect of large disturbances. It is the ability to maintain synchronism of a
power system when subjected to a sever disturbance such as three phase short circuit, single-line
to earth faults, short circuit in transmission lines etc. [18, 19].
Transient stability depends on both the severity of the disturbance and the initial state of the
operating system. The instability in most cases manifests itself as the first swing instability in the
form of aperiodic angular separation due to insufficient synchronizing torque. Methods of analysis
include [15]:
Swing equation
Equal Area Criterion
Numerical Integration methods such as Euler method, Runge-Kutta etc.
Lypanov Analysis
Small signal (or small disturbance) Stability
This is the ability of a power system to maintain synchronism under small disturbances. This
disturbances occur continually on the system because of small variations in loads and generation
[18]. The rotor-angle small signal instability may occur in two forms:
Rotor oscillations of increasing amplitude due to lack of sufficient damping torque
Steady increase in rotor angle due to lack of sufficient synchronizing torque
Small signal stability is largely a problem of insufficient damping oscillations. The following types
are of concern [1, 16]:
I. Control modes. They are associated with generating units and other controls
10
II. Torsional modes. They are associated with the turbine governor shaft system rotation
III. Inter-area modes. They are associated with the swinging of many machine in one part of
the system against machines in other parts.
IV. Local modes. They are associated with the swinging units at a generating station with
respect of the power system.
2.1.2 Frequency Stability
Frequency stability is the ability of a power system to balance generation and load following a
severe system upset hence maintaining a steady frequency. It indicates whether there is equilibrium
or no equilibrium between power generation and consumption [17]. With regard to frequency
stability considerations, there must be sufficient power available to cover transient power
fluctuation to maintain the system frequency within required ranges. Instability that may occur
results in the form of sustained frequency swings leading to tripping of generator units or loads
[21].
Frequency stability issues may occur in different time frames ranging from a few seconds (inertia
problems) to several minutes or even hours (load following reserve). If a load is suddenly
connected to the system or if a generator is suddenly disconnected there will be a long term
distortion in power balance between that delivered by the turbines and that consumed by the loads.
The imbalance is initially covered from kinetic energy of rotating turbines, generators and motors
and as a result the frequency in the system will change [16, 17].
The stored kinetic energy of all generators (and spinning loads) on the system is given by [13]:
𝑲.𝑬 =𝟏𝑰𝒘𝟐
𝟐
Where I = moment of inertia of all generators (kgm2)
W= rotational speed of all generators (rad/s)
Any imbalance between the produced and consumed power may lead to frequency change. When
the generated power is less than the load power, the electrical torque of generator is less than the
mechanical input torque. The speed of the generator will slow down decreasing the frequency.
Otherwise when the generated power is greater than the load power, the speed of the generator will
increase, increasing the frequency [17, 20].
11
The torque balance of any spinning masses determines the rotational speed [13];
𝑇𝑚 − 𝑇𝑒 = 𝐼𝑑𝑤
𝑑𝑡
In power system it is conventional to express the inertia in per unit as H constant
𝑯 =𝟏𝟐𝑰𝒘𝒔
𝟐
𝑺𝒓𝒂𝒕𝒆𝒅 [𝑾𝒔 𝑽𝑨⁄ ]
Where Srated is the MVA rating of either and individual generator or the whole system
Ws is the angular velocity (rads/sec) at synchronous speed
A generator’s rotating mass provides kinetic energy to the grid or absorbs it due to the
electromechanical coupling in case of a frequency deviation. The grid frequency is directly
coupled to the rotational speed of a synchronous generator and this to the active power balance.
The inertia constant H that is the rotational inertia minimizes the rate change of frequency in case
of frequency deviations. This renders frequency dynamics slower and thus increases the available
response time to react to faults.
For stable operations of power system it is a necessary requirement to maintain grid frequency
within an acceptable range. Low levels of rotational inertia in power system caused in particular
by shares of inverters connected renewable energy sources that is wind-turbine and PV units have
implications on frequency dynamics because they do not normally provide any rotational inertia.
Hence leading to frequency control scheme being to slow in respect with to the disturbance
dynamic for preventing large frequency deviations. Hence leading to the frequency instability
phenomena [17].
2.1.3 Voltage Stability
Voltage stability is the ability of a power system to maintain steady acceptable voltages at all buses
in the system under normal operating condition and after being subjected to a disturbance. When
there is a progressive and uncontrollable drop of voltage due to a disturbance, increase in load or
change in system condition, a system enters s state of voltage instability [1, 16, 22]. The inability
of the power system to meet demand for reactive power is the main factor causing voltage stability
12
[5]. Instability that may result occurs in the form of a progressive fall or rise of voltages of some
buses. Due to voltage instability the following may occur [16]:
Tripping of transmission lines and other elements by their protective systems leading to
cascading outages
Loss of load in an area
If voltages after disturbance are close to voltages at normal operating condition we say a power
system is voltage stable. At a given operating condition for every bus in the system, the bus voltage
magnitude increases as the reactive power injection at the same bus is increased, this is the criteria
for voltage stability. A system is voltage unstable, if the bus voltage magnitude (V) decreases as
the reactive power injection (Q) at the same bus is increased, for at least one bus in the system [1].
The term voltage collapse is the process by which the sequence of events accompanying voltage
instability leads to a blackout or abnormally low voltages in a significant part of a power system
leading to a low voltage profile [5].
We can classify voltage stability into;
a. Large-disturbance voltage stability
It is concerned with a system’s ability to maintain steady voltages following large disturbances
such as faults, loss of generators. This ability is determined by the system load characteristics and
interactions of both discrete and continuous controls and protections. The study period may last
from a few seconds to tens of minutes.
b. Small-disturbance voltage stability
It is concerned with a system’s ability to maintain steady voltages following small disturbances
such as changes in load demands.
2.2 INTRODUCTION TO FACTS
Flexible Alternating Current Transmission System (FACTS) is a system incorporating power
electronic based and other static controllers for the AC transmission of the electrical energy. They
are used to enhance controllability and increase power transfer capability [25, 26]. The advantages
of FACTS include [25]:
13
a. Control power flow and voltage as desired.
b. It decreases overall generation reserve requirements.
c. Increased system security and reliability.
d. Overall enhancement of the quality of electric energy delivered to customers.
Types of FACTS [9]
1. Series Controllers
2. Shunt Controllers
3. Combined series-series Controllers
4. Combined Series-Shunt Controllers
2.2.1 Series Controller
The series compensator could be a variable impedance such as a capacitor or variable source. All
series controllers inject voltage in series with the line [25]. The series compensators include:
Static Synchronous Series Compensator
Thyristor Controlled Series Capacitor
Thyristor Switched Series Capacitor
Thyristor Controlled Series Reactor
Thyristor Switched Series Reactor
Static Synchronous Series Compensator (SSSC)
It is implemented using GTO based voltage source inverter that can provide controllable
compensating voltage over an identical capacitive or inductive range independent of the line
current. In principle, the inserted series voltage can be regulated to change the impedance (more
precisely reactance) of the transmission line. It is capable of handling power flow control,
improvement of transient stability margin and improve damping of transient [11, 2, 28].
Thyristor Controlled Series Compensator (TCSC)
It is a capacitive reactive compensator consisting of a series capacitor bank shunted by a thyristor
controlled reactor in order to provide a smoothly variable series capacitance, the bi-direction
thyristor valve is fired with an angle ranging between 900 and 1800 with respect to capacitor voltage
[25, 29]. It can operate in
14
I. Bypass-thyristor mode
II. Blocked-thyristor mode
III. Vernier mode
In bypass thyristor mode, thyristors are made to conduct with conduction angle of 1800. There is
a continuous flow through the thyristor valves because gate pulses are applied as soon as the
voltage across the thyristor reaches zero and becomes positive. It behaves like a parallel capacitor-
inductor combination. In blocked thyristor mode firing pulses of TCSC are blocked. The net TCSC
reactance is capacitive. The vernier mode allows TCSC to behave as a continuously controllable
capacitive reactance or continuously controllable inductive reactance [28].
Thyristor Switched Series Capacitor
It consists of a capacitor shunted by a pair of reversely parallel connected thyristor. The operation
of a TSSC is to make use of a thyristor to act as a valve or switch for the capacitor connected in
parallel to it such that when thyristor is triggered the capacitor will be activated to start
compensator [29].
Thyristor Controlled Series Reactor (TCSR)
It is an inductive reactance compensator consisting of a series reactor shunted by a thyristor
controlled reactor in order to provide a smoothly variable series inductive reactance. When the
firing angle of the thyristor controlled reactor is 1800, it stops conducting and the controlled reactor
acts as a current limiter.
As the angle decreases below 1800, the net inductance decreases until firing angle of 900, when net
inductance in the parallel combination of the two reactors.
Thyristor Switched Series Reactor
TSSR is an inductive reactance compensator which consists of a series reactor shunted by a
thyristor-controlled switched reactor in order to provide a stepwise control of series inductive
reactance [28].
2.2.2 Shunt Controllers
They maybe variable source, variable impedance or a combination. At the point of connection, all
shunt controllers inject current in the system [25]. They are
15
Static Var Compensators (SVCs)
Static Synchronous Compensator (STATCOM)
Thyristor Controlled reactor
Thyristor Switch Capacitor
Static Var Compensator
This is a shunt connected static Var absorber or generator whose output is adjusted to exchange
capacitive or inductive current so as to maintain or control specific parameters of the electrical
system [25]. It consists of fixed capacitors or reactance, Thyristor Switched Capacitors (TSC) for
lagging VAR and Thyristor Controlled Reactors (TCR) for leading VAR are connected in parallel
with electrical system [8]. The basic idea of the TSC is to split up capacitor bank into sufficiently
small capacitor steps and switch the steps on and off, using anti-parallel connected Thyristors as
switching elements. It is based on thyristors with GTO capability. In a power system the load
varies from time to time. This may change reactive power balance. The SVC may be installed at
various points in the system to maintain the voltage at accepted levels by providing sufficient
reactive power to the system thus maintain reactive power balance and further reducing losses [25,
28].
Static Synchronous Compensator (STATCOM)
STATCOM is a self-commutated switching power converter supplied from an appropriate electric
energy source and operated to produce a set of adjustable multiphase voltage which maybe coupled
to an AC power system for the purpose for exchanging independently controllable real and reactive
power. It consist of thyristors with gate turn-off capability (GTO) or many IGBT’s. It is a solid-
state based power converter of the SVC. They do not require large inductive and capacitive
components to provide inductive or capacitive power to high voltage transmission networks as
required by SVC [11, 28].
A STATCOM comprises of three main parts;
Voltage Source Converter
Step-up Coupling transformer
A controller
16
From a DC input Voltage source, provided by the charged capacitor, the converter produces a set
of controllable 3 phase output voltages with frequency of the AC system. Via a relatively small tie
reactance, each output voltage of the converter is in phase with the corresponding AC system
voltage. By varying the output voltage produced, the reactive power exchange between the
convertor and the AC system can be controlled. For the voltage source converter, its AC output
voltage is controlled so that it is just right for the required reactive current flow for any AC bus
voltage. The DC capacitor voltage is automatically adjusted as required to serve as a voltage source
for the converter [25].
If the amplitude of the output voltage is increased above the AC system voltage, then the current
flows through the reactance from the converter to the AC system and the converter generates
reactive power. If the amplitude of the output voltage is decreased to that of the AC system voltage,
then the reactive flows from the AC system to the converter and the convertor absorbs reactive
power. If the amplitude of the output voltage is equal to that of the AC system voltage, the reactive
power exchange is zero.
Thyristor Controlled Reactor (TCR)
This is a shunt connected thyristor controlled inductor whose effective reactance is varied in a
continuous manner by partial control of the thyristor valve. TCR is a subset of SVC in which
conduction time and hence shunt in a reactor is controlled by a thyristor based ac switch with firing
angle control. It consists of a fixed reactor of inductance and a bidirectional thyristor valve [8, 28].
Thyristor Switch Capacitor
It is a shunt connected capacitor whose effective reactance is varied in a stepwise manner by full
or zero conduction operation of the thyristor valve [28].
2.2.3 Combined Series-Series Controller
This may be a combination of separate series controllers, which are controlled in a coordinated
manner, in a multi-line transmission system. The IPFC (Interline Power Flow Controller) is the
combination of two or more Static Synchronous Series Compensators which are coupled via a
common dc link to facilitate bi-directional flow of real power between the ac terminals of the
SSSCs [28].
17
2.2.4 Combined Series-Shunt Controller
UPFC is combination of shunt connected device (STATCOM) and a series branch (SSSC) in the
transmission line via its DC link [8].
2.3 Effects of Renewable Energy on Power Stability
Due to the unpredictable nature of the renewable energy that is solar energy and wind energy there
may be a mismatch between the generation of power and the demand of power. This causes
deviations in the system frequency. In the case of a power deficit, the generation is less than the
power demand leading to a reduction of speed and hence the system frequency goes down. While
if the generation of power is more than the demand, it will cause an increase in speed and hence
an increase in the system frequency thus leading to frequency instability.
The inertia dictates how large the frequency deviations would be due to a sudden change in the
generation and load power balance which plays a significant role in maintaining the stability of a
power system during a transient scenario. The larger the inertia of a system, the smaller the rate of
change in rotor speed in the generator during a power imbalance. In wind energy power generation,
when fixed speed induction generators are used it contributes to the inertia of a power system
because the stator is directly connected to the grid and thus changes in frequency manifests as a
change in speed. These speeds are resisted by the rotating masses leading to rotating energy
transfer. While in variable speed wind turbines, its rotational speed is decoupled from the grid
frequency by electronic converter. Thus variation in grid frequency does not alter the turbine
output power. With high wind penetration there is a risk that the power system inertial effect
decreases thus aggravating the frequency of the grid. In solar power generation, the solar power
plant consists of the solar cell and DC to AC converter. Hence they do not possess inertia hence
won’t release energy to grid when frequency. This can thus can cause frequency instability.
In traditional power systems, the rotor angles of synchronous generators are impacted by the
changes in active power flow in the system. When there is a change in active power, the
synchronizing generators will respond with an electromagnetic torque that will dampen and
minimize the rotor angle deviations thus they have synchronizing power. Renewable Energy
technology such as wind turbines generator and photovoltaic (PV) are asynchronous machines.
They are integrated to the grid via inverters which makes them lose the ability to maintain
18
synchronism hence they lack synchronizing power and hence this may bring about rotor angle
instability.
Voltage instability is brought about by the inability of the power system to meet the demand of
reactive power. In the case of wind energy, if the wind turbine technology utilizes the induction
machines it may lead to a power system’s inability to meet the demand of reactive power this is
because induction machine consume reactive power thus leading to voltage instability. While in
solar energy, photovoltaic use inverters which are designed to operate at unity power factor hence
reactive power is neither produced nor absorbed. Hence there is a need to implement a way of
voltage control otherwise it may lead to issues with voltage stability.
19
CHAPTER THREE
SOLUTION TO POWER STABILITY WITH RENEWABLE ENERGY
3.1 Methods of Solving Power Stability with Renewable Energy Problem
Optimization is the art of achieving the best possible solution to an optimization problem where
there are a number of competing or conflicting parameters.
Categories of Optimization Methods
Due to the optimization problems there a number of techniques that are available to solve the
problems of power system operation. The techniques have been classified into three groups [14];
1. Conventional Methods
2. Intelligence Search Methods
3. Non-quality approaches to address uncertainties in objective constraints
Conventional Methods
The conventional optimization methods include:
Gradient Method
Linear Search
Lagrange Multiplier Method
Newton Raphson Method Optimization
Quasi-Newton Method
Trust-Region Optimization
Linear Programming
Non-linear Programming
Quadratic Programming
Newton’s Methods
Interior Point Method
Mixed Integer Programming
Network Flow Programming
Intelligent Search Methods
20
The Intelligent Search Programming Methods include:
Optimization Neural Network
Evolutionary Algorithms
Tabu Search
Particle Swarm Optimization
Non-quality approaches to address uncertainties in objective constraints
The Non-quality approaches to address uncertainties in objective constraints methods include:
Probabilistic Optimization
Fuzzy Set Application
Analytical Hierarchal Process (AHP)
3.2 Load Flow
3.2.1 Bus Classifications
A bus is a node at which one or many lines, one or many loads and generators are connected. In
the power system there are four potentially unknown quantities, they include [23]:
P- Real Power
Q-Reactive Power
|V|- Voltage magnitude
δ - Voltage angle
Buses are classified according to which two out of the four variables are specified. They are
1. Load bus(P-Q bus)
No generator is connected to the bus. At this bus the real and reactive power are specified. It is
desired to find out the voltage magnitude and phase angle through load flow solutions. It is required
to specify only Pd (real power demand) and Qd (reactive power demand) at such a bus as at a load
bus, voltage can be allowed to vary within the permissible values. Pg (real power generated) and
Qg (reactive power generated) are taken to be zero. The load bus is referred to as the P-Q bus since
the scheduled values are known and the real and reactive power mismatches, ΔP and ΔQ
respectively can be defined [1, 5].
21
2. Generator bus or voltage controlled bus (P-V bus)
The voltage magnitude and real power are specified. It is required to find out the reactive power
generation Q and phase angle of the bus voltage. A generator bus is usually called a voltage
controlled or PV bus. This is because a generator connected to a bus, the megawatt generation can
be controlled by adjusting prime mover, and voltage magnitude can be controlled by adjusting
generator excitation [1, 5].
3. Slack (swing) bus:
This is used as a reference bus in order to meet the power balance condition. Slack bus is usually
a generating unit that can be adjusted to take up whatever is needed to ensure power balanced. For
the Slack Bus, it is assumed that the voltage magnitude |V| and voltage phase Θ are known, whereas
real and reactive powers P and Q are obtained through the load flow solution [23].
3.2.2 The Load Flow Problem
The first step in performing load flow analysis is by using the transmission line and transformer
input data to form the Y-bus admittance. The nodal equation for a power system network using Y
bus can be written as follows [4, 5, 6]:
𝐼 = 𝑌𝑏𝑢𝑠𝑉 ………1)
The nodal equation can be written in a generalized form for an n bus system
𝐼𝑖 = ∑𝑌𝑖𝑗𝑉𝑗 𝑓𝑜𝑟 𝑖 = 1,2,3…𝑛 ………2)
𝑛
𝑗=1
The real and reactive power injected to bus i
𝑃𝑖 + 𝑗𝑄𝑖 = 𝑉𝑖𝐼𝑖∗ ……… .3)
𝐼𝑖 =𝑃𝑖 − 𝑗𝑄𝑖
𝑉𝑖∗ ……… . .4)
Substituting for Ii yields
𝑃𝑖 − 𝑗𝑄𝑖
𝑉𝑖∗ = 𝑉𝑖 ∑𝑌𝑖𝑗 − ∑ 𝑌𝑖𝑗𝑉𝑗 ………5)
𝑛
𝑗=1
𝑛
𝑗=1
22
The above equation uses iterative techniques to solve load flow problems.
The complex conjugate of the power injected at bus i formulated in polar form are
𝑃𝑖 − 𝑗𝑄𝑖 = 𝑉𝑖∗ ∑𝑌𝑖𝑗𝑉𝑗
𝑛
𝑗=1
……… .6)
Which thus becomes [14]
𝑃𝑖 − 𝑗𝑄𝑖 = ∑|𝑌𝑖𝑗𝑉𝑖𝑉𝑗| < 𝜃𝑖𝑗 + 𝛿𝑖 − 𝛿𝑗
𝑛
𝑗=1
……… .7)
Expanding this equation and equating real and reactive parts we obtain
𝑃𝑖 = ∑|𝑌𝑖𝑗𝑉𝑖𝑉𝑗|cos (𝜃𝑖𝑗 + 𝛿𝑖 − 𝛿𝑗)
𝑛
𝑗=1
………8)
𝑄𝑖 = −∑|𝑌𝑖𝑗𝑉𝑖𝑉𝑗| sin(𝜃𝑖𝑗 + 𝛿𝑖 − 𝛿𝑗)
𝑛
𝑗=1
……… .9)
Solving Load flow problems [5, 14]
The load flow problems are non-linear in nature hence cannot be explicitly solved by linear
methods thus application of Iterative methods are used. The most common methods of solving the
load flow problem include:
I. Newton Raphson Method
II. Gauss Seidel Method
III. Fast Decoupled Load Flow method
3.2.3 Newton Raphson Method
The Newton Raphson Method is an iterative method which approximates a set of non-linear
simultaneous equations into a set of linear equations using the Taylor series expansion. The
Newton Raphson (NR) method is applied in load flow studied due to its practicality in large power
systems and its efficiency. It has the advantage of the number of iterations required to obtain a
23
solution is independent of the size of the problem in load flow studies. Computationally, it is also
very fast [4, 5, 14, 23].
A non-linear equation with single variable can be expressed as
𝑓(𝑥) = 0
An initial value x is selected for solving this equation. The difference between the initial value and
the final solution will be ∆ X0. Then X0 +∆ X0 is the solution of non-linear equation above i.e.
𝑓(𝑥0 + ∆𝑥0) = 0…… . . 𝑖𝑖)
Expanding the above equation with Taylor series, we obtain
𝑓(𝑥0 + ∆𝑥0) = 𝑓(𝑥0) + 𝑓′(𝑥0)∆𝑥0 + 𝑓′′(𝑥0)(∆𝑥0)2
2!+ ⋯…… .+𝑓𝑛(𝑥0)
(∆𝑥0)𝑛
𝑛!
= 0……… 𝑖𝑖𝑖)
Where f’(x0) … f n(x0) are the derivatives of the function f(x).
If the difference ∆x0 is very small, the second and higher derivatives terms can be neglected. Thus
equation becomes linear as shown below:
𝑓(𝑥0 + ∆𝑥0) = 𝑓(𝑥0) + 𝑓′(𝑥0)∆𝑥0 = 0
f(x°) = − f ’(x°)∆x°
Hence
∆𝑥0 =𝑓(𝑥0)
𝑓′(𝑥0)
Then new solution will then become
𝑥′ = 𝑥0 + ∆𝑥0 −𝑓(𝑥0)
𝑓′(𝑥0)
Since the above equation is an approximate equation hence the solution x is not a real solution.
Further iterations are required. The iteration equation is:
𝑥𝑘+1 = 𝑥𝑘 + ∆𝑥𝑘+1 = 𝑥𝑘 −𝑓(𝑥𝑘)
𝑓′(𝑥𝑘)
24
If the conditions below are met the iterations will be stopped
|∆𝑥𝑘| < 휀1
Where 휀1 is the prescribed convergence precision.
This newton method may be expanded to a non-linear equation with n variables
𝑓1(𝑥1, 𝑥2 …………𝑥𝑛) = 0
𝑓2(𝑥1, 𝑥2 …………𝑥𝑛) = 0
⋮
𝑓𝑛(𝑥1, 𝑥2 …………𝑥𝑛) = 0
For a given set of initial values x10, x2
0……xn0, then the corrected values ∆ x1
0, ∆ x20….∆ xn
0.
Hence our equation becomes,
𝑓1(𝑥10 + ∆𝑥1
0, 𝑥20 + ∆𝑥2
0 ……𝑥𝑛0 + ∆𝑥𝑛
0) = 0
𝑓2(𝑥10 + ∆𝑥1
0, 𝑥20 + ∆𝑥2
0 ……𝑥𝑛0 + ∆𝑥𝑛
0) = 0
⋮
𝑓𝑛(𝑥10 + ∆𝑥1
0, 𝑥20 + ∆𝑥2
0 ……𝑥𝑛0 + ∆𝑥𝑛
0) = 0
Expanding this we obtain
𝑓1(𝑥10, 𝑥2
0 …𝑥𝑛0) +
𝜕𝑓1𝜕𝑥1
|𝑥1
0
∆𝑥10 +
𝜕𝑓2𝜕𝑥2
|𝑥2
0
∆𝑥20 + ⋯
𝜕𝑓𝑛𝜕𝑥𝑛
|𝑥𝑛
0
∆𝑥𝑛0+= 0
𝑓2(𝑥10, 𝑥2
0 …𝑥𝑛0) +
𝜕𝑓1𝜕𝑥1
|𝑥1
0
∆𝑥10 +
𝜕𝑓2𝜕𝑥2
|𝑥2
0
∆𝑥20 + ⋯
𝜕𝑓𝑛𝜕𝑥𝑛
|𝑥𝑛
0
∆𝑥𝑛0+= 0
⋮
𝑓𝑛(𝑥10, 𝑥2
0 …𝑥𝑛0) +
𝜕𝑓1𝜕𝑥1
|𝑥1
0
∆𝑥10 +
𝜕𝑓2𝜕𝑥2
|𝑥2
0
∆𝑥20 + ⋯
𝜕𝑓𝑛𝜕𝑥𝑛
|𝑥𝑛
0
∆𝑥𝑛0+= 0
The above equation can be written in matrix from as
25
[ 𝑓1(𝑥1
0, 𝑥20 …𝑥𝑛
0)
𝑓2(𝑥10, 𝑥2
0 …𝑥𝑛0)
⋮𝑓𝑛(𝑥1
0, 𝑥20 …𝑥𝑛
0)] = −
[ 𝜕𝑓1𝜕𝑥1
|𝑥1
0
𝜕𝑓1𝜕𝑥2
|𝑥2
0
…𝜕𝑓1𝜕𝑥𝑛
|𝑥𝑛
0
𝜕𝑓2𝜕𝑥1
|𝑥1
0
𝜕𝑓2𝜕𝑥2
|𝑥2
0
…𝜕𝑓2𝜕𝑥𝑛
|𝑥𝑛
0
⋮𝜕𝑓𝑛𝜕𝑥1
|𝑥1
0
⋮𝜕𝑓𝑛𝜕𝑥1
|𝑥2
0
… ⋮
…𝜕𝑓𝑛𝜕𝑥1
|𝑥𝑛
0 ]
[ ∆𝑥1
0
∆𝑥20
⋮∆𝑥𝑛
0]
From above equation, we can get Δx01, Δx0
2…Δx0n.Then the new solution can be obtained. The
iteration equation can be written as follows:
[ 𝑓1(𝑥1
𝑘, 𝑥2𝑘 …𝑥𝑛
𝑘)
𝑓2(𝑥1𝑘, 𝑥2
𝑘 …𝑥𝑛𝑘)
⋮𝑓𝑛(𝑥1
𝑘 , 𝑥2𝑘 …𝑥𝑛
𝑘)]
= −
[ 𝜕𝑓1𝜕𝑥1
|𝑥1
𝑘
𝜕𝑓1𝜕𝑥2
|𝑥2
𝑘
…𝜕𝑓1𝜕𝑥𝑛
|𝑥𝑛
𝑘
𝜕𝑓2𝜕𝑥1
|𝑥1
𝑘
𝜕𝑓2𝜕𝑥2
|𝑥2
𝑘
…𝜕𝑓2𝜕𝑥𝑛
|𝑥𝑛
𝑘
⋮𝜕𝑓𝑛𝜕𝑥1
|𝑥1
𝑘
⋮𝜕𝑓𝑛𝜕𝑥1
|𝑥2
𝑘
… ⋮
…𝜕𝑓𝑛𝜕𝑥1
|𝑥𝑛
𝑘 ]
[ ∆𝑥1
𝑘
∆𝑥2𝑘
⋮∆𝑥𝑛
𝑘]
𝑥𝑘+1 = 𝑥𝑘 + ∆𝑥𝑘+1
The equations can be written as
𝐹(𝑋𝑘) = −𝐽𝑘∆𝑋𝑘
𝑋𝑘+1 = 𝑋𝑘+1 + ∆𝑋𝑘
Where J is an n*n matrix called a Jacobian matrix.
The NR method is better formulated using polar coordinates and is very accurate. In power flow
studies, the elements in the Jacobean matrix are calculated by differentiating the power & reactive
power expression and substituting values of voltage magnitude and phase angle.
The next stage of Newton Raphson Solution, the Jacobean is inverted. Matrix inversion is a
computationally complex task with the resources of time and storage increasing rapidly with order
of (J).
Assuming all buses are PQ and using estimated values, we obtain the difference between them and
the calculated values and represent them as:
26
∆𝑃𝑖 = 𝑃𝑖𝑠𝑐ℎ− 𝑃𝑖𝑐𝑎𝑙𝑐
∆𝑄𝑖 = 𝑄𝑖𝑠𝑐ℎ− 𝑄𝑖𝑐𝑎𝑙𝑐
Note, the Jacobean matrix gives the linearized relationship between small changes in ∆Pi (k) and
voltage magnitude ∆ (Vik) with the small changes in real and reactive power ∆Pi
(k) and ∆Qi (k).
Formulating the load follow problem in polar form,
[ ∆𝑃2
(𝑘)
⋮
∆𝑃𝑛(𝑘)
∆𝑄2(𝑘)
⋮
∆𝑄𝑛(𝑘)
]
=
[
[
[ 𝜕𝑃2
(𝑘)
𝜕𝛿2⋯
𝜕𝑃2(𝑘)
𝜕𝛿𝑛
⋮ ⋱ ⋮
𝜕𝑃𝑛(𝑘)
𝜕𝛿𝑛⋯
𝜕𝑃𝑛(𝑘)
𝜕𝛿𝑛 ]
⋯
[ 𝜕𝑃2
(𝑘)
𝜕|𝑉2|⋯
𝜕𝑃2(𝑘)
𝜕|𝑉𝑛|⋮ ⋱ ⋮
𝜕𝑃2(𝑘)
𝜕|𝑉2|⋯
𝜕𝑃2(𝑘)
𝜕|𝑉𝑛|]
⋮ ⋱ ⋮
[ 𝜕𝑄2
(𝑘)
𝜕𝛿2⋯
𝜕𝑄2(𝑘)
𝜕𝛿2
⋮ ⋱ ⋮
𝜕𝑄𝑛(𝑘)
𝜕𝛿2⋯
𝜕𝑄𝑛(𝑘)
𝜕𝛿𝑛 ]
⋯
[ 𝜕𝑄2
(𝑘)
𝜕|𝑉2|⋯
𝜕𝑄2(𝑘)
𝜕|𝑉𝑛|⋮ ⋱ ⋮
𝜕𝑄𝑛(𝑘)
𝜕|𝑉2|⋯
𝜕𝑄𝑛(𝑘)
𝜕|𝑉𝑛| ]
]
]
[ ∆𝛿2
(𝑘)
⋮
∆𝛿𝑛(𝑘)
∆|𝑉|2(𝑘)
⋮
∆|𝑉|𝑛(𝑘)
]
The matrix can be simplified to
[∆𝑃∆𝑄
] = [𝐽1 𝐽2𝐽3 𝐽4
] [∆𝛿
∆|𝑉|]
3.2.4 Gauss Seidel Method
This method is developed based on the Gauss method. It is an iterative method of solving
simultaneous non-linear equations. The type of data specified for different kinds of buses makes
it complex to obtain formal solutions for power flow in a power system. The method makes use of
an estimated value of voltage, to obtain a new calculated value for each bus voltage. The estimated
value is replaced by a calculated value and both real power and reactive power are specified. . A
new set of bus voltages is hence available, these voltages are used to calculate yet another set of
voltages at the buses. The process is then repeated until the iteration solution converges. The
convergence is quite sensitive to the starting values assumed. But this method suffers from poor
convergence characteristics [4, 24].
27
This is an iterative method used to solve the load flow problem for the value of Vi, and the iterative
sequence hence becomes
𝑉𝑖(𝑘+1)
=
𝑃𝑖𝑠𝑐ℎ − 𝑗𝑄𝑖
𝑠𝑐ℎ
𝑉𝑖∗ + ∑𝑌𝑖𝑗 𝑉𝑗
(𝑘)
∑𝑌𝑖𝑗 𝑗 ≠ 1
Where Pisch= Pgi-Pdi and Qi
sch= Qgi-Qdi
Pisch and Qi
sch is the net scheduled real and reactive power being injected into bus i
Pgi and Qgi denotes the real and reactive power being generated
Pdi and Qdi denotes the real and reactive power demand at that load bus
It is assumed that the current injected into bus i is positive, then the real and the reactive powers
supply into the buses, such as generator buses, Pisch and Qi
sch have a positive value. The real and
the reactive powers flowing away from the buses, such as load buses Pisch and Qi
sch have a negative
values. Pi and Qi are solved from Equation (5) which gives
𝑃𝑖(𝑘+1)
= 𝑅𝑒𝑎𝑙 [𝑉𝑖∗(𝑘)
{∑𝑌𝑖𝑗
𝑛
𝑖−0
− ∑𝑉𝑖(𝑘)
𝑛
𝑗𝑖
}]……… 𝑗 ≠ 1
𝑄𝑖(𝑘+1)
= 𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 [𝑉𝑖∗(𝑘)
{∑𝑌𝑖𝑗
𝑛
𝑖−1
− ∑𝑉𝑖(𝑘)
𝑛
𝑗𝑖
}]……… 𝑗 ≠ 1
The power flow equation is usually expressed in terms of the bus admittance matrix, using the
diagonal elements of the bus admittance and the non-diagonal elements of the matrix, then the
Equation 10) becomes,
𝑉𝑖(𝑘+1)
=
𝑃𝑖𝑠𝑐ℎ − 𝑗𝑄𝑖
𝑠𝑐ℎ
𝑉𝑖∗ − ∑𝑌𝑖𝑗 𝑉𝑗
(𝑘)
𝑌𝑖𝑗
And
𝑃𝑖(𝑘+1)
= 𝑅𝑒𝑎𝑙 [𝑉𝑖∗(𝑘)
{𝑉𝑖∗(𝑘)
𝑌𝑖𝑖 − ∑ 𝑌𝑖𝑗 𝑉𝑖(𝑘)
𝑛
𝑖=1,𝑗=1
}]……… 𝑗 ≠ 1
28
𝑄𝑖(𝑘+1)
= 𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 [𝑉𝑖∗(𝑘)
{𝑉𝑖∗(𝑘)
𝑌𝑖𝑖 − ∑ 𝑌𝑖𝑗 𝑉𝑖(𝑘)
𝑛
𝑖=1,𝑗=1
}]……… 𝑗 ≠ 1
3.2.5 The Fast De-coupled Method
In the Newton Raphson power flow method, for every iteration the jacobian matrix has been
recalculated. The Fast Decoupled Power Flow Method is based on a simplification of the Newton-
Raphson method. This method, offers calculation simplifications, fast convergence and reliable
results and became a widely used method in load flow analysis. However in some cases, where
there is a high resistance-to-reactance (R/X) ratios or heavy loading (low voltage) at some buses,
it does not converge well because many assumptions are made to simplify the Jacobian matrix and
it is an approximation method. In a practical power system, the resistance of the branch is much
lower than the reactance of the branch, therefore, there exists strong coupling between the real
power and voltage angles whereas the coupling between the voltage magnitude and the real power
is weak hence changes in the voltage magnitude have little effect in the real power. On the other
hand, the reactive power has a strong coupling relationship with the voltage magnitude while it
has a weak coupling relationship with the voltage angle [5, 33].
The Jacobian matrix is reduced by half by ignoring the element of J2 and J3 and it simplifies to:
[∆𝑃∆𝑄
] = [𝐽1 00 𝐽4
] [∆𝛿
∆|𝑉|]
Expanding the equation above gives two separate equation
∆𝑃 = 𝐽1∆𝛿 = [𝜕𝑃
𝜕𝛿] ∆𝛿
∆𝑄 = 𝐽4∆|𝑉| = [𝜕𝑃
𝜕|𝑉|] ∆|𝑉|
∆𝑃
𝑉𝑖= −𝐵′∆𝛿
∆𝑄
𝑉𝑖= −𝐵′′∆|𝑉|
29
B' and B'' are the imaginary parts of the bus admittance. It is better to ignore all shunt connected
elements, as to make the formation of J1 and J4 simple. This will allow for only one single matrix
than performing repeated inversion. The successive voltage magnitude and phase angle changes
are
∆𝛿 = −[𝐵′]−1∆𝑃
|𝑉|
∆|𝑉| = −[𝐵′′]−1∆𝑄
|𝑉|
A summary of comparison of the methods of power flow solution is given in the table below:
PROPERTIES NEWTON -RAPHSON GAUSS – SEIDEL DECOUPLED METHODS
Storage Requirements Requires the most
Memory Space as the
Jacobian matrix are
more.
It has the minimum
memory space
requirement.
It has a less memory
requirement as compared to
NR as it stores less Jacobian
elements.
Accuracy It is the most accurate. It is moderately
accurate.
Due to the general
assumptions made it is the
least accurate.
Speed It is relatively fast as
compared to Gauss-
Seidel
It is the slowest. It is the fastest.
System Size It is Independent of
system size.
The number of
iterations increase
with increase of
system size.
It is independent of system
size.
Complexity It is the most complex It is the least
complex
It is relatively complex as
compared to GS.
Table 3.1: Comparison of load Flow Methods
30
3.3 Power Stability with Renewable Energy Problem Formulation
Voltage stability is directly associated with reactive power deficiency with renewable energy. The
gradual changes in power systems can lead to shortage of reactive power leading to a reduction of
power stability. Deterioration of power system increases as load and generation grows. There is a
decrease in voltage at the bus with increased power flow. Further increase in loading leads to
shortage of reactive power and any further increase in active power causes a quick decrease in
magnitude of voltage at the buses. Hence we can include a reactive power support on the weakest
buses which can immediately provide relief and enhance voltage stability. The disturbance was
implemented by via load change by increasing the load in the PV buses by 50%.
The STATCOM can be used to provide voltage support through controlled reactive power
injection. A STATCOM is chosen as a voltage control bus. The STATCOM internal losses are
neglected and as a result the specified real power at those PV buses are set to zero. The problem
to be formulated is that of computing load flow incorporating a STATCOM device. It also includes
the implementation of real participation factors where a Newton Raphson Solver is used.
The basic power flow equations are obtained by applying Kirchhoff’s law to the network
represented by the bus/branch model. The real and reactive power injection at each bus is given
by [14];
𝑃𝑖 = 𝑉𝑖 ∑𝑉𝑖(𝐺𝑖𝑗 cos 𝛿𝑖𝑗 + 𝐵𝑖𝑗 sin 𝛿𝑖𝑗)…………… .1)
𝑛
𝑗=1
𝑄𝑖 = 𝑉𝑖 ∑𝑉𝑖(𝐺𝑖𝑗 sin 𝛿𝑖𝑗 − 𝐵𝑖𝑗 cos 𝛿𝑖𝑗)…………… .2)
𝑛
𝑗=1
Where Pi is the real power injection at bus i
Qi is the real power injection at bus i
δij =δi-δj
δi is the bus voltage angle at bus i
δj is the bus voltage angle at bus j
31
Vi is the bus voltage magnitude at bus i
Vj is the bus voltage magnitude at bus j
Gij is the conductance
Bij is the suspectance
Assuming all buses are PQ and using estimated values, we obtain the difference between them and
the calculated values and represent them as:
∆𝑃𝑖 = 𝑃𝑖𝑠𝑐ℎ− 𝑃𝑖𝑐𝑎𝑙𝑐
∆𝑄𝑖 = 𝑄𝑖𝑠𝑐ℎ− 𝑄𝑖𝑐𝑎𝑙𝑐
Note, the Jacobean matrix gives the linearized relationship between small changes in ∆Pi (k) and
voltage magnitude ∆ (Vik) with the small changes in real and reactive power ∆Pi
(k) and ∆Qi (k).
Formulating the load follow problem in polar form,
[ ∆𝑃2
(𝑘)
⋮
∆𝑃𝑛(𝑘)
∆𝑄2(𝑘)
⋮
∆𝑄𝑛(𝑘)
]
=
[
[
[ 𝜕𝑃2
(𝑘)
𝜕𝛿2⋯
𝜕𝑃2(𝑘)
𝜕𝛿𝑛
⋮ ⋱ ⋮
𝜕𝑃𝑛(𝑘)
𝜕𝛿𝑛⋯
𝜕𝑃𝑛(𝑘)
𝜕𝛿𝑛 ]
⋯
[ 𝜕𝑃2
(𝑘)
𝜕|𝑉2|⋯
𝜕𝑃2(𝑘)
𝜕|𝑉𝑛|⋮ ⋱ ⋮
𝜕𝑃2(𝑘)
𝜕|𝑉2|⋯
𝜕𝑃2(𝑘)
𝜕|𝑉𝑛|]
⋮ ⋱ ⋮
[ 𝜕𝑄2
(𝑘)
𝜕𝛿2⋯
𝜕𝑄2(𝑘)
𝜕𝛿2
⋮ ⋱ ⋮
𝜕𝑄𝑛(𝑘)
𝜕𝛿2⋯
𝜕𝑄𝑛(𝑘)
𝜕𝛿𝑛 ]
⋯
[ 𝜕𝑄2
(𝑘)
𝜕|𝑉2|⋯
𝜕𝑄2(𝑘)
𝜕|𝑉𝑛|⋮ ⋱ ⋮
𝜕𝑄𝑛(𝑘)
𝜕|𝑉2|⋯
𝜕𝑄𝑛(𝑘)
𝜕|𝑉𝑛| ]
]
]
[ ∆𝛿2
(𝑘)
⋮
∆𝛿𝑛(𝑘)
∆|𝑉|2(𝑘)
⋮
∆|𝑉|𝑛(𝑘)
]
The matrix can be simplified to
[∆𝑃∆𝑄
] = [𝐽1 𝐽2𝐽3 𝐽4
] [∆𝛿
∆|𝑉|]
When the STATCOM model below is incorporated at a weak bus s. The bus S is heavily affected
by voltage regulation. The controllable voltage source of STATCOM is denoted as Vt in series
with Zt impedance [30].
32
Figure 3: STATCOM Representation
The summation terms of the injected active and reactive power due to the STATCOM
𝑃𝑡 = 𝑉𝑠2𝐺𝑡 − 𝑉𝑠𝑉𝑡(𝐺𝑡 cos 𝛿𝑠𝑡 + 𝐵𝑡 sin 𝛿𝑠𝑡)…………3)
𝑄𝑡 = −𝑉𝑠2𝐵𝑡 + 𝑉𝑡𝑉𝑠(𝐺𝑡 sin 𝛿𝑠𝑡 + 𝐵𝑡 cos 𝛿𝑠𝑡)…………4)
One more equation is needed to solve the power flow problem. This equation is needed to find the
power consumed by Vt. The power must be zero in steady state condition.
𝑃𝑣𝑡 = 𝑅𝑒𝑎𝑙[𝑉𝑡𝐼𝑡∗] = −𝑉𝑡
2𝐺𝑡 + 𝑉𝑠𝑉𝑡(𝐺𝑡 cos 𝛿𝑠𝑡 + 𝐵𝑡 sin 𝛿𝑠𝑡) = 0……… .5)
The load flow problem is formulated as a set of non-linear algebraic equations.
3.4 Solution of Power Stability with RE using Newton-Raphson Method
As per the above equations, the equation below is used when applying Newton Raphson method
to solve the equation 1) to 5)
[
∆𝑃∆𝑄∆𝑃𝑣𝑡
∆𝐹
] = 𝐽(𝑥𝑘) [
∆𝛿∆𝑉∆𝛿𝑡
∆𝑉𝑡
]………6)
Where control variable for jacobian matrix, F=Vs.
33
Where k is the iteration counter.
J(xk) is the problem’s Jacobian matrix
The Jacobian matrix is given by
𝐽(𝑥𝑘) = [
𝐽11 𝐽12 𝐽13 𝐽14
𝐽21 𝐽21 𝐽23 𝐽24
𝐽31
𝐽41
𝐽32
𝐽42
𝐽33
𝐽43
𝐽34
𝐽44
]……… .7)
=
[ 𝜕𝑃
𝜕𝛿
𝜕𝑃
𝜕𝑉
𝜕𝑃
𝜕𝛿𝑡
𝜕𝑃
𝜕𝑉𝑡
𝜕𝑄
𝜕𝛿
𝜕𝑄
𝜕𝑉
𝜕𝑄
𝜕𝛿𝑡
𝜕𝑄
𝜕𝑉𝑡
𝜕𝑃𝑣𝑡
𝜕𝛿𝜕𝐹
𝜕𝛿
𝜕𝑃𝑣𝑡
𝜕𝑉𝜕𝐹
𝜕𝑉
𝜕𝑃𝑣𝑡
𝜕𝛿𝑡
𝜕𝐹
𝜕𝛿𝑡
𝜕𝑃𝑣𝑡
𝜕𝑉𝑡
𝜕𝐹
𝜕𝑉𝑡 ]
………8)
The solution can be iteratively solved by solving the system represented by 6). The voltage
magnitude and voltage phase angles are updated as;
𝛿𝑘+1 = 𝛿𝑘 + ∆𝛿
𝑉𝑘+1 = 𝑉𝑘 + ∆𝑉
𝛿𝑠𝑡𝑎𝑘+1 = 𝛿𝑠𝑡𝑎𝑡
𝑘 + ∆𝛿𝑠𝑡𝑎𝑡
𝑉𝑠𝑡𝑎𝑡𝑘+1 = 𝑉𝑠𝑡𝑎𝑡
𝑘 + ∆𝑉𝑠𝑡𝑎𝑡
Until convergence is obtained.
3.5 Formulation of The Real Participation factors
For the unbalanced power flows with distributed generators, the distributed slack bus will be
proposed to accommodate growth of distributed generators. The single slack bus is used as the
reference bus for voltage phase angles and to absorb system real power loss Ploss. However in an
actual system, there is no slack bus. Hence the single slack bus may significantly distort computed
power flows. Hence we introduce a distributed slack bus [7].
34
The distributed slack model is based upon distributing the burden of the slack among other
generator buses in the power system. This is to accommodate the growth of distributed generators
due to geographical and environmental constraints, stability and security problems of large
generation plants, competitive energy makers and emergence of advanced techniques with small
ratings employed resulting in environmental and increased profitability [2, 7].
To distribute Ploss a participation factor will be implemented which means that the system loss is
shared by several generator buses during power flow calculation based on their assigned
participation factors. In [3, 7] the participation factors are related to the characteristics of turbines
on each generator bus and load allocation. In [10], the author provides a method of choosing
participation based on the scheduled generator outputs. However, these previous publication
focused on balanced transmission lines and for varying reasons, they may not suitable for
distributed systems with DG’s.
In [9] the network based participation factor for distributed slack bus is modelled. Using scalar
participation factors the distributed slack bus is modelled to assess the unknown system losses. In
distribution systems participating sources include the sub-station and distributed generators whose
real output can be adjusted. The distributed slack bus model for three phase power flow, the system
real power loss Ploss is treated as an unknown and distributed to participating sources according to
their assigned participation factors Ki. According to S.Tong and K.Miiu [7] they applied the
distributed slack bus participation to distribute real power losses to participating source. In these
cases, the participation factor Ki for the source i is calculated as
𝐾𝑖 =𝑃𝐺𝑖
𝑙𝑜𝑠𝑠
𝑃𝑙𝑜𝑠𝑠 𝑖 = 0, 1,2……𝑚
Where
∑𝐾𝑖 = 1
𝑚
𝑖=0
Where m = number of participating distributed generators
By applying the participation factors, the total area power outputs of participating sources can be
expressed as
35
𝑃𝐺𝑖 = 𝑃𝐺𝑖𝑙𝑜𝑎𝑑 + 𝐾𝑖𝑃𝑙𝑜𝑠𝑠 𝑖 = 0, 1, 2…𝑚
Where Ploss is the total real power loss in the system
𝑃𝐺𝑖𝑙𝑜𝑎𝑑 Is the real power load associated with participating sources.
Not all DGs in distribution systems are allowed to adjust their real power outputs, since most are
small machines and may not have the necessary control technologies.
This is the formulation that is to be added to equation 6) to improve the weakness of Newton-
Raphson especially when distributed systems (wind and PV systems) are involved. Hence the
improved matrix becomes
𝐽(𝑥𝑘) = [
𝐽11 𝐽12 𝐽13 𝐽14 𝐽15 𝐽16
𝐽21 𝐽21 𝐽23 𝐽24 𝐽25 𝐽26
𝐽31
𝐽41
𝐽32
𝐽42
𝐽33
𝐽43
𝐽34 𝐽35 𝐽36
𝐽44 𝐽46 𝐽46
]
Where J15-J45 are the derivative of real power injection with the loss i.e. participation factors.
J16-J46 are the derivative of reactive power injection with the loss.
3.6 Solution Algorithm for Solving Stability with Renewable Energy
Step 1: Read System data and formulate the Ybus.
Step 2: initialize bus voltage, phase angles and set initial Ploss=0.
Step 3: Set the iteration counter k=o and convergence criteria.
Step 4: Define the participating buses and participating factors.
Step 5: Compute Pi(k), Qi
(k), Pstat(k), Qstat
(k) for the system buses.
Step 6: Evaluate Power Mismatches.
Step 7: Evaluate the jacobian matrix.
Step 8: Evaluate the increments of bus voltage magnitude, voltage angle, STATCOM voltage
magnitudes and phase angles.
Step 9: This process continues until the power mismatches are less than tolerance.
36
Step 10: Evaluate the real and reactive and reactive power flow and the losses.
37
3.7 Flow Chart
Flowchart for Power Stability with Renewable Energy
Figure 4: Flowchart for Power Stability with Renewable Energy
Start
Input system data, bus,
line and STATCOM data
Generate System Admittance
Matrix
Define participating buses and factors
Compute Pik and Qik for k=1,2…
Compute mismatch equations
Evaluate the Jacobian Matrix
Update bus voltages and angles
Has it
converged? Output load flow
& line flows Stop
38
CHAPTER FOUR
RESULTS AND ANALYSIS
The proposed algorithm was tested on IEEE-30 bus system. The results were compared with those
obtained in the absence and presence of the STATCOM. The STATCOMs were placed in the
buses 6, 7 and 8.
4.1 Case study: IEEE 30-Bus System
Below is a one line diagram of IEEE 30-Bus System.
Figure 5: IEEE 30 Bus Test Network
39
4.2 Results and Validation
Bus Data for IEEE 30 Bus Test Network
Bus Type Specifie
d
voltage
Angl
e
Real
Power
Gen
(MW)
Reactiv
e Power
Gen
(MVAR
)
Real Power
Load(MW)
Load Ql
(MVAR
)
Qma
x
Qmi
n
1 SLACK 1.00 0 23.54 0.00 0 0 -20 150
2 PV 1.00 0 60.97 0.00 21.7 12.7 -20 60
3 PQ 1.00 0 0 0.00 2.4 1.2 0 0
4 PQ 1.00 0 0 0.00 7.6 1.6 0 0
5 PQ 1.00 0 0 0.00 94.2 19 -40 40
6 PQ 1.00 0 0 0.00 0 0 0 0
7 PQ 1.00 0 0 0.00 22.8 10.9 0 0
8 PQ 1.00 0 0 0.00 30 30 -10 40
9 PQ 1.00 0 0 0.00 0 0 0 0
10 PQ 1.00 0 0 0.00 5.8 2 0 0
11 PQ 1.00 0 0 0.00 0 0 0 0
12 PQ 1.00 0 0 0.00 11.2 7.5 0 0
13 PV 1.00 0 37.00 0.00 0 0 -15 44.7
14 PQ 1.00 0 0 0.00 6.2 1.6 0 0
15 PQ 1.00 0 0 0.00 8.2 2.5 0 0
16 PQ 1.00 0 0 0.00 3.5 1.8 0 0
17 PQ 1.00 0 0 0.00 9 5.8 0 0
18 PQ 1.00 0 0 0.00 3.2 0.9 0 0
19 PQ 1.00 0 0 0.00 9.5 3.4 0 0
20 PQ 1.00 0 0 0.00 2.2 0.7 0 0
21 PQ 1.00 0 0 0.00 17.5 11.2 0 0
22 PV 1.00 0 21.59 0.00 0 0 -15 62.5
23 PV 1.00 0 19.20 0.00 3.2 1.6 -10 40
24 PQ 1.00 0 0 0.00 8.7 6.7 0 0
25 PQ 1.00 0 0 0.00 0 0 0 0
26 PQ 1.00 0 0 0.00 3.5 2.3 0 0
27 PV 1.00 0 26.91 0.00 0 0 -15 48.7
28 PQ 1.00 0 0 0.00 0 0 0 0
29 PQ 1.00 0 0 0.00 2.4 0.9 0 0
30 PQ 1.00 0 0 0.00 10.6 1.9 0 0
Table 4.1: Bus Data
Line Data IEEE 30 Bus
40
From Bus To Bus Resistance
(p.u)
Reactance (p.u) Half-line
susceptance(B/2)
Transformer
tap Settings 1 2 0.0192 0.0575 0.0528 1
1 3 0.0452 0.1652 0.0408 1
2 4 0.057 0.1737 0.0368 1
3 4 0.0132 0.0379 0.0084 1
2 5 0.0472 0.1983 0.0418 1
2 6 0.0581 0.1763 0.0374 1
4 6 0.0119 0.0414 0.009 1
5 7 0.046 0.116 0.0204 1
6 7 0.0267 0.082 0.017 1
6 8 0.012 0.042 0.009 1
6 9 0 0.208 0 0.978
6 10 0 0.556 0 0.969
9 11 0 0.208 0 1
9 10 0 0.11 0 1
4 12 0 0.256 0 0.932
12 13 0 0.14 0 1
12 14 0.1231 0.2559 0 1
12 15 0.0662 0.1304 0 1
12 16 0.0945 0.1987 0 1
14 15 0.221 0.1997 0 1
16 17 0.0524 0.1923 0 1
15 18 0.1073 0.2185 0 1
18 19 0.0639 0.1292 0 1
19 20 0.034 0.068 0 1
10 20 0.0936 0.209 0 1
10 17 0.0324 0.0845 0 1
10 21 0.0348 0.0749 0 1
10 22 0.0727 0.1499 0 1
21 22 0.0116 0.0236 0 1
15 23 0.1 0.202 0 1
22 24 0.115 0.179 0 1
23 24 0.132 0.27 0 1
24 25 0.1885 0.3292 0 1
25 26 0.2544 0.38 0 1
25 27 0.1093 0.2087 0 1
28 27 0 0.396 0 0.968
27 29 0.2198 0.4153 0 1
27 30 0.3202 0.6027 0 1
29 30 0.2399 0.4533 0 1
8 28 0.0636 0.2 0.0428 1
6 28 0.0169 0.0599 0.013 1
Table 4.2: Line Data
41
Comparison of Voltages without Renewable Energy and With Renewable Energy.
Bus No. Voltages(p.u) without Renewable
Energy
Voltages(p.u) with Renewable
Energy
1 1.0000 1.0000
2 1.0000 1.0000
3 0.9780 0.9781
4 0.9724 0.9726
5 0.9817 0.9817
6 0.9684 0.9686
7 0.9651 0.9652
8 0.9554 0.9556
9 0.9894 0.9894
10 0.9890 0.9889
11 0.9894 0.9894
12 1.0019 1.0019
13 1.0000 1.0000
14 0.9906 0.9906
15 0.9906 0.9906
42
16 0.9888 0.9888
17 0.9833 0.9832
18 0.9772 0.9772
19 0.9726 0.9725
20 0.9758 0.9758
21 0.9938 0.9938
22 1.0000 1.0000
23 1.0000 1.0000
24 0.9873 0.9872
25 0.9897 0.9897
26 0.9715 0.9715
27 1.0000 1.0000
28 0.9700 0.9701
29 0.9796 0.9796
30 0.9679 0.9679
Table 4.3: Comparison between voltages with and without Renewable Energy
43
Comparison of Voltages with Renewable Energy and With Renewable Energy and
STATCOM under Normal Operating Conditions.
Bus
Number
Voltage (p.u) with
Renewable Energy
under normal operating
conditions
Voltage (p.u) with
Renewable Energy and
STATCOM under normal
operations
1 1.0000 1.0000
2 1.0000 1.0000
3 0.9781 0.9943
4 0.9726 0.9925
5 0.9817 1.0036
6 0.9686 1.0000
7 0.9652 1.0000
8 0.9556 1.0000
9 0.9894 1.0052
10 0.9889 0.9961
11 0.9894 1.0052
12 1.0019 1.0083
13 1.0000 1.0000
14 0.9906 0.9960
15 0.9906 0.9952
16 0.9888 0.9955
17 0.9832 0.9903
44
18 0.9772 0.9827
19 0.9725 0.9787
20 0.9758 0.9822
21 0.9938 0.9955
22 1.0000 1.0000
23 1.0000 1.0000
24 0.9872 0.9873
25 0.9897 0.9898
26 0.9715 0.9717
27 1.0000 1.0000
28 0.9701 1.0000
29 0.9796 0.9796
30 0.9679 0.9679
Table 4.4: Comparison of Voltages with renewable with and without STATCOM in normal
operating Conditions.
Comparison of voltages with renewable energy without STATCOM and with renewable
energy and STATCOM after disturbance (50% load change).
Bus Number
Voltage (p.u) with
Renewable Energy after
disturbance
Voltage (p.u) with
Renewable Energy and
STATCOM after disturbance.
1 1.0000 1.0000
2 0.9900 1.0000
3 0.9567 0.9909
45
4 0.9465 0.9886
5 0.9574 1.0044
6 0.9370 1.0000
7 0.9309 1.0000
8 0.9161 1.0000
9 0.9656 1.0034
10 0.9702 0.9938
11 0.9656 1.0034
12 0.9854 1.0038
13 1.0000 1.0000
14 0.9677 0.9870
15 0.9675 0.9872
16 0.9672 0.9880
17 0.9606 0.9836
18 0.9485 0.9701
19 0.9424 0.9650
20 0.9481 0.9709
21 0.9722 0.9932
22 1.0000 1.0000
23 0.9800 1.0000
24 0.9742 0.9810
25 0.9818 0.9845
46
26 0.9540 0.9573
27 1.0000 1.0000
28 0.9390 0.9998
29 0.9686 0.9686
30 0.9504 0.9504
Table 4.5: Comparison of voltages with renewable energy without STATCOM and with renewable
energy and STATCOM after disturbance (50% load change).
4.3 Analysis and Discussion
Comparison of Total Real and Reactive Power Injected and Generated Without Renewable
Energy, with Renewable Energy and with Renewable Energy and STATCOM.
Total Injected
Power
Total Generated
Power
Total Load
MW MVAR MW MVAR MW MVAR
Without Renewable Energy 2.438 -33.559 191.638 73.641 189.200 107.200
With Renewable Energy 2.412 -33.602 191.612 73.598 189.200 107.200
With Renewable Energy and
STATCOM
2.140 -34.910 191.340 72.290 189.200 107.200
With Renewable Energy after
disturbance
8.611 -10.613 279.961 143.137 271.350 153.750
With Renewable Energy and
STATCOM after disturbance.
7.956 -15.133 279.306 138.617 271.350 153.750
Table 4.6: Comparison of Total Real and Reactive Power Injected and Generated Without
Renewable Energy, with Renewable Energy and with Renewable Energy and STATCOM.
47
Comparison of Total Real and Reactive Losses without Renewable Energy, with Renewable
Energy and with Renewable Energy and STATCOM.
Total Losses
MW MVAR
Without Renewable 2.438 9.403
With Renewable Energy 2.412 9.352
With Renewable Energy and
STATCOM
2.140 7.884
With Renewable Energy after
disturbance.
8.611 31.768
With Renewable Energy and
STATCOM after disturbance.
7.956 27.629
Table 4.7: Comparison of Total Real and Reactive Losses without Renewable Energy, with
Renewable Energy and with Renewable Energy and STATCOM.
48
Figure 6: Voltage Profile in Normal Operating Conditions
Figure 7: Comparison of Losses under Normal Operating Conditions
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Vo
ltag
e in
p.u
.
Bus Number
VOLTAGE PROFILE IN NORMAL OPERATING
CONDITIONS
With Renewable Energy With Renewable Energy and STATCOM
0
1
2
3
4
5
6
7
8
9
10
Total Real Power Losses (MW) Total Reactive Power Losses (MVAR)
TOTAL LOSSES IN NORMAL OPERATING
CONDITIONS
With Renewable Energy With Renewable Energy and STATCOM
49
Figure 8: Voltage Profile after Disturbance
Figure 9: Comparison of Total Losses after Disturbance
Renewable Energy was incorporated in the load flow analysis by including participating factors
whereby a distributed slack model is utilized this is to accommodate the growth of distributed
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
VOLTAGE PROFILE AFTER DISTURBANCE
Voltage (p.u) with Renewable Energy Voltage (p.u) with Renewable Energy and STATCOM
0
5
10
15
20
25
30
35
Total Real Power Losses (MW) Total Reactive Power Losses (MVAR)
TOTAL LOSSES AFTER DISTURBANCE
With Renewable Energy With Renewable Energy and STATCOM
50
generators such as Wind energy and Photovoltaic. The distributed slack model is based upon
distributing the burden of the slack among other generator buses in the power system. To distribute
the losses real participation factor was implemented which means the system loss is shared by
several generator buses during power flow calculations based on their assigned participation.
When renewable is introduced to the power system we observe an improvement of voltage in the
range of 0-0.0002, a reduction of real power loss from 2.438MW to 2.412MW and a reduction of
reactive power loss from 9.403MVAR to 9.352MVAR.
In normal operating conditions, it is observed when the STATCOM is introduced in the system by
placing them in the weakest buses which were the buses 6, 7, and 8, we observe a great
improvement in the voltage magnitudes thus improving the voltage stability of the power system.
In terms of losses, when the STATCOM was introduced to the renewable energy incorporated
system, the total real power losses was reduced by 0.002MW and total reactive power losses was
reduced by 1.468 MVAR.
After disturbance, we observe the voltage magnitudes dropped that is as compared to normal
operating conditions the voltage magnitudes reduced. There is a decrease in voltage at the bus with
increased power flow. Further increase in loading leads to shortage of reactive power and any
further increase in active power causes a quick decrease in magnitude of voltage at the buses.
When the STATCOM was introduced to the disturbed system by placing them in the weakest buses
we observed an improvement in voltage magnitude this thus improving the voltage stability after
the system is subjected to a disturbance. In terms of losses, when the STATCOM was introduced
to the renewable energy incorporated system, the total real power losses was reduced by 0.655MW
and total reactive power losses was reduced by 4.139 MVAR.
Hence we can conclude that a STATCOM is used to generate reactive power which is injected in
the power system which greatly improved the voltage stability on the weak buses which further
improved the voltage stability in the whole power system during normal operating conditions and
after a disturbance. Furthermore the real and reactive power losses were also reduced.
51
CHAPTER FIVE
CONCLUSION AND RECOMMENDATON FOR FURTHER WORK
5.1 Conclusion
This project applied Newton Raphson with real participating factors incorporating a STATCOM
device for the solution of Power Stability with Renewable Energy. The proposed algorithm was
successfully tested on IEEE 30 bus system.
Introduction of Renewable Energy was done by using real participation factors by distributing the
losses hence accommodating distributed generation. By introducing the STATCOM in the weakest
buses both under normal operating conditions and after a disturbance the voltage magnitudes were
greatly improved hence improving the voltage stability of the system and it further decreased the
real and reactive power losses.
5.2 Recommendation
The project only considered improving voltage stability in normal operating conditions and after
a disturbance when Renewable Energy is introduced in power system. In future, the project could
be extended to consider improving frequency and rotor angle stability after a disturbance has
occurred in a power system.
52
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55
APPENDIX
APPENDIX 1: STATCOM Data function statdata = statdata30()
% |Bus | Vsp | theta |Qsmx | Qsmn | statdata = [ 6 1 0 0.5 -0.5; 7 1 0 0.5 -0.5; 8 1 0 0.5 -0.5];
56
APPENDIX 2: Program for formation of Bus Admittance Matrix Program to form Admittance And Impedance Bus Formation.... % with Transformer Tap setting..
function ybus = ybusppg() % Returns ybus
linedata = linedata30(); % Calling "linedata3.m" for Line Data... fb = linedata(:,1); % From bus number... tb = linedata(:,2); % To bus number... r = linedata(:,3); % Resistance, R... x = linedata(:,4); % Reactance, X... b = linedata(:,5); % Ground Admittance, B/2... a = linedata(:,6); % Tap setting value.. z = r + i*x; % Z matrix... y = 1./z; % To get inverse of each element... b = i*b; % Make B imaginary...
nbus = max(max(fb),max(tb)); % no. of buses... nbranch = length(fb); % no. of branches... ybus = zeros(nbus,nbus); % Initialise YBus...
% Formation of the Off Diagonal Elements... for k = 1:nbranch ybus(fb(k),tb(k)) = ybus(fb(k),tb(k))-y(k)/a(k); ybus(tb(k),fb(k)) = ybus(fb(k),tb(k)); end
% Formation of Diagonal Elements.... for m = 1:nbus for n = 1:nbranch if fb(n) == m ybus(m,m) = ybus(m,m) + y(n)/(a(n)^2) + b(n); elseif tb(n) == m ybus(m,m) = ybus(m,m) + y(n) + b(n); end end end
57
APPENDIX 3: Newton Raphson Solver % Program for Improved Newton-Raphson Load Flow Analysis tic nbus = 30;% IEEE-30 num = nbus; nb = nbus; Y = ybusppg(); % Calling ybusppg.m.. busdt = busdatas(num); % Calling Busdata. BMva = 100; % Base MVA.. bus = busdt(:,1); % Bus Number.. type = busdt(:,2); % Type of Bus 1-Slack, 2-PV, 3-PQ.. V = busdt(:,3); del = busdt(:,4); % Voltage Angle.. Pg = busdt(:,5); % PGi.. Qg = busdt(:,6); % QGi.. Pl = busdt(:,7); % PLi.. Ql = busdt(:,8); % QLi.. Qmin = busdt(:,9); % Minimum Reactive Power Limit.. Qmax = busdt(:,10); % Maximum Reactive Power Limit.. nbus = max(bus); % To get no. of buses.. P = Pg - Pl; % Pi = PGi - PLi.. Q = Qg - Ql; % Qi = QGi - QLi.. Psp = P; % P Specified.. Qsp = Q; % Q Specified.. G = real(Y); % Conductance matrix.. B = imag(Y); % Susceptance matrix.. pv = find(type == 2); % PV Buses.. pq = find(type == 3); % PQ Buses.. npv = length(pv); % No. of PV buses.. npq = length(pq); % No. of PQ buses..
Tol = 1; Iter = 1; while (Tol > 1e-2) % Iteration starting..
P = zeros(nbus,1); Q = zeros(nbus,1); LTpij = zeros(1, 1); Pgsum = zeros (1,1); Ki = zeros(nbus,1);
% Calculate P and Q for i = 1:nbus for k = 1:nbus P(i) = P(i) + V(i)* V(k)*(G(i,k)*cos(del(i)-del(k)) +
B(i,k)*sin(del(i)-del(k)))+ Ki(i)*(LTpij); Q(i) = Q(i) + V(i)* V(k)*(G(i,k)*sin(del(i)-del(k)) -
B(i,k)*cos(del(i)-del(k))); end end
%Defining Participating Buses Pg = P + Pl; for i = 1:nbus; Pgsum = sum(Pg(1) +Pg(2)+ Pg(13) + Pg(22) +Pg(23) +Pg(27));
58
end
% Defining Participation Factors for i = 1:nbus for m = bus(i); if type(m) == 2 Ki(i) = Pg(i) / Pgsum; else Ki(i) = 0; end end end
% Calculate change from specified value dPa = Psp-P; dQa = Qsp-Q; k = 1; dQ = zeros(npq,1); for i = 1:nbus if type(i) == 3 dQ(k,1) = dQa(i); k = k+1; end end dP = dPa(1:nbus); M = [dP; dQ]; % Mismatch Vector
% Jacobian % J1 - Derivative of Real Power Injections with Angles.. J1 = zeros(nbus,nbus); for i = 1:(nbus) m = bus(i); for k = 1:(nbus) n = bus(k); if n == m for n = 1:nbus J1(i,k) = J1(i,k) + V(m)* V(n)*(-G(m,n)*sin(del(m)-
del(n)) + B(m,n)*cos(del(m)-del(n))); end J1(i,k) = J1(i,k) - V(m)^2*B(m,m); else J1(i,k) = V(m)* V(n)*(G(m,n)*sin(del(m)-del(n)) -
B(m,n)*cos(del(m)-del(n))); end end end
% J2 - Derivative of Real Power Injections with V.. J2 = zeros(nbus,npq); for i = 1:(nbus) m = bus(i); for k = 1:npq n = pq(k); if n == m for n = 1:nbus
59
J2(i,k) = J2(i,k) + V(n)*(G(m,n)*cos(del(m)-del(n)) +
B(m,n)*sin(del(m)-del(n))); end J2(i,k) = J2(i,k) + V(m)*G(m,m); else J2(i,k) = V(m)*(G(m,n)*cos(del(m)-del(n)) +
B(m,n)*sin(del(m)-del(n))); end end end
% J3 - Derivative of Reactive Power Injections with Angles.. J3 = zeros(npq,nbus); for i = 1:npq m = pq(i); for k = 1:(nbus) n = bus(k); if n == m for n = 1:nbus J3(i,k) = J3(i,k) + V(m)* V(n)*(G(m,n)*cos(del(m)-del(n))
+ B(m,n)*sin(del(m)-del(n))); end J3(i,k) = J3(i,k) - V(m)^2*G(m,m); else J3(i,k) = V(m)* V(n)*(-G(m,n)*cos(del(m)-del(n)) -
B(m,n)*sin(del(m)-del(n))); end end end
% J4 - Derivative of Reactive Power Injections with V.. J4 = zeros(npq,npq); for i = 1:npq m = pq(i); for k = 1:npq n = pq(k); if n == m for n = 1:nbus J4(i,k) = J4(i,k) + V(n)*(G(m,n)*sin(del(m)-del(n)) -
B(m,n)*cos(del(m)-del(n))); end J4(i,k) = J4(i,k) - V(m)*B(m,m); else J4(i,k) = V(m)*(G(m,n)*sin(del(m)-del(n)) -
B(m,n)*cos(del(m)-del(n))); end end end
% J5 - Derivative of Real Power Injections with Loss i.e. Participation
Factors J5 = zeros (nbus, 1); for i = 1:(nbus) m = bus(i); if type(m) == 2 J5(i) = Ki(i);
60
else J5(i) = 0; end end
%J6 - Derivative of Reactive Power Injection with Loss J6 = zeros (npq, 1);
J = [J1 J2 J5; J3 J4 J6]; % Jacobian Matrix.. X = pinv(J) * M; % Correction Vector dTh = X(1:nbus); % Change in Voltage Angle.. dV = X(nbus+1:end - 1); % Change in Voltage Magnitude.. dLTpij = X(end : end); % Updating State Vectors.. del(1:nbus) = dTh + del(1:nbus); % Voltage Angle.. k = 1; for i = 1:nbus if type(i) == 3 V(i) = dV(k) + V(i); % Voltage Magnitude.. k = k + 1; end end LTpij = dLTpij + LTpij; %Participation Factor Term
Iter = Iter + 1; % Checking Generator Limits for n = 1:nbus if Q(n) < Qmin(n) Q(n) = Qmin(n); busd(n,2) = 3; end if Q(n) > Qmax(n) Q(n) = Qmax(n); busd(n,2) = 3; end end
Tol = max(abs(M)); % Tolerance..7
end loadflow(nbus,V,del,BMva); % Calling Loadflow.m..
61
APPENDIX 4: Newton Raphson Solver with STATCOM % Program for Newton-Raphson Load Flow with STATCOM
nbus = 30;% IEEE-30 num = nbus; nb = nbus; Y = ybusppg(); % Calling ybusppg.m.. busdt = busdatas(num); % Calling Busdata.. statdata = statdata30(); % Statcom Data.. BMva = 100; % Base MVA.. bus = busdt(:,1); % Bus Number.. type = busdt(:,2); % Type of Bus 1-Slack, 2-PV, 3-PQ.. V = busdt(:,3); % Specified Voltage.. del = busdt(:,4); % Voltage Angle.. Pg = busdt(:,5); % PGi.. Qg = busdt(:,6); % QGi.. Pl = busdt(:,7); % PLi.. Ql = busdt(:,8); % QLi.. Qmin = busdt(:,9); % Minimum Reactive Power Limit.. Qmax = busdt(:,10); % Maximum Reactive Power Limit.. nbus = max(bus); % To get no. of buses.. P = Pg - Pl; % Pi = PGi - PLi.. Q = Qg - Ql; % Qi = QGi - QLi.. P = P/BMva; % Converting to p.u.. Q = Q/BMva; Qmin = Qmin/BMva; Qmax = Qmax/BMva; Tol = 1; Iter = 1; Psp = P; Qsp = Q; G = real(Y); % Conductance.. B = imag(Y); % Susceptance.. Vsp = V;
% Details of STATCOM statb = statdata(:,1); % Buses at which statcoms are placed.. Vsh = statdata(:,2); Thst = statdata(:,3); Qsmx = statdata(:,4); Qsmn = statdata(:,5); gsh = 0.9901; bsh = -9.901; Vshmx = 1.1; Vshmn = 0.9; Thstmx = pi; Thstmn = -pi; np = length(statb); % Number of STATCOMs..
pv = find(type == 2); % Index of PV Buses.. pq = find(type == 3); % Index of PQ Buses..
npv = length(pv); % Number of PV buses.. npq = length(pq); % Number of PQ buses..
while (Tol > 1e-1 && Iter <= 50) % Iteration starting..
62
P = zeros(nbus,1); Q = zeros(nbus,1); LTpij = zeros(1, 1); Pgsum = zeros (1,1); Ki = zeros(nbus,1); % Calculate P and Q for i = 1:nbus for k = 1:nbus P(i) = P(i) + V(i)* V(k)*(G(i,k)*cos(del(i)-del(k)) +
B(i,k)*sin(del(i)-del(k)))+ Ki(i)*(LTpij); Q(i) = Q(i) + V(i)* V(k)*(G(i,k)*sin(del(i)-del(k)) -
B(i,k)*cos(del(i)-del(k))); end m = find(statb == i); if ~isempty(m) P(i) = P(i) + V(i)^2*gsh - V(i)*Vsh(m)*(gsh*cos(del(i)-Thst(m)) +
bsh*sin(del(i)-Thst(m)))+ Ki(i)*(LTpij); Q(i) = Q(i) - V(i)^2*bsh + V(i)*Vsh(m)*(bsh*cos(del(i)-Thst(m)) -
gsh*sin(del(i)-Thst(m))); end end
%Defining Participating Buses for i = 1:nbus-1 m = i+1; if type(m) == 2 Pg = P+ Pl; Pgsum = sum (Pg); end end % Defining Participation Factors for i = 1:nbus-1 for m = i+1; if type(m) == 2 Ki(i) = Pg(i) / Pgsum; Ki = abs(Ki); else Ki(i) = 0; end end end % Checking Q-limit violations.. if Iter >= 2 for n = 1:nbus if type(n) == 2 if Q(n) < Qmin(n) V(n) = V(n) + 0.01; elseif Q(n) > Qmax(n) V(n) = V(n) - 0.01; end end end end
% Calculating PE, PE = zeros(np,1); for i = 1:np
63
m = statb(i); PE(i) = Vsh(i)^2*gsh - V(m)*Vsh(i)*(gsh*cos(del(m)-Thst(i)) -
bsh*sin(del(m)-Thst(i))); end
% Calculate F, Control Variables F = zeros(np,1); for i = 1:np m = statb(i); F(i) = V(m) - Vsp(m); end
% Calculate change from specified value dPa = Psp-P; dQa = Qsp-Q; dQ = zeros(npq,1); k = 1; for i = 1:nbus if type(i) == 3 dQ(k,1) = dQa(i); k = k+1; end end dP = dPa(2:nbus); dPE = -PE; dF = -F; M = [dP; dQ; dPE; dF]; % Mismatch Vector
% Jacobian % J11 - Derivative of Real Power Injections with Angles.. J11 = zeros(nbus-1,nbus-1); for i = 1:(nbus-1) m = i+1; p = find(statb == m); for k = 1:(nbus-1) q = k+1; if q == m for n = 1:nbus J11(i,k) = J11(i,k) + V(m)* V(n)*(-G(m,n)*sin(del(m)-
del(n)) + B(m,n)*cos(del(m)-del(n))); end J11(i,k) = J11(i,k) - V(m)^2*B(m,m); if ~isempty(p) J11(i,k) = J11(i,k) + V(m)*Vsh(p)*(gsh*sin(del(m)-
Thst(p)) - bsh*cos(del(m)-Thst(p))); end else J11(i,k) = V(m)* V(q)*(G(m,q)*sin(del(m)-del(q)) -
B(m,q)*cos(del(m)-del(q))); end end end
% J12 - Derivative of Real Power Injections with V.. J12 = zeros(nbus-1,npq); for i = 1:(nbus-1)
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m = i+1; p = find(statb == m); for k = 1:npq q = pq(k); if q == m for n = 1:nbus J12(i,k) = J12(i,k) + V(n)*(G(m,n)*cos(del(m)-del(n)) +
B(m,n)*sin(del(m)-del(n))); end J12(i,k) = J12(i,k) + V(m)*G(m,m); if ~isempty(p) J12(i,k) = J12(i,k) + 2*V(m)*gsh -
Vsh(p)*(gsh*cos(del(m)-Thst(p)) + bsh*sin(del(m)-Thst(p))); end else J12(i,k) = V(m)*(G(m,q)*cos(del(m)-del(q)) +
B(m,q)*sin(del(m)-del(q))); end end end
% J13 - Derivative of Real Power Injections with Vsh.. % J14 - Derivative of Real Power Injections with Thsh.. J13 = zeros(nbus-1,np); J14 = zeros(nbus-1,np); for i = 1:(nbus-1) m = i+1; for k = 1:np p = statb(k); if m == p J13(i,k) = -V(m)*(gsh*cos(del(m)-Thst(k)) + bsh*sin(del(m)-
Thst(k))); J14(i,k) = -V(m)*Vsh(k)*(gsh*sin(del(m)-Thst(k)) -
bsh*cos(del(m)-Thst(k))); end end end
% J15 - Derivative of Real Power Injections with Loss i.e. Participation
Factors J15 = zeros (nbus-1, 1); for i = 1:(nbus-1) m = i+1; if type(m) == 2 J15(i) = Ki(i); else J15(i) = 0; end end %J6 - Derivative of Reactive Power Injection with Loss J16 = zeros (nbus-1, 1);
% J21 - Derivative of Reactive Power Injections with Angles.. J21 = zeros(npq,nbus-1); for i = 1:npq m = pq(i); p = find(statb == m);
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for k = 1:(nbus-1) q = k+1; if q == m for n = 1:nbus J21(i,k) = J21(i,k) + V(m)* V(n)*(G(m,n)*cos(del(m)-
del(n)) + B(m,n)*sin(del(m)-del(n))); end J21(i,k) = J21(i,k) - V(m)^2*G(m,m); if ~isempty(p) J21(i,k) = J21(i,k) - V(m)*Vsh(p)*(gsh*cos(del(m)-
Thst(p)) + bsh*sin(del(m)-Thst(p))); end else J21(i,k) = -V(m)* V(q)*(G(m,q)*cos(del(m)-del(q)) +
B(m,q)*sin(del(m)-del(q))); end end end
% J22 - Derivative of Reactive Power Injections with V.. J22 = zeros(npq,npq); for i = 1:npq m = pq(i); p = find(statb == m); for k = 1:npq q = pq(k); if q == m for n = 1:nbus J22(i,k) = J22(i,k) + V(n)*(G(m,n)*sin(del(m)-del(n)) -
B(m,n)*cos(del(m)-del(n))); end J22(i,k) = J22(i,k) - V(m)*B(m,m); if ~isempty(p) J22(i,k) = J22(i,k) - 2*V(m)*bsh -
Vsh(p)*(gsh*sin(del(m)-Thst(p)) - bsh*cos(del(m)-Thst(p))); end else J22(i,k) = V(m)*(G(m,q)*sin(del(m)-del(q)) -
B(m,q)*cos(del(m)-del(q))); end end end
% J23 - Derivative of Reactive Power Injections with Vsh.. % J24 - Derivative of Reactive Power Injections with Thsh.. J23 = zeros(npq,np); J24 = zeros(npq,np); for i = 1:npq q = pq(i); m = i+1; for k = 1:np p = statb(k); if q == p J23(i,k) = -V(m)*(gsh*sin(del(m)-Thst(k)) - bsh*cos(del(m)-
Thst(k)));
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J24(i,k) = V(m)*Vsh(k)*(gsh*cos(del(m)-Thst(k)) +
bsh*sin(del(m)-Thst(k))); end end end % J25 - Derivative of Real Power Injections with Loss i.e. Participation
Factors J25 = zeros (npq, 1); for i = 1:(npq) m = i+1; if type(m) == 2 J25(i) = Ki(i); else J25(i) = 0; end end %J26 - Derivative of Reactive Power Injection with Loss J26 = zeros (npq, 1);
% J31 - Derivative of PE with Angles.. % J41 - Derivative of F with Angles.. J31 = zeros(np,nbus-1); J41 = zeros(np,nbus-1); for i = 1:np m = statb(i); for k = 1:(nbus-1) if m == k+1 J31(i,k) = V(m)*Vsh(i)*(gsh*sin(del(m)-Thst(i)) +
bsh*cos(del(m)-Thst(i))); end end end
% J32 - Derivative of PE with V.. % J42 - Derivative of F with V.. J32 = zeros(np,npq); J42 = zeros(np,npq); for i = 1:np m = statb(i); for k = 1:npq if m == pq(k) J32(i,k) = -Vsh(i)*(gsh*cos(del(m)-Thst(i)) - bsh*sin(del(m)-
Thst(i))); end if m == pq(k) J42(i,k) = 1; end end end
% J33 - Derivative of PE with Vsh.. % J34 - Derivative of PE with Thsh.. % J43 - Derivative of F with Vsh.. % J44 - Derivative of F with Thsh.. J33 = zeros(np,np); J34 = zeros(np,np); J43 = zeros(np,np);
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J44 = zeros(np,np); for i = 1:np m = statb(i); for k = 1:np p = statb(k); if m == p J33(i,k) = 2*Vsh(k)*gsh - V(m)*(gsh*cos(del(m)-Thst(k)) -
bsh*sin(del(m)-Thst(k))); J34(i,k) = -V(m)*Vsh(k)*(gsh*sin(del(m)-Thst(k)) +
bsh*cos(del(m)-Thst(k))); end end end % J35 - Derivative of Real Power Injections with Loss i.e. Participation
Factors J35 = zeros (np, 1); for i = 1:np m = i+1; if type(m) == 2 J35(i) = Ki(i); else J35(i) = 0; end end %J36 - Derivative of Reactive Power Injection with Loss J36 = zeros (np, 1);
% J45 - Derivative of Real Power Injections with Loss i.e. Participation
Factors J45 = zeros (np, 1); for i = 1:np m = i+1; if type(m) == 2 J45(i) = Ki(i); else J45(i) = 0; end end %J46 - Derivative of Reactive Power Injection with Loss J46 = zeros (np, 1); J = [J11 J12 J13 J14 J15 J16; J21 J22 J23 J24 J25 J26; J31 J32 J33
J34 J35 J36; J41 J42 J43 J44 J45 J46]; % Jacobian clear J11 J12 J13 J14 J21 J22 J23 J24 J31 J32 J33 J34 J41 J42 J43 J44
X = pinv(J)*M; % Correction Vector dTh = X(1:nbus-1); dV = X(nbus:nbus+npq-1); dLTpij = X(end:end); dVsh = X(nbus+npq:nbus+npq+np-1); dThst = X(nbus+npq+np:nbus+npq+2*np-1); del(2:nbus) = dTh + del(2:nbus); k = 1; for i = 2:nbus if type(i) == 3 V(i) = dV(k) + V(i); k = k+1; end end
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LTpij = dLTpij + LTpij; %Participation Factor Term
Iter = Iter + 1; % Checking Generator Limits for n = 1:nbus if Q(n) < Qmin(n) Q(n) = Qmin(n); busd(n,2) = 3; end if Q(n) > Qmax(n) Q(n) = Qmax(n); busd(n,2) = 3; end end Vsh = Vsh + dVsh; Thst = Thst + dThst;
% Calculate Qsh.. Qsh = zeros(np,1); for m = 1:np i = statb(m); Qsh(m) = -V(i)^2*bsh + V(i)*Vsh(m)*(bsh*cos(del(i)-Thst(m)) -
gsh*sin(del(i)-Thst(m))); end Iter = Iter + 1; Tol = max(abs(M)); end
Iter = Iter - 1; % Number of Iterations took.. V; Del = 180/pi*del; Thst = 180/pi*Thst; E2 = [V Del]; % Bus Voltages and angles..
loadflow(nb,V,del,BMva);
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APPENDIX 5: Program for Calculating Power Outputs, Line Flows and Losses % Program for Bus Power Injections, Line & Power flows (p.u)...
function [Pi Qi Pg Qg Pl Ql] = loadflow(nb,V,del,BMva)
Y = ybusppg(); % Calling Ybus program.. lined = linedatas(nb); % Get linedats.. busd = busdatas(nb); % Get busdatas.. Vm = pol2rect(V,del); % Converting polar to rectangular.. Del = 180/pi*del; % Bus Voltage Angles in Degree... fb = lined(:,1); % From bus number... tb = lined(:,2); % To bus number... nl = length(fb); % No. of Branches.. Pl = busd(:,7); % PLi.. Ql = busd(:,8); % QLi..
Iij = zeros(nb,nb); Sij = zeros(nb,nb); Si = zeros(nb,1);
% Bus Current Injections.. I = Y*Vm; Im = abs(I); Ia = angle(I);
%Line Current Flows.. for m = 1:nl p = fb(m); q = tb(m); Iij(p,q) = -(Vm(p) - Vm(q))*Y(p,q); % Y(m,n) = -y(m,n).. Iij(q,p) = -Iij(p,q); end Iij = sparse(Iij); Iijm = abs(Iij); Iija = angle(Iij);
% Line Power Flows.. for m = 1:nb for n = 1:nb if m ~= n Sij(m,n) = Vm(m)*conj(Iij(m,n))*BMva; end end end Sij = sparse(Sij); Pij = real(Sij); Qij = imag(Sij);
% Line Losses.. Lij = zeros(nl,1); for m = 1:nl p = fb(m); q = tb(m); Lij(m) = Sij(p,q) + Sij(q,p); end Lpij = real(Lij); Lqij = imag(Lij);
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% Bus Power Injections.. for i = 1:nb for k = 1:nb Si(i) = Si(i) + conj(Vm(i))* Vm(k)*Y(i,k)*BMva; end end Pi = real(Si); Qi = -imag(Si); Pg = Pi+Pl; Qg = Qi+Ql;
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