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OPTIMAL TUNING OF POWER SYSTEM STABILIZERS BASED ON EVOLUTION ALGORITHM
Prepared for
Professor K.A. Folly
Associate professor
Department of Electrical Engineering
University of Cape Town
Prepared by
Tshina Mulumba
Electrical Engineering Student
Department of Electrical Engineering
University of Cape Town
13th May 2008
Prepared as a prerequisite and in partial fulfilment for the awarding of a
Bachelor of Science (BSc) degree in Electrical Engineering at the University
of Cape Town
DECALRATION
I, Tshina Mulumba, hereby declare that this thesis project, submitted for the fulfilment
of the Bachelor of Science degree in Engineering, is my own work.
I have not plagiarised from any sources. References and acknowledgments of
sources are given and cited.
Date…..………………………………………….
Signature………………………………………..
i
ACKNOWLEDGMENTS
Firstly, I am grateful to The One who has given me life, supported me through
hardship and good time, the source of my strength, The Almighty God. I dedicate this
work to Him.
I would like to thank my supervisor, Professor K.A. Folly, for his inspirational
guidance and encouragement demonstrated throughout my research. He has
influenced me to develop particular interests in the Power Systems stability and
Control.
A special word of thank to Professor J. Greene, for the time and help he gave me by
answering all my questions.
The biggest thank and gratitude goes to my parents, Mr. and Mrs. Mulumba, for their
everlasting love and advices given, their supports and encouragements throughout
my studies.
I owe a special thank to my Aunts (Ivette and Nadine) and my big sister (Sophie), for
their continual supports. My three brothers (Dilan, Glory and Christian), for being a
source of inspiration, as well as my little sisters.
Last but not least, a special word of gratitude to all my friends who have contributed
in completing this report, I could not have done this without you.
ii
TERMS OF REFERENCE
Associate Professor K.A Folly of the Department of Electrical Engineering, University
of Cape Town, initiated this research on the Optimal Tuning of Power System
Stabilizer (PSS) using Evolution Algorithm.
His specific directives were:
• To review the modelling of a single machine infinite bus (SMIB) equipped with a
PSS.
• To investigate the tuning method of a Power System Stabilizer (PSS) in SMIB
system by using specifically the Differential Evolution (a variant of Genetic
Algorithm).
• To develop a program in MATLAB and implement the Differential Evolution (DE)
• To chose an appropriate objective function for the DE.
• To analyse and compare the simulations with the CPSS, the simple GA and the
DE .
iii
SYNOPSIS
The problem of damping low frequency oscillations in the range of 0.2 – 3 Hz
observed in power systems, have been the subject of many researches over the past
few years. These low frequencies are mainly caused by a heavy power transmission
on weak transmission lines and the exciter’s high gain.
The exciter, also known as the automatic voltage regulator (AVR), helps to improve
the system voltage during faults conditions. However, the AVR adversely reduces the
damping of the system causing oscillations. These oscillations limit the power
transmission capability of a network and, sometimes, even cause a loss of
synchronism and an eventual breakdown of the entire system [13].
To remediate this problem, the Power System Stabilizer (PSS) is used to improve the
system stability by providing supplementary damping. The PSS is mainly constituted
of a gain stage K, washout stage Tw or high pass filter and a lead – lag compensator
T1 – T4. These parameters are tuned at a particular operating condition with
conventional techniques such as phase compensation and root locus to compensate
for the system’s phase lag. Although many modern control techniques with different
structures such as adaptive control have been developed, the conventional lead – lag
PSS (CPSS) remains widely used by power industry because of its simple structure
and reliability. However, the main problem that faces the CPSS is the nonlinearity of
power systems of which the operating conditions change constantly. Under these
conditions the CPSS performance becomes inadequate considering that it is only
tuned for a particular operating condition.
The above problem has led to many researchers to investigate methods to improve
the PSS performance over the entire range of operating conditions. In the past 15
years, interests have been focused on the optimization of the PSS parameters to
provide adequate performance for all conditions. Hence, many optimizations
techniques such as Genetic Algorithms (GAs) have been used to find the optimum
set of parameters to effectively tune the PSS.
Recently, an optimization technique similar to GA, known as Differential Evolution
(DE), has been proposed.
iv
DE is parallel direct search method that uses a differential mutation scheme and
greedy selection process (the better one of new solution and its parents wins the
competition) to direct its search toward the prospective regions of search space. DE
is a new heuristic approach that present some advantages over GAs:
• Fast convergence
• Finds true global minimum regardless of the initial parameters
• Ease of use
• Efficient memory utilization
• Lower computational complexity
In this thesis, DE is used to tune the PSS’s parameters by optimizing a frequency
domain objective function.
The objective function consists of finding, shifting the poorly damped or unstable
polesinto the left side of the s – plane (stability plane). The efficacy of the method
was tested on a single machine infinite bus (SMIB) and compared with GA based
PSS (GAPSS) and the CPSS. The results were as follows:
• DEPSS provides better damping than GAPSS and CPSS.
• DEPSS is more robust than GAPSS and CPSS. As the stressed level increases,
DEPSS provides better results.
• DEPSS is able to restrain all eigenvalues in the desired region, indicate by the
relative stability, for most of the operating conditions.
In order to contribute to the research in the area of PSS tuning strategies, the
following recommendations were made:
• Investigate the application of DE to tune the PSS using an objective function
based on the phase compensation technique and compare with DE based
eigenvalue shifting method.
v
• Investigate the application of DE to tune the PSS in the area of multimachine
system.
• Investigate the application of other Evolution Algorithm to tune the PSS with
faster convergence time.
vi
TABLE OF CONTENTS
Page
DECALRATION ....................................... ......................................................... i
ACKNOWLEDGMENTS ................................... ................................................ i
TERMS OF REFERENCE ............................................................................... ii
SYNOPSIS ..................................................................................................... iii
TABLE OF CONTENTS ................................. ................................................ vi
LIST OF TABLES .................................... ........................................................ x
LIST OF FIGURES ......................................................................................... xi
1 INTRODUCTION ................................................................................ 1
1.1 Subject of the thesis ............................. ............................................ 1
1.2 Problem definition ................................ ............................................ 1
1.3 Objectives of the thesis........................... ......................................... 3
1.4 Scope and Limitation............................... ......................................... 3
1.5 Thesis outline .................................... ................................................ 4
2 REVIEW OF POWER SYSTEM STABILITY .................. .................... 5
2.1 Small signal stability ............................ ............................................ 7
2.2 Low Frequency Oscillations in Power System ........ ....................... 8
2.2.1 Local modes ....................................... ....................................................... 8
2.2.2 Inter – area modes ................................ .................................................... 9
2.3 Power System Stabilizer ........................... ....................................... 9
2.3.1 Conventional Power System Stabilizer structure and design .............. 10
2.3.2 Genetic Algorithm based PSS (GAPSS) ............... ................................. 12
3 POWER SYSTEM MODELLING ............................ .......................... 15
vii
3.1 Small Signal Dynamic Modelling .................... ............................... 15
3.1.1 State – Space representation ...................... ........................................... 15
3.1.2 Linearization ..................................... ....................................................... 16
3.2 Machine modelling ................................. ........................................ 17
3.2.1 Machine Modelling with Power System Stabilizer .... ............................ 18
3.3 Stability analysis ................................ ............................................. 19
4 GENETIC ALGORITHM ................................. .................................. 22
4.1 Encoding of individuals ........................... ...................................... 23
4.2 Objective and Fitness functions ................... ................................. 23
4.3 SGA operators ..................................... ........................................... 24
4.3.1 Selection ......................................... ......................................................... 24
4.3.2 Crossover ......................................... ....................................................... 25
4.3.3 Mutation .......................................... ......................................................... 25
4.3.4 Reinsertion ....................................... ....................................................... 25
4.3.5 Convergence and termination ....................... ......................................... 25
5 DIFFERENTIAL EVOLUTION (DE) ....................... ........................... 27
5.1 Advantage of DE over GA ........................... ................................... 27
5.2 Reason for using DE ............................... ....................................... 27
5.3 Population structure .............................. ......................................... 28
5.4 Initialization .................................... ................................................. 29
5.5 Mutation .......................................... ................................................. 29
5.6 Recombination or crossover ........................ ................................. 30
5.7 Selection ......................................... ................................................. 31
6 PARAMETERS SETTING OF GA & DE ..................... ..................... 32
6.1 Objective function ................................ .......................................... 32
viii
6.2 PSS tuning approach................................ ...................................... 35
6.2.1 Simple Genetic Algorithm .......................... ............................................ 35
6.2.2 Differential Evolution (DE) ....................... ............................................... 36
6.2.3 Tuning process .................................... ................................................... 37
7 SIMULATION RESULTS AND DISCUSSIONS ................ ............... 40
7.1 Power System to be investigated ................... ............................... 40
7.2 Operating conditions .............................. ........................................ 40
7.3 PSS parameters .................................... .......................................... 41
7.3.1 CPSS parameter selection .......................... ............................................ 41
7.3.2 DEPSS and GAPSS parameters selection .............. ............................... 42
7.4 Simulations results ............................... .......................................... 43
7.4.1 Eigenvalues analysis .............................. ................................................ 44
7.4.2 Time domain response .............................. ............................................. 50
7.4.3 Robustness tests of PSSs .......................... ............................................ 52
8 CONCLUSION .................................................................................. 58
9 RECOMMENDATIONS .................................................................... 59
REFERENCES .............................................................................................. 60
APPENDIX A: SYSTEM EQUATIONS ...................... .................................... 66
A.1 EQUATIONS OF STATE SQUARE MATRIX VARIABLES ........ ..... 66
A.2 HEFFRON – PHILLIPS DIAGRAM AND CONSTANTS .......... ........ 67
APPENDIX B: TUNING GUIDELINES FOR A CPSS .......... ......................... 69
APPENDIX C: SYSTEM OPERATION ...................... .................................... 70
C.1 SINGLE MACHINE INFINITE BUS DATA .................. ..................... 70
C.2 LOAD FLOW REPORTS ................................. ................................. 71
APPENDIX D: MATLAB CODES & SIMULATION TOOLS ....... ................... 73
ix
D.1 SIMULATION TOOLS .................................. .................................... 73
Power System Toolbox .............................. ................................................. 73
Operations ........................................ ............................................................ 73
smibDriver.m ...................................... ................................................................... 74
Model_maker & Step_testerMod.m .................... .................................................. 74
pssOptimizer.m .................................... ................................................................. 75
dePSS.m ........................................... ..................................................................... 76
gaPSS.m ........................................... ..................................................................... 76
eigenShiftDE.m and eigenShiftGA.m ................. .................................................. 77
Program flow chart ................................ ................................................................ 77
D.2 MATLAB CODES ...................................... ....................................... 78
PSSOPTIMIZER.M ........................................................................................ 78
DEPSS.M ....................................................................................................... 80
EIGENSHIFTDE.M ........................................................................................ 82
GAPSS.M ...................................................................................................... 84
EIGENSHIFTGA.M ........................................................................................ 86
APPENDIX E: SIMULATION RESULTS DATA & GRAPHS ...... .................. 88
E.1 DATA .............................................. .................................................. 88
E.2 GRAPHS .......................................................................................... 94
APPENDIX F: SOFTWARE CD ........................... ........................................ 101
x
LIST OF TABLES
Table 6.1 SGA parameters .................................................................................... 35
Table 6.2 DE parameters ....................................................................................... 37
Table 7.1 Operating conditions ............................................................................... 41
Table 7.2 CPSS parameters ................................................................................... 42
Table 7.3 Parameters boundaries .......................................................................... 43
Table 7.4 DEPSS and GAPSS parameters ............................................................. 43
Table 7.5 No PSS at minimum condition ................................................................ 44
Table 7.6 No PSS at nominal condition ................................................................... 45
Table 7.7 CPSS at nominal condition ...................................................................... 45
Table 7.8 GAPSS at nominal condition .................................................................. 46
Table 7.9 DEPSS at nominal condition .................................................................. 47
Table 7.10 No PSS maximum condition .................................................................. 47
Table 7.11 CPSS at maximum condition ................................................................. 48
Table 7.12 GAPSS at maximum condition .............................................................. 49
Table 7.13 DEPSS at maximum condition ............................................................. 49
Table 7.14 CPSS eigenvalues and Damping ration under robust test .................... 52
Table 7.15 GAPSS eigenvalues and damping ratio under robust test ..................... 53
Table 7.16 DEPSS eigenvalues and damping ratio under robust test .................... 53
Table 7.17 Average PSSs damping ratio ............................................................... 54
xi
LIST OF FIGURES
Figure 2.1 Power system classification [2, 6] ........................................................... 6
Figure 2.2 CPSS structure ..................................................................................... 10
Figure 3.1 S – plane representation [50] ............................................................... 21
Figure 4.1 Chromosome structure constituted of 2 variables .................................. 23
Figure 4.2 The roulette wheel ................................................................................. 24
Figure 4.3 Single point crossover ........................................................................... 26
Figure 4.4 Mutated individual ................................................................................. 26
Figure 5.1 Differential Evolution cycles ................................................................... 28
Figure 5.2 Differential Evolution: the weighted differential, 1, 2,( )r g r gF x x⋅ − , is added
to the base vector, 0,r gx , to produce a mutant, ,i gv [48, 49]. ......................... 30
Figure 5.3 A flow chart of DE’s operation and test loop [48] .................................... 31
Figure 6.1 Relative stability region on the left hand side of the line at α− ............ 34
Figure 6.2 Flow chart representation of the tuning process ..................................... 39
Figure 7.1 : SMIB system ....................................................................................... 40
Figure 7.2 : Step response at Nominal condition ..................................................... 50
Figure 7.3: Step response at Maximum condition ................................................... 51
Figure 7.4: Open - loop eigenvalues ....................................................................... 55
Figure 7.5: CPSS closed - loop eigenvalues .......................................................... 55
Figure 7.6: GAPSS closed-loop eigenvalues .......................................................... 56
xii
Figure 7.7 : DEPSS closed-loop eigenvalues .......................................................... 57
Figure A.9.1 Heffron – Philips 3rd model of SMIB system with PSS included .......... 67
Figure D.9.2 Program flow chart ............................................................................ 77
1
1 INTRODUCTION
1.1 Subject of the thesis
This thesis explores the possibility of tuning the Power System Stabilizer (PSS) using
the Differential Evolution (DE). The main objective focuses on finding the optimal
parameters that ensures a robust system over a wide range of operating conditions.
1.2 Problem definition
Small signal disturbances observed on the power system are caused by many
factors such as heavy power transmitted over weak tie – line and the effect of fast
acting, high gain automatic voltage regulator (AVRs) [6, 7, 52].
The main function of the AVR is to improve the transient stability during faults
conditions. However, its high gain and fast acting effect, have an adverse effect on
the system damping which is reduced to a negative value [2, 15, 52]. The
underdamped system exhibits low frequency oscillations also known as
electromechanical oscillations. These oscillations limit the power transfer over the
network and if not properly damped, they can grow in magnitude to cause system
separation.
To counteract the adverse effects of the AVRS, Power system stabilizer (PSS) is
used in the auxiliary feedback to provide supplementary damping to the system to
damp these low frequency oscillations on the rotor [5].
The PSS, also referred to as conventional PSS (CPSS), is made of gain stage K, a
high pass filter and the lead – lag compensators, with T1 – T4 as time constants.
These parameters require fine tuning at a particular set of operating conditions,
usually nominal, in order to improve the system damping. As the power system is
extremely nonlinear, operating conditions are constantly changing. Therefore, the
CPSS’s parameters may not provide adequate performance and may need to be
retuned.
Thus, finding a set of parameters that guarantee adequate damping and good
system performance over the entire range of operating conditions is essential.
2
To overcome this problem, several approaches based on modern control theory,
such as Optimal control, Variable control and Intelligent control were simulated and
tested with satisfactory results. But these stabilizers have been proved to be difficult
to implement in real systems [2, 13]. Thus, CPSS remains widely used by power
utilities for its simple structure and reliability [13].
Over the past 15 years, interests have been focused on the optimization of the PSS
parameters to provide adequate performance for all operating conditions. Hence,
many optimizations techniques such as Genetic Algorithms (GAs) have been used to
find the optimum set of parameters to effectively tune the PSS.
These optimization techniques have demonstrated to be slow when converging
toward optimum values. They require complex computation and memory.
Recently, an optimization technique similar to GA, known as Differential Evolution
(DE), has been proposed.
In fact, DE is new heuristic approach that uses differential mutation scheme, to direct
its search toward the prospective regions of search space [47, 48], and present some
advantages over GAs:
• Fast convergence
• Finds true global minimum regardless of the initial parameters
• Ease of use
• Efficient memory utilization
• Lower computational complexity
For these reasons, this thesis aims to investigate the application of DE to optimally
tune PSS’s parameters in order to ensure robust system performance. The results
will be compared to GA based PSS and the CPSS.
3
1.3 Objectives of the thesis
The objectives of this research are formulated as follow:
• Review of the system modelling of single machine infinite bus (SMIB) equipped
with conventional power system stabilizer (CPSS)
• Select an appropriate objective function for the DE
• Design and tune the PSS parameters using DE in MATLAB
• Check for the robustness of the system over a wide range of operating conditions
• Simulate and compare the results to the simple GA as well as the CPSS
1.4 Scope and Limitation
This thesis examines the application of Differential Evolution (DE) to tune a power
system stabilizer with a set of optimum parameters that ensures adequate
performance for a robust operation.
The PSS is installed on a single machine infinite bus (SMIB), which is implemented in
MATLAB using the Power System Toolbox package (PST). The tuning of the PSS
with optimal parameters is accomplished with the aid of the DEMAT package [48].
These optimum parameters are obtained by evaluating an eigenvalue based
objective function.
The DE based PSS performance is compared to the genetic algorithm based PSS as
well as the conventional PSS over a wide range of operating conditions.
This thesis is limited to the use of PST, DEMAT and GAOT toolbox, to optimize the
parameters and simulate the results.
4
1.5 Thesis outline
The thesis chapters are outlined as follows:
• Chapter 1 introduces the thesis topic and identifies the problem to be examined.
• Chapter 2 reviews the power system stability with emphasis on the small signal
stability problem and also present the conventional power system stabilizer
structure. A review of the relevant work in the area of tuning of PSS is discussed.
The present work tuning method is briefly introduced.
• Chapter 3 reviews the mathematical modelling of a synchronous machine, the
linearization of the small signal in a single machine infinite bus system. The
system analysis based on the eigenvalues is also discussed.
• Chapter 4 reviews the basic concepts of the genetic algorithms and the genetic
operators.
• Chapter 5 reviews the Differential Evolution, outlines the main differences with
GAs and discusses in depth the DE operators.
• Chapter 6 describes the implementation of DEPSS and GAPSS. The eigenvalue
shift objective function is presented as well as the tuning procedure.
• Chapter 7 describes the software tools with its main program files used to
simulate the operation of the DEPSS.
• Chapter 8 discusses the results obtained from the simulation and provides a
comparative study between the DEPSS, GAPSS and the CPSS.
• Chapter 9 summarizes the present work and draws conclusions on the proposed
tuning method.
• Chapter 10 covers the scope for the future work in the tuning area of PSSs.
5
2 REVIEW OF POWER SYSTEM STABILITY
The stability of power system is one of the most important aspects in electric system
operations. It determines whether or not the system can settle down to a new
operating point after the occurrence of a disturbance [5 – 7]. Power system stability
is defined in [9] as follows:
“The ability of an electric power system, for a given initial operating condition, to
regain a state of operating condition after being subject to a physical disturbance,
with most system variables bounded so that practically the entire system remains
intact.”
Over the past few decades, power system stability problems have received a great
deal of attention. Many studies and techniques have been conducted and developed
to help power systems maintain the frequency and the voltage level under any
disturbances. These disturbances can occur due to a sudden increase in the load,
loss of a generator, switching of a transmission line or a fault, etc [1, 2].
The rapid growth of power demand has been recorded since the advance of
industrialization. Subsequently, more power generation infrastructures are needed.
On the other hand, severe economical and environmental restrictions are also
reinforced to preserve an ecological balance. These restrictions have limited the
generation and the expansion of the power systems transmission networks.
Consequently, modern power systems are more heavily loaded than before [3 – 5].
These constraints entail the power systems to be operated under intensive stress
conditions and near their stability limits. This implies that the tight stability margins
imposed on the power systems can be a limiting factor in the transmission of power
[1, 3 – 6].
The problem of stability in power systems is classified into three categories [8]:
i. Angle stability – refers to the ability of a power system to maintain synchronism
when subject to small and severe disturbances. These perturbations can cause
a loss of generating capacity, a system separation or blackout if no proper
actions are initiated such as load shading etc.
ii. Voltage stability – also referred to as load stability. The Voltage stability is
concerned with the ability of a power system to maintain acceptable voltage at all
6
the buses under normal conditions and after perturbations [6, 8 – 10]. Voltage
instability can lead to a low voltage profile observed in major parts of the power
system.
iii. Frequency stability – is the ability of a power system to maintain a steady
frequency within an acceptable variation range following a disturbance. The
Frequency instabilities can lead to a large generation – load imbalance [9].
The above classification provides a convenient system analysis and enhances the
understanding of the nature of instability [2, 11]. This classification allows suitable
grounds to develop solutions related to the disturbances in power system.
The power system stability is further depicted into subclasses represented by the
block diagram in Figure 2.1. The highlighted blocks show the areas of direct interest
and significance to this research.
This thesis emphasizes the Rotor angle stability by paying particular attention to the
effects of small signal instability.
Power SystemStability
FrequencyStability
VoltageStability
AngleStability
TransientStability
Small signalStability
Non-oscillatoryinstability
Oscillatoryinstability
Small disturbanceStability
Large disturbanceStability
Localmodes
Control modes
Inter-area modes
Torsionalmodes
Figure 2.1 Power system classification [2, 6]
7
From the power system stability point of view, power systems are subjected to
different types of disturbances. There are small scale disturbances, which occur for
the vast majority of time, and large scale for the more severe ones. Therefore, rotor
angle stability is subdivided into two categories known as transient stability and small
signal stability. A system is said to be transiently stable if it can withstand large
disturbances and remains stable after the perturbations. The transient stability
problem is mainly concerned with the way the system responds to a severe
disturbance such as short circuit on power line. The transient stability is related to the
short term or transient period which is usually limited to the first few seconds
following the disturbance [2, 11].
On the other hand, the power system is said to be small signal stable if the
generators are able to maintain synchronism with each other after being subject to
small disturbances. These disturbances arise with the switching of capacitors, small
and gradual generation changes [11] etc.
Small signal stability is further discussed in the next point.
2.1 Small signal stability
Small signal stability is defined in [14] as follows:
“A power system is said to be small signal stable for a particular steady – state
operating condition if, following any small disturbance, it reaches a steady – state
operating condition which is identical or close to the pre – disturbance operating
condition.”
A disturbance is considered to be of small signal if the equations describing the
system response can be linearized for the purpose of analysis [6].
The system instability resulting from small disturbances can be of two forms: (i) the
steady increase in the rotor angle due to the lack of synchronizing torque, or (ii) the
increase in rotor oscillation due to the lack of damping torque [6]. These disturbances
are mainly caused by, the exciter’s high gain and the weak links in interconnected
power systems [6, 52].
Therefore, the restructuring of the electric power industry as well as the complexity
of the networks both contribute to the deterioration of the stability margin in power
systems [6, 12 – 13]. Consequently, the systems are more vulnerable to small signal
8
disturbances nowadays than they were before. In fact, many studies on stability of
small signals have been conducted to ensure sufficient stability margins in addition to
system security and reliability.
If the small signal oscillations are not damped properly, they can build up through out
the network to cause transient instability [7].
Modern generators are equipped with high gain and fast response exciters. These
exciters, also known as Automatic Voltage Regulators (AVR), enhance the transient
stability and prevent voltage fluctuation [2, 15, 52]. It achieves the above mentioned
by simply increase the synchronising torque which reduces the generator angle and
avoid non – oscillatory instability [52] .
The AVR, conversely, contributes to low frequency oscillations by decreasing the
damping torque to a negative value [16, 52]. These oscillations lead to an unstable
condition even without the existence of severe fault [2, 14 – 15].
2.2 Low Frequency Oscillations in Power System
Low frequency oscillations are often observed when large power systems are
connected with weak tie – line [18]. In fact, when bulk power is transmitted over long
distances and weak transmission lines, oscillations of low frequency in the range of
0.2 to 3 Hz can be detected [14]. The AVRs also contribute to these low frequency
oscillations.
Large electric power systems usually have poorly damped electromechanical
oscillations associated with the rotor angle of the synchronous machines [15]. The
insufficient damping of electromechanical dynamics causes oscillations of low
frequencies and negative damping to grow in magnitude [16 – 17].
Depending on the system, Low frequency oscillations are often classified into two
modes.
2.2.1 Local modes
For this mode, oscillations are in the range of 0.8 – 2.0 Hz, caused by one generator
swinging against the rest of the system [17].
9
2.2.2 Inter – area modes
For the inter – area modes, groups of generators in different areas swing against
each other with oscillations in the range of 0.2 – 0.7 Hz [17]. These oscillations are
usually observed in a large interconnection between power systems with weak tie-
lines.
To remediate the small signal instabilities caused by the AVR and other factors, the
Power System Stabilizer (PSS) was introduced to stabilize the system and increase
the system’s security. PSS is further discussed in the next section.
2.3 Power System Stabilizer
One problem that faces power systems nowadays is the low frequency oscillations
arising from interconnected systems. Sometimes, these oscillations sustain for
minutes and grow to cause system separation. The separation occurs if no adequate
damping is available to compensate for the insufficiency of the damping torque in the
synchronous generator unit [19]. This insufficiency of damping is mainly due to the
AVR exciter’s high speed and gain and the system’s loading.
In order to overcome the problem, PSSs have been successfully tested and
implemented to damp low frequency oscillations [16, 28]. The PSS provides
supplementary feedback stabilizing signal in the excitation system [5]. The feedback
is implemented in such a way that electrical torque on the rotor is in phase with
speed variations [20]. PSS parameters are normally fixed for certain values that are
determined under particular operating conditions. Once the system’ operating
conditions are changed, PSS may not produce adequate damping [18] into an
unstable system.
Since PSSs are tuned at the nominal operating point, the damping is only adequate
in the vicinity of those operating points. But power systems are highly nonlinear
systems, therefore, the machine parameters change with loading and time. The
dynamic characteristics also vary at different points [21].
Hence, several approaches based on modern control theory have been applied to
design different power system stabilizer structures [5, 22]. This includes optimal
control, adaptive control, variable structure control and intelligent control which are
further developed in [24 – 26].
10
Figure 2.2 CPSS structure
In [23], S. Panda and N.P. Padhy presented a systematic procedure for modelling
and designing of a power system equipped with PSS and Flexible AC transmission
system (FACTS)-based controller. Further, they evaluated the impact of the PSS and
FACTS – based controller on the power system.
Alberto Del Rosso et al examined in [27] the use of Thyristor Controlled series
Capacitors (TCSC) for stability improvement of power systems. An appropriate model
of TCSC was used to design a simple controller based primarily on the dynamics
response of the power system.
Despite the numerous approaches of modern control techniques with different
structures, power system utilities still prefer the conventional lead – lag PSS (CPSS).
The CPSS has a simple structure and is considered to be reliable for actual power
system applications [5, 22].
2.3.1 Conventional Power System Stabilizer structur e and design
The basic function of CPSS is to damp electromechanical oscillations. To achieve
the damping, the CPSS proceeds by controlling the AVR excitation using auxiliary
stabilizing signal. The CPSS’s structure is illustrated in Figure 2.2.
PSSK1
w
w
s T
s T +1
2
1
1
s T
s T
++
3
4
1
1
s T
s T
++
_PSS MAXV
_PSS MINV
PSSVω∆
Gain W ashout Lead Lag− Compensator
L im iter
F ilte r
11
2.3.1.1 CPSS input
The CPSS classically uses the following inputs:
• The shaft speed deviation ω∆
• Active power output, aP∆ (Change in accelerating power) and
• eP∆ (change in electric power),
• Bus frequency f∆ .
Since the main action of PSS is to damp electromechanical (or rotor) oscillations,
thus ω∆ is used as the input signal to the PSS.
2.3.1.2 Gain
The gain determines the amount of damping introduced by the stabilizer. Therefore,
increasing the gain can move unstable oscillatory modes into the left – hand complex
plane [7]. Ideally, the gain should be set to a value corresponding to a maximum
damping. However, in practice the gain KPSS is set to a value satisfactory to damp the
critical mode without compromising the stability of other modes [2, 7].
2.3.1.3 Washout
The washout stage is a High Pass Filter (HPF) with purpose to respond only to
oscillations in speed and block the dc offsets. The Washout filter prevents the
terminal voltage of the generator to drift away due to any steady change in speed.
2.3.1.4 Phase compensation
This stage consists of two lead – lag compensators as shown in Figure 1 (lead – lag
compensation stage). The lead stage is used to compensate for the phase lag
introduced by the AVR and the field circuit of the generator [30, 29]. The lead – lag
parameters are tuned in such as way that speed oscillations give a damping
torque on the rotor [7, 29].
When the terminal voltage is varied, the PSS affects the power flow from the
generator, which efficiently damps the local modes [29]. Larsen and Swann express
in [30] the difficulty of tuning the Lead – Lag parameters to compensate the dynamic,
which varies according to the operating points and the network reactance.
1 4T T−
12
2.3.1.5 Torsional Filter
This stage is added to reduce the impact on the torsional dynamics of the generator
while preventing the voltage errors due to the frequency offset [7].
2.3.1.6 Limiter
The PSS output requires limits in order to prevent conflicts with AVR actions during
load rejection. The AVR acts to reduce the terminal voltage while it increases the
rotor speed and the bus frequency. Thus, the PSS is compelled to counteract and
produce more positive output [31]. As described in by P. Kundur in [7], the positive
and negative limit should be around the AVR set point to avoid any counteraction.
The positive limit of the PSS output voltage contributes to improve the transient
stability in the first swing during a fault. The negative limit appears to be very
important during the back swing of the rotor. Indeed, after the initial acceleration is
over, the system requires a large amount of synchronizing torque to return to
equilibrium in the post – fault state [2, 7and 39].
The tuning of the PSS parameters remains a complex task. Kundur et al. presented
in [9] a full analysis of CPSS and different effects of its parameters on the dynamic
performance of the system. They demonstrated that appropriate selection of washout
time, compensator parameters and PSS limits, provide satisfactory performance.
Bikash Pal and & Balarko Chaudhuri in [31] outline criteria and guidelines, based on
Larsen & Swann studies in [30], to choose the compensator parameters T1 – T4.
2.3.2 Genetic Algorithm based PSS (GAPSS)
GAs are global search techniques equipped with powerful tools used to solve
optimization problems. GAs are based on mechanics of natural selection and
genetics. They apply the principle of survival of the fittest on a population of potential
solution to generate increasingly better approximations of solution until optimization
is reached [2].
2.3.2.1 GAPSS review
Several techniques of tuning PSS have been developed and tested over the recent
years. Komsa Hongesombut et al. in [20] incorporated the use of an analytical
method known as phase control loop and intelligent method. This procedure was
performed using micro – GA to select PSS parameters combined with Hierarchical
GA (HGA) in the process of reinitialization.
13
M.A. Abida and Y.L. Abdel Magid in [5] employed Evolution Programming (EP)
techniques to search for optimal setting of PSS parameters. These settings had shift
the system eigenvalues associated with electromechanical modes to the left in the s-
plane. Tested under different disturbances, loading conditions and system
configurations, the methods were found to be effective.
J. Lu et al. in [33] applied a selection of fuzzy rules used from the operating point
settings, to tune the stabilizer parameters online according to real-time
measurements. The membership functions of the fuzzy parameter tuner were
optimized using a genetic algorithm (GA).
In [34], Genetic Local Search (GLS) was presented. The proposed approach
hybridized GA with heuristic local search and used to tune PSS parameters on
different operating conditions. The simulation showed the effectiveness and
robustness of the GLSPSS.
M.A. Abido and Magid in [35] investigated the effect of tuning PSS using classical GA
and compared the results with those of CPSS over a wide range of parameters.
GAPSS demonstrated to be more robust over the CPSS.
In [37], the authors presented a novel approach to combine GA with a new recurrent
neural network (RNN). The method included the design of a genetic algorithm based
on recurrent neural networks power system stabilizer (GARNNPSS) for multi
machine power system. The GARNNPSS consists of a recurrent neural network
identifier (RNNI) that tracks and identifies the power generator and a recurrent neural
network controller (RNNC). It supplies an adaptive signal to the governor and exciter
to damp the power system oscillation. Both RNNI and RNNC are firstly trained offline
by GA to find the optimal learning rates, and then online to damp the oscillations. The
simulation results demonstrated the effectiveness of the proposed GARNNPSS and
its optimal performance.
K.A. Folly presented in [40] a simplified version of GA called Population Based
Incremental Learning (PBIL) to design a PSS for multimachine power system. The
control problem was converted into a optimization problem solved with PBIL. The
resulting controllers ensured robust stability and good performance for both the
nominal and off-nominal operating conditions.
Asante Phiri in [38] investigated the application of Breeder Genetic Algorithm (BGA)
tuning PSS parameters. A comparative analysis between BGA and GA was
14
developed and simulated, which revealed that BGAPSS performed better than
GAPSS.
2.3.2.2 Proposed optimization technique
Although many researches have proposed GAs to tune the PSS, it still presents
some disadvantages such as, the speed of convergence which remains time
consuming and the parameters’ encoding that takes up a lot of memory [48]. In fact,
classical GA use bit string, better suited for combinatorial optimization [48], to encode
their parameters and to modify them with logical operators. This property requires
heavy computational effort and memory [47, 48].
To overcome the above setbacks, the Differential Evolution (DE), a new heuristic
approach that uses differential mutation and greedy selection process [47,48], is
investigated in an attempt to tune PSS’s paramters.
In fact, DE, classified as Evolutionary Algorithm, provides significant converging
performance over GAs by using the principle of greedy selection: “the better one of
new solution and its parents wins the competition” [47].
The resulting DE based PSS will then be compared to GAPSS, which uses Classical
GA to tune the PSS parameters. The comparison will focus on the robustness of the
system, speed of convergence toward the optimum solution and the overall
performance of the system.
15
u
t
3 POWER SYSTEM MODELLING
As mentioned in chapter two, power systems are highly nonlinear and consequently,
difficult to analyse. They are also constantly subject to small signal instability for most
of the time. These small disturbances can be linearized around the operating point
and therefore analysed. The system analysis is accomplished by using control
theories such as modal analysis, root locus etc. They provide valuable information
about the inherent dynamic characteristics of the power system important for the
system design.
3.1 Small Signal Dynamic Modelling
3.1.1 State – Space representation
The state – space representation of a system is a fundamental concept in control
theory. It gives some information about the system at any instant in time [6].
To achieve the state representation, the power system is represented by a set of first
order nonlinear ordinary differential equations. They describe the behaviour of the
dynamic system. These equations are of the form:
Where ‘ ‘ represents the state vector, ‘ ‘ is the vector of input to the system
and ‘ ‘ denotes the time. However, if the derivatives are not function explicit of
time, the system is said to be autonomous. The equation (3.1) becomes
The system output variables may be expressed in terms of the state vector and input
vector as follows:
The variable ‘Y ’ is referred to as the output vector and ‘ g ’ is the vector of nonlinear
functions relating the state and the input variable to output.
X
( ) ( , , )d
X t X f X u tdt
•= = (3.1)
( ),X f X u•
= (3.2)
( ),Y g X u= (3.3)
16
0X X X• • •
= + ∆
X•
∆
3.1.2 Linearization
The equations describing the dynamic of the system can be linearized around the
equilibrium point where the system is at rest. All the variables are constant and
unvarying with time.
Let X0 be the initial state vector and U0 the input vector corresponding to the
equilibrium point that is under investigation. Equation (3.2) becomes
When the perturbation is introduced, the above state becomes:
0X X X= + ∆ & 0u u u= + ∆
Where ‘ ∆ ’ denotes a small deviation. Therefore, the new state is defined as:
( )0 0( ),f X X u u• = + ∆ + ∆
Since the deviations are very small around the equilibrium point, the function f (X, u)
can be developed into Taylor expression and solved for . Thus, the linearized
form of the system equations 3.2 and 3.3 obtained are:
Where
X∆ is the state vector of the system
Y∆ is the output vector of the system
u∆ is the vector of input to the system
A is the state square matrix
B is the control matrix also called the input matrix
(3.5)
X A X B u•
∆ = ∆ + ∆ (3.6)
Y C X D u∆ = ∆ + ∆ (3.7)
( )0 0 0, 0X f X u•
= = (3.4)
17
0δ ω ω•
= − (3.8)
C is the output matrix
D is the feed-forward matrix, which defines the proportion of the input that appears
directly in the output.
3.2 Machine modelling
The system dynamics of the synchronous machine can be expressed as a set of four
first order linear differential equations given in equations 3.8 – 3.11. These equations
represent a fourth order generator model suggested by the IEEE 1986 task force
[53]. Higher machine’s models are also proposed based on the varying degrees of
complexity [13] which provide better results. But it is also adequate to use the fourth
order machine with data correctly determined [13, 52]. The above cited machine is a
two axis model, includes the AVR, PSS, turbine governors and excitation system
necessary for this particular research.
Where,
,d qi i = d-q components of armature current
fdE = voltage proportional to field voltage
'dE = voltage proportional to damper winding flux
'qE = voltage proportional to field flux
'0dT = d-axis transient time constant
'0qT = q-axis transient time constant.
( )1
2 m eD T TH
ω ω•
= − + − (3.9)
( )' ' ''0
1q q d d d fd
d
E E x x i ET
• = − + − + (3.10)
( )' ' ''0
1d d q q q
q
E E x x iT
• = − + − (3.11)
18
X A X B u•
∆ = ∆ + ∆
Hence, the state space model of the synchronous machine can be expressed from
the above equations to . .. In a more complete form, the state space
is as follows:
The equations describing the variables 11 44a a− of the state square matrix A are
given in Appendix A1.
The power system model used for this thesis is the single machine infinite bus
(SMIB), as in shown in figure 8.1
3.2.1 Machine Modelling with Power System Stabilize r
As mentioned in the precedent chapter, section 2.3.1, the PSS basic function is to
provide supplementary damping torque to the system by controlling the excitation.
Figure A.1 in appendix A, illustrates the role and shows the place of the PSS in the
SMIB system from the Heffron – Phillips model.
The function includes different stages of the PSS, also portrayed in Figure A.1, has
been discussed in section 2.3.1. Hence, the state space representation obtained
from the block diagram is:
11 12 13 1
21'
' 32 33 34'
42 43 44'
0
0 0 0 0
0 0
0 0
mq
q
d
d
a a a b
aT
a a a EE
a a a EE
ωω
δ δ
•
•
•
•
∆ ∆ ∆ ∆ = + ∆ ∆ ∆ ∆ ∆
(3.12)
1311 12
21' '
33 3412'
' 4712 12 44
51 52 53 55 1
161 62 63 65 66
71 72 73 75 76 772
0 0 0 0
0 0 0 0 0 0
0 0 0 0
0 0 0
0 0 0
0 0
0
q q
dd
aa a
aE a a Ea
a Ea a aEa a a a
a a a a a
a a a a a a
u
ωωδδ
νν νν
•
•
•
•
•
•
•
∆ ∆ ∆ ∆ ∆ ∆ = ∆ ∆ ∆ ∆ ∆ ∆
∆
2
u
∆
(3.13)
19
It can be observed that three more variables 1ν•
∆ , 2ν•
∆ and u•
∆ are added to the
state matrix. These variables represent respectively the output of the washout time
and the phase compensation. 11 77a a− equations are given in Appendix A1.
The state – space does not give any information on the stability of the system.
3.3 Stability analysis
The stability of a system can be determined by analysing the state – space matrix
properties such as eigenvalues, eigenvectors, modes shape, participation factor as
well as controllability and observability. These concepts are detailed in [7].
Participation factor
The participation factor is used as a measure of the association between the state
variable and the mode [6]. It evaluates the relative participation of the ith eigenvalue
or mode in the jth state [6].
Eigenvalues
The eigenvalues of the state – space matrix are obtained from equation 3.14
Where, A is the state matrix
I is the identity matrix and ‘S’ the eigenvalues
The resulting eigenvalues can either be real or complex.
For a real eigenvalues, equation 3.15, the system is corresponding to a non-
oscillatory mode. If the real part of the eigenvalue is positive, then the system is
unstable. If the eigenvalues are negative, the system is stable in decaying mode [7].
The further the eigenvalues are in the negative s – plane, the faster will be the
system response, see figure 3.3 below.
Where ‘σ ’is the real eigenvalue.
[ ]det 0A sI− = (3.14)
s σ= ± (3.15)
20
As for the complex form, like in equation 3.16, the eigenvalues occur in conjugate
pairs [7]. The real component ,‘σ ’, depending on whether it is positive or negative,
increases the oscillation amplitude to complete instability or damps out the
oscillation. The imaginary part ‘ω ’ ,represents the frequency of oscillation.
Thus, the frequency of oscillation can be express as follows
The damping ratio of the oscillation is given by equation 3.17 determines how fast the
oscillation is damped.
Where the amplitude decay 1
Tσ
= in seconds
Hence, applying appropriate control theory, the space – state matrix can be mapped
onto the s – plane. This plane will determine whether the system is stable or not by
simply mapping the eigenvalues to the satisfaction of figure 3.3 below. The
eigenvalues represent the system poles.
s jσ ω= ± (3.16)
2f
ωπ
= (3.17)
2 2
σςσ ω
=+
(3.18)
21
Hence, a stable system will have its eigenvalues restrained on the left hand side of
the s-plane, as in figure 3.3 above. Furthermore, if the system has damped
oscillatory modes (complex eigenvalues on left hand side of the origine), a damping
ratio between 0.05 and 0.7071 is adequate for the power system [50] to operate
normally under small perturbations.
Figure 3.1 S – plane representation [50]
22
4 GENETIC ALGORITHM
Genetic Algorithms (GAs) are heuristic search procedures inspired by the
mechanism of evolution and natural genetic. They combine the survival of the fittest
principle with information exchange among individuals. GAs are simple yet powerful
tools for system optimization and other applications [41].
This technique has been pioneered few decades ago by Holland, basing the
approach on the Darwin’s survival of the fittest hypothesis. In GAs, candidates’
solutions to a problem are similar to individuals in a population. A population of
individuals is maintained within the search space of GAs, each representing a
possible solution to a given problem [42]. The individuals are randomly collected to
form the initial population from which improvement is sought [26, 42]. The individuals
are then selected according to their level of fitness within the problem domain and
breed together. The breeding is done by using the operators borrowed from the
natural genetic, to form future generations (offsprings). The population is
successively improved with respect to the search objective. The least fit individuals
are replaced with new and fitter offspring [2, 43] from previous generation.
Over the recent years, GAs have been at the centre of researches. Especially in the
optimization problems where GAs provide better solutions with simpler techniques
than other optimization methods. According to Goldberg in [43], GAs differs from
other optimization methods in four ways:
• GAs search from a population of points in parallel, no single point
• GAs use probabilistic transition rules, not deterministic ones
• GAs work on encodings of parameters set rather than the parameter set itself
(except where real – valued individuals are used)
• GAs do not require derivative information or other auxiliary knowledge; only the
objective function and corresponding fitness levels influence the directions of the
search.
• These differences give an edge to the GAs search techniques with respect to
other methods. In addition, GAs can provide a number of potential solutions to a
23
particular problem that does not have a singular solution such as the Pareto –
optimal solutions. In this case, the final choice is left to the user [44].
There are many variations of GAs but the basic form is the Classical genetic
algorithm (CGA). The SGA is considered in this research as well as the Differential
Evolution, which is reviewed in the subsequent chapter.
Thus, the working principle [13, 46] of CGA can be described as follows:
4.1 Encoding of individuals
Individuals or current approximations are encoded as strings also called
chromosomes. They are constituted of genes joined together. The chromosomes are
represented in different manners that allow the genotypes (chromosome value) to be
uniquely mapped onto the decision variable (phenotypes) domain [44]. In fact, CGA
can be represented by different alphabets, such as binary alphabet {0, 1}, integers,
real – valued etc [43, 44]. For example, if 10 bits are used to code each variable in a
two-variable function optimization problem, the chromosome would contain two
genes, and it would consist of 20 binary digits [14]. Figure 4.1 from [44] illustrates
how the variables are mapped onto a chromosome structure.
where x1 is encoded with 10 bits and x2 with 15 bits, possibly reflecting the level of
accuracy or range of the individual decision variables [44]. The above chromosome
representation does not yield any information. Thus, to obtain valuable information,
the chromosome has to be decoded into his phenotype value. The performance of an
individual can then be assessed.
4.2 Objective and Fitness functions
The objective function measures the ability of an individual to perform in the problem
domain. The individuals are assigned values also called fitness values, from higher to
lower relative to their performances. In the natural world, this can be interpreted as
Figure 4.1 Chromosome structure constituted of 2 variables
24
an individual’s ability to survive in its present environment. Therefore, the objective
function establishes the basis for selection of individuals that will be mated together
during reproduction.
The fitness function is used to transform the objective function value into measure of
relative fitness [45]. Equation 4.1 reflects the operation of the fitness function
Where f is the objective function and g is the fitness function transforming the value
from the objective function and F is the relative fitness.
4.3 SGA operators
The SGA works with a set of ‘N’ initial individuals constituting a randomly generated
population. ‘N’ denotes the size of the population. This is usually achieved by
generating the required number of individuals using a random number generator that
uniformly distributes numbers in the desired range [44]. The individual’s fitness is
then calculated for every member. These individuals will undergo a transformation in
stages to form a new current population for the next iteration. To achieve the
transformation, the following genetic operators are applied in sequence:
4.3.1 Selection
In this stage, individuals are selected from the current population according to their
fitness value, obtained from the objective function previously described. The purpose
of the selection is to choose individuals to be mated.
The selection can be performed in several ways. But many selection techniques
employ a “roulette wheel”. It is a mechanism to probabilistically select individuals
based on some measure of their performances [44]. Figure 4.2 clearly illustrates the
wheel. The segment sizes on the wheel correspond to the individual fitness value.
The larger the segment, the higher the fitness value for the individual. This wheel will
be spun several times until enough offsprings are produced to populate the next
generation.
( ) ( ( ))F x g f x= (4.1)
25
4.3.2 Crossover
In this stage, the individuals retained (in pairs), from the above stage, exchange
genetic information to form new individuals (offsprings). This process helps the
optimization search to escape from possible local optima and search different zones
of the search space [2, 43]. The combination or crossover is done by randomly
choosing a cutting point where both parents are divided in two. Then the parents
exchange information to form two offsprings that may replace them if the children are
fitter. Figure 4.3 demonstrates how the single point crossover is done. There are
other methods of combination, such as multi – point crossover, uniform crossover,
discrete crossover and various other that are discussed in [43].
4.3.3 Mutation
Mutation is a process where one random allele of the gene is randomly replaced to
produce another new genetic structure [44]. This process increases the probability of
a complete search that will allow an investigation in the vicinity of the local optima.
The effect of the mutation, as shown in Figure 4.4, is applied with a low probability in
the range of 0.001–0.1 [44].
4.3.4 Reinsertion
It is in this process that children populate the next generation by replacing parents, if
fitter. Reinsertion can be made partially or completely, uniformly (offspring replace
parents uniformly at random) or fitness-based [2, 44].
4.3.5 Convergence and termination
As the population evolves over successive generations, the best and average
individual increase toward global optimum [14]. Therefore, the termination is set for a
Figure 4.2 Roulette wheel
26
fixed number of iterations after which, the best individual of the current population is
taken as the optimum solution.
Figure 4.3 Single point crossover
Figure 4.4 Mutated individual
27
5 DIFFERENTIAL EVOLUTION (DE)
The Differential Evolution (DE) was first proposed by Kenneth Price. He was
attempting to solve the Chebyshev Polynomial fitting Problem that had been posed to
him by Rainer Storm in 1994 [48]. The breakthrough occurred when Kenneth came
up with the idea of using vector difference to perturb the vector population [48]. This
method grew rapidly and made DE versatile and a robust optimization tool in today’s
world.
DE is a parallel direct search method that uses a population of points to search for a
global minimum of a function over wide search space [55]. Like all GAs, DE is a
population based genetic that uses similar operators; crossover, mutation and
selection. However DE search methods differ from GAs in some aspects. The main
difference between the two search methods is that, GAs rely on the crossover to
escape from local optima and search in different zones of the search space.
Whereas, DE relies on the mutation parameters as a search mechanism and
selection operation to direct the search toward the prospective regions in the search
space [47].
5.1 Advantage of DE over GA
In DE, all solutions have the same chance of being selected as parents regardless of
their fitness value. DE is known to use the greedy selection process whereby the
better one of the new solution and its parents wins the competition. This principle
provides a better convergence performance over GAs [47].
5.2 Reason for using DE
DE encodes parameters in floating – point regardless of their type [48]. This
encoding offers a great malleability with arithmetic operators and provides significant
advantages over the other optimizations methods, including [48]:
• Ease of use
• Efficient memory utilization
• Lower computational complexity – scales better on large problems
28
• Faster convergence
• Greater freedom in designing a mutation distribution iteration
Nevertheless, DE is a very simple Evolutionary Algorithm (EA), which follows a
sequence, presented in figure 5.1, until optimization is reached or termination occurs.
5.3 Population structure
DE starts with a population of NP vectors of D – dimensional real – valued
parameters as represented in equation 5.1
gX represents the population and the index g denotes the generation. The
population is constituted of Np vectors denoted by ,i gx where the index i refers to
the vector within the population. The vector is also constituted of parameters , ,j i gz
where j is the position of the parameter within the vector.
In the mutation stage, DE creates an intermediate population gV of the same size as
the initial population composed of ,i gv vectors. The intermediate population proceeds
to the next stage. DE also creates a second intermediate population ,i gU which is
also of the size Np with , ,j i gu vectors. The population is created after the
recombination stage.
, max
, , ,
( ) where 1,2,..., 1,...,
( ) 1,2,...,g i g
i g j i g
X x i Np g g
x z j D
= = =
= =(5.1)
Figure 5.1 Differential Evolution cycles
29
5.4 Initialization
The population in DE is initialized by specifying the Upper and Lower bound for each
parameter of a vector. Equation 5.2 is used to generate the vector parameter so that
, ,L Uj j i g jz z z≤ ≤ .
Where jrand generates number in the range of [0, 1] for the j th parameter.
5.5 Mutation
After initialization, DE mutates the population to produce a population of trial vectors.
As previously mentioned, DE relies on the mutation stage, also called differential
mutation, to expand the search space. It is worth highlighting that this operation is
performed differently than in the conventional GAs where an allele was replaced.
In DE, four vectors from the initial population are randomly sampled where one is
chosen as the target vector mentioned in the next stage, and another as the base
vector. The difference of the remaining two vectors scaled by a factor is added to the
base vector to form the trial vector. Equation 5.3 and Figure 5.2 show how the
process of creating the intermediate vector is achieved.
The scale factor, F, is a positive real number that controls the rate at which the
population evolves [48]. The base vector, denoted by 0r , is randomly chosen, in
such a way that 0 1 2r r r≠ ≠ where 1& 2r r are also randomly chosen. Vi,g is the trial
vector
, , (0,1) ( )U L Lj i g j j j jz rand z z z= ⋅ − + (5.2)
, 0, 1, 2,( )i g r g r g r gv x F x x= + ⋅ − (5.3)
30
Figure 5.2 Differential Evolution: the weighted differential, 1, 2,( )r g r gF x x⋅ − , is added to the
base vector, 0,r gx , to produce a mutant, ,i gv [48, 49].
(Xr1 – Xr2)
Xr2
F.(Xr1 – Xr2)
Xr0
Vi
5.6 Recombination or crossover
DE uses the crossover, also referred as discrete recombination, to complement the
differential mutation strategy mentioned in previous section [48]. In this stage, DE
crosses each vector with a mutant vector to form a second intermediate population
as shown by Equation 5.4.
[0,1]Cr ∈ is the crossover probability defined by the user within the specified range,
which control the parameter values that are copied from the mutants. Therefore, if
jrand , the random number generated, is lower than the crossover probability Cr ,
the corresponding j parameter is copied from the mutant vector. On the hand, if
jrand is higher thanCr , the parameter will become , ,j i gz , the target vector. DE can
also apply different type of crossovers: uniform crossover, one – point crossover, N –
point crossover or exponential crossover detailed in [48].
, ,
, , ,, ,
[ (0,1) or ]
.j i g j rand
i g j i gj i g
v if rand Cr j jU u
z otherwise
≤ == =
(5.4)
Xr1
31
5.7 Selection
The selection of vectors to populate the next generation is accomplished by
comparing each vector ,i gu of the second intermediate population gU to its target
vector ,i gx from which it inherits parameters. The values of the vectors are obtained
using the function as illustrated in equation 5.5.
As soon as the new population is installed, the cycle is repeated until the optimum is
located or termination criterion is satisfied [48].
Figure 5.3 clearly illustrates all the steps and various populations used to achieve the
cycle necessary to find the optimum values.
, , ,
, 1,
( ) ( )
.i g i g i g
i gi g
u if f u f xx
x otherwise+
≤=
(5.5)
Figure 5.3 A flow chart of DE’s operation and test loop [48]
32
6 PARAMETERS SETTING OF GA & DE
The main objective of the PSS is to provide additional damping in order to stabilize
an oscillatory unstable system. The stabilization is achieved by fine tuning the PSS’s
parameters to optimum values, using various techniques stated in chapter two.
The widely used conventional PSS (CPSS) is tuned by using two basic techniques:
phase compensation and root locus. The phase compensation consists of adjusting
the stabilizer to compensate for the phase lag through the generator, excitation
system and the power system [30, 52].The root locus on the other hand consists of
shifting the eigenvalues associated with the power system modes to the stable region
[16, 30]. As mentioned before, the CPSS is tuned for a particular operating and
system condition. As the condition changes, the CPSS can no longer preserve the
quality of performance, therefore, it needs to be retuned.
The alternative tuning methods investigated in this thesis are the Differential
Evolution (DE) and the Classical Genetic Algorithms (CGA). They are implemented
to optimize the PSS. Unlike the CPSS, the DE based PSS (DEPSS) and GA based
PSS (GAPSS) parameters guarantee a minimum performance for all operating
conditions. The processes of finding these parameters are described in later
sections.
Some factors, such as the choice of the objective function and the optimization
problem discussed in the next sections, must be taken into account when tuning the
PSS’s parameters in order to achieve the desired performance over the entire range
of operating conditions.
6.1 Objective function
The objective function, given by equations 6.1 & 6.2, was applied to determine the
optimum values of the PSS’ parameters. The function is based on the eigenvalues
that are associated with the unstable modes of oscillation. It consists of shifting the
eigenvalues corresponding to the undamped mechanical modes to the stable region
of the s – plane, therefore stabilizing the system. This approach gives considerable
insight to the closed – loop poles performance of the system such as stability,
damping and frequency. This function aims to restrain these closed – loop
eigenvalues to lie to the stable region of the s – plane, at a specific damping factor.
33
It is important that the variation of the damped mode’s frequency resulting from the
eigenvalues’ shifting remains within the acceptable range of the undamped mode’s
frequency. This criterion will both provide adequate performance and meet the
design requirement stated in [30]. Most importantly, the shifting of eigenvalues
should be executed without affecting other modes.
The objective function is given as:
Where z C∈ . ‘C ’ is the complex plane and ‘ z ’ a point in the plan, with 0α <
The function in equation 6.1 is the distance from the point to the
The PSS’ parameters are selected with the purpose of minimizing the objective
function, which is the distance between the eigenvalue to be shifted and the relative
stability point. Therefore, the function is defined as follows:
Where n is the number of eigenvalues. iλ is the thi closed – loop eigenvalue of the
power system to be restrained to lie on the left hand side of the vertical line at α as
shown in figure 6.1. ‘α ’ determines the relative stability corresponding to the desired
damping factor, at a certain frequency of the power system.
( ) Re( )f x z α= − (6.1)
min[Re( ) ] 1, 2,...,iJ i nλ α= − = (6.2)
34
The optimization problem is formulated as follows:
Minimize the function J subject to the constraints:
Where 1 2 3 4, , , & K T T T T are the PSS’ parameters to be tuned within their
boundaries. The optimal value of these parameters will guarantee a satisfying time –
domain performance and relative stability.
1min 1 1maxT T T≤ ≤m in m a xK K K≤ ≤
2min 2 2 maxT T T≤ ≤
3min 3 3maxT T T≤ ≤
4 min 4 4 maxT T T≤ ≤
(6.3)
Figure 6.1 Relative stability region on the left hand side of the line at α−
α
35
6.2 PSS tuning approach
The PSS’s parameters were tuned using two implementation methods in addition to
the cited objective function in equation 6.2:
6.2.1 Simple Genetic Algorithm
The GAPSS was implemented using the Genetic Algorithm for Optimization Toolbox
(GAOT) developed by C.R. Houk, J.A. Joines and M.G. Kay in 1995 [52].The GAOT
implements a simulated evolution in MATLAB environment using both binary and real
representations [52]. The toolbox is comprehensive and very flexible in the genetic
operators, selection functions, termination functions and the evaluation functions that
can be used.
The SGA’ parameters, summarized in Table 6.1, were carefully chosen and set as to
avoid premature termination resulting in solutions caught in local optima. As in [14],
the main criteria in the choice of the SGA parameters values were:
• Accuracy in the solution,
• Convergence to global optimum values.
Table 6.1 SGA parameters
Encoding Binary
Population size 100
Selection function Roulette
Crossover operation single crossover
Crossover probability 0.7
Mutation rate 0.1
Maximum generation 100
36
6.2.2 Differential Evolution (DE)
The DEPSS was implemented using the DeMat package, the MATLAB version
developed by J.V. Zandt and A. Neumaier. DeMat [48] provides a framework for
solving function optimization problems, which was adapted to suite the objective of
this work. The structure is further discussed in the subsequent chapter.
DEPSS employs genetic operators described in chapter five by following a sequence
illustrated in figure 5.1.
DE’s optimization is strongly influenced by the mutation scale factor ‘F’ and the
mutation rate ‘Cr’. Thus, appropriate values of F and Cr will guarantee a good
performance of the PSS.
6.2.2.1 Mutation scale factor
DEPSS employs differential mutation technique as detailed in section 5.3. This
technique consists of scaling the difference between two vectors by a factor F that is
then added to the base vector to form the trial vector (equation 5.3).
The scaling factor F is any number ranging from (0, 1+). This factor controls the rate
at which the population evolves. As the selection operator has a tendency to reduce
the diversity of population, the mutation on the other hand, increases it [48].
Therefore, to avoid premature convergence, it is imperative that F is properly
selected to counteract the selection’s effect [48].
6.2.2.2 Mutation rate
The mutation rate also known as the crossover probability Cr, is the likelihood that a
parameter will be inherited from a mutant as pointed out in section 5.4.
The mutation rate ranges from [0, 1]. Accurate Cr value is crucial for the
performance of the DE. Strom and Price proposed in [48] after extensive test beds
that, optimization is best achieved with 0 0.2Cr≤ ≤ or 0.9 1Cr≤ ≤ for all functions.
DE parameters are summarized in Table 6.2. These parameters were set to values
that ensure optimal performance of DE.
37
Table 6.2 DE parameters
Population size 100 DE step - size F 0.8 Crossover probability Cr 1 Number of parameters 5 Minimum boundary K & T1-T4 Maximum boundary K & T1-T4 Maximum generation 100
6.2.3 Tuning process
The tuning process of the PSS involves a number of steps, given below, to find the
optimal parameters.
Step 1. Set the minimum and maximum boundaries where the optimal values of the
PSS’ parameters will be found.
Step 2. Obtain the system operating conditions and select the desired relative
stability.
Step 3. Generate an initial population within the constraints given by the set of
equations in 6.3, using DE and SGA.
Step 4. Run the load flow for each individual to check if the system converges. If
not, discards and then change the operating conditions.
Step 5. Get the state space matrices; calculate the eigenvalues vector with their
participation factors.
Step 6. Check for system controllability. If the system is controllable, then the
eigenvalue shift can be performed, therefore the system can be improved. If
the system is not controllable, then the oscillation cannot be damped,
therefore the system parameters need to be changed.
Step 7. Identify the electromechanical modes and retain the eigenvalues associated
to the above modes by using the participation factor for each individual.
Step 8. Evaluate the objective function J defined by equation 6.2.
Step 9. Check if any eigenvalues are on the right hand side of the s-plane.
38
Step 10. Check if the damping is within the specify range
Step 11. Check if the electromechanical mode’s frequency is within the acceptable
range of the undamped mode’ frequency.
Step 12. Repeat step 4 to step 11 until maximum generation is reached.
After intense performance checks (Step 1 – Step 12) on each individual, the best
performing one is selected as the optimal set of PSS parameters.
The flow chart in figure 6.2 summarized the above steps.
39
Figure 6.2 Flow chart representation of the tuning process
40
7 SIMULATION RESULTS AND DISCUSSIONS
Simulations were performed using Power system Toolbox (PST), a MATLAB
package further discussed in appendix D.
7.1 Power System to be investigated
The system considered for the purpose of this thesis is the single machine connected
to a large system or infinite bus (SMIB), via transmission lines.
The general configuration is represented in Figure 8.1.
Where,
Pand Q are the power and reactive power at the generator terminal.
is the line reactance of the system.
7.2 Operating conditions
The DEPSS’s performance was tested over a range of operating conditions defined
by the power P and reactive power Q at the generator terminal along with the
equivalent line reactance Xe.
Three operating conditions, defined in Table 8.1, were set to encompass practically
all system’s conditions, ranging from light to heavy loaded system.
eX
Infinite bus
eXG
P jQ+uuuuuuuv
Figure 7.1 : SMIB system
41
Table 7.1 Operating conditions
P Q Xe
Minimum 0.1 0.06 0.5
Nominal 1.0 0.44 0.7
Maximum 1.0 0.62 0.9
7.3 PSS parameters
The performance analysis of the SMIB equipped with DEPSS is conducted for the all
range of operating conditions in Table 8.1, to evaluate the efficacy of the proposed
stabilizer. The results are compared with the GAPSS and the CPSS, all tuned with
single set of parameters. The comparison includes the damping and frequency
performance of each PSS including the system response to a disturbance of one
percent (1%) in the input voltage. The desired damping in this research is as follows:
1.0
0.2 0.5damping ratio
ας
= −= ≤ ≤
Where ‘α ’ is the relative stability mentioned in chapter 6.
The CPSS’s design is described in the subsequent section.
7.3.1 CPSS parameter selection
The CPSS was designed at nominal operating condition using the phase
compensation technique by following guidelines provided in Appendix B.
The parameters are specified in Table 8.2.
42
The CPSS consists of a lead – lag compensator, washout stage and the gain, as
described in chapter two.
The selected CPSS’ parameters ensure a stable system at nominal operating
condition and provide grounds to tune the DEPSS and GAPSS as discussed in the
next section.
7.3.2 DEPSS and GAPSS parameters selection
By minimizing the objective function, min[Re( ) ]iJ λ α= − described in chapter six,
DEPSS and GAPSS, maximize the parameters.
In fact, the CPSS’ parameters allow to define a search region where the optimum
PSS’ parameters are expected to be found. Hence, the DEPSS and GAPSS
parameters were obtained by solving the constraint given in equation 6.3 by setting
the search boundaries as given in table 8.3.
To provide a fair comparison between the different tuning methods, the DEPSS and
GAPSS were also tuned at nominal operating condition.
After some trial and error, and intensives tests, the two PSS (DEPSS & GAPSS)
provided good results within the boundaries given in table 8.3 below.
The washout time was chosen to be by following criteria in [52], Tw = 2.0
Table 7.2 CPSS parameters
K 29.5
T1 0.8
T2 0.1
T3 0.1
T4 0.3
43
Table 7.3 Parameters boundaries
Bounds K T 1 T2 T3 T4
Min 15 0.6 0.02 0.02 0.02
Max 20 5 2 2 2
The optimum parameters found by DE and GA are given in Table 8.4 below.
Table 7.4 DEPSS and GAPSS parameters
DEPSS GAPSS
K 19.811 18.6
T1 3.526 3.88
T2 0.409 0.801
T3 0.288 0.505
T4 0.901 0.787
7.4 Simulations results
Having found the optimum parameters, the system was tested at various stress level
by varying the generator power (P) and the reactance of the line (Xe) independently
from each other over the selected range defined in equation 7.1and 7.2 below. The
following range was considered for the particular reason that the system is only
unstable in the cited ranges. The left out ranges have been discarded because the
system was able to stabilize after disturbances, as analysed in table 7.5 at minimum
condition, with adequate damping.
The selected testing range is as follows:
44
The power is varied from:
The line reactance as:
The DEPSS is expected to provide adequate performance over the selected testing
range. Load flow reports are provided in Appendix C for the nominal and maximum
operating conditions.
7.4.1 Eigenvalues analysis
7.4.1.1 Minimal system condition
The system’s performance of the open loop is given in Table 7.5 below. To track the
electromechanical modes, as mentioned earlier, the participation factor, provided in
Appendix E.1, was used to identify the highest participating eigenvalue(s) in the
particular state.
Table 7.5 No PSS at minimum condition
Eigenvalue Damping Frequency (rad/s)
-0.422 + j5.037 0.084 5.055
-0.422 - j5.037 0.084 5.055
-1.205 1.00 1.205
-5.542 + j18.179 0.292 19.00
-5.542 - j18.179 0.292 19.00
-59.16 1.00 59.16
The system without PSS is stable and is able to damp oscillations of frequency 5.055
rad/s associated with the rotor angle and the speed, with associated eigenvalues
-0.422 j5.037± . Although the damping performance is just above to the lower
bound (0.05) but is acceptable for a power system. Therefore, No PSSs were tested
for this operating condition. Two eigenvalues, -5.542 j18.179 ± are associated
with the AVR, exhibit damped oscillations of 19 rad/s. Two non-oscillatory
0.7 0.9 p.u.eX = − (7.2)
0.6 1.0 p.uP = − (7.1)
45
eigenvalue, -1.205 is associated with the AVRS system and -59.16 is associated
with the control mode.
7.4.1.2 Nominal condition
The open - loop system in table 7.6 displays under damped electromechanical
modes with eigenvalues 0.0212 j3.98 ± therefore oscillatory at a frequency of 3.98
rad/s.
Table 7.6 No PSS at nominal condition
Eigenvalues Damping Frequency
(Rad/s)
0.0212 + j3.98 -0.0053 3.98
0.0212 - j3.98 -0.0053 3.98
-1.728 1 1.728
-5.906 +j17.24 0.323 17.23
-5.9046 -j17.24 0.323 17.23
-58.5794 1 58.579
The CPSS tuned at this operating condition, stabilizes the system by improving the
under damped electromechanical modes and moving the associated eigenvalues to
-0.64 j3.69± at a damped frequency of 3.7 rad/s, as shown in Table 7.7.
Table 7.7 CPSS at nominal condition
Eigenvalues Damping Frequency (Rad/s)
-0.245 1 0.245
-0.64 + j3.69 0.1705 3.70
-0.64 - j3.69 0.1705 3.70
-1.74 1 1.738
-3.97 1 3.978
-4.90 +j17.13 0.275 17.13
-4.90 -j17.13 0.275 17.13
-10 1 10
-58.61 1 58.61
46
The GAPSS in Table 7.8 considerably improves the damping factor of the mode
associated to the speed and rotor angle and eigenvalues -1.49 j3.79 ± with a slight
increase in the frequency to 4.07 rad/s.
Table 7.8 GAPSS at nominal condition
Eigenvalues Damping Frequency (Rad/s)
-0.45 1 0.45
-1.21 +j 0.57 0.905 1.34
-1.212 - j0.57 0.905 1.34
-1.49 + j3.79 0.364 4.07
-1.49 - j3.79 0.364 4.07
-1.77 1 1.77
-4.43 +j16.98 0.252 17.54
-4.43 -j16.98 0.252 17.54
-58.62 1 58.63
DEPSS, in Table 7.9, also has three oscillatory modes: one associated with
rotor angle and speed or the electromechanical mode, with eigenvalues at
-1.58 j3.93± with a frequency at 4.13 rad/s; the two other with frequencies of
1.588 and 17.53 respectively associated with the excitation system,
-1.45 j0.65 ± and the PSS -4.51 j16.94± . DEPSS demonstrates better
damping performance.
47
Table 7.9 DEPSS at nominal condition
Eigenvalues Damping Frequency (Rad/s)
-0.44 1 0.449
-1.45 +j0.65 0.912 1.588
-1.45 -j0.65 0.912 1.588
-1.58 +j3.93 0.374 4.130
-1.581-j3.93 0.374 4.130
-1.978 1 1.978
-4.51 +j16.94 0.257 17.53
-4.51-j16.94 0.257 17.53
-58.62 1 58.62
7.4.1.3 Maximum condition
The open – loop system is unstable. The electromechanical modes are under
damped as summarized in table 7.10. The associated eigenvalues 0.0885 j2.85±
have further moved into the unstable region of the s – plane, thus, resulting in an
unstable oscillatory mode at a frequency of 2.84 rad/s.
Table 7.10 No PSS maximum condition
Eigenvalues Damping Frequency(rad/s)
0.09 + j2.85 -0.031 2.84
0.09 - j2.85 -0.031 2.84
-1.77 1 1.76
-5.72 +j17.72 0.307 18.62
-5.72 -j17.72 0.307 18.62
-58.89 1 58.89
48
The CPSS, as expected, poorly performs for this condition due to its low damping.
The electromechanical modes, with associated eigenvalues -0.233 j2.64± , are
damped at a frequency of 2.66 rad/s.
Table 7.11 CPSS at maximum condition
Eigenvalues Damping Frequency (rad/s)
-0.2429 1 0.242
-0.33 + j2.64 0.121 2.66
-0.33 - j2.64 0.121 2.66
-1.765 1 1.76
-4.09 1 4.09
-4.93 +j17.65 0.268 18.32
-4.93 -j17.65 0.268 18.32
-10 1 10
-58.92 1 58.92
The GAPSS has a relatively good damping at maximum condition. The resulting
system is stable. The oscillatory mode associate with the electromechanical has a
frequency of 2.72 rad/s, with associated eigenvalues at -1.0681 j 2.50± .
49
Table 7.12 GAPSS at maximum condition
Eigenvalues Damping Frequency (rad/s)
-0.429 1 0.429
-1.07 +j 2.50 0.393 2.72
-1.07 - j2.50 0.393 2.72
-1.30 + j0.685 0.885 1.47
-1.30 - j0.685 0.885 1.47
-1.75 1 1.75
-4.55 +j17.52 0.251 18.10
-4.55 –j17.52 0.251 18.10
-58.9294 1 58.93
With the addition of the DEPSS, the system becomes stable. The PSS exhibits a
good damping of the electromechanical modes at a frequency of 2.7 rad/s and
eigenvalues -1.2598 j2.498± . The system is expected to stabilize faster than
GAPSS and CPSS.
Table 7.13 DEPSS at maximum condition
Eigenvalues Damping Frequency (Rad/s)
-0.43 1 0.43
-1.25 + j2.498 0.450 2.70
-1.25 - j2.498 0.450 2.70
-1.58 + j1.035 0.836 1.88
-1.58 - j1.035 0.836 1.88
-1.70 1 1.70
-4.62+j17.49 0.255 18.09
-4.62 -j17.49 0.255 18.09
-58.93 1 58.93
50
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=1 and Q=0.43617 at Xe=0.7
time(s)
chan
ge in
Vt
(p.u
)
DE PSS
GA PSSCPSS
7.4.2 Time domain response
The comparison between the different PSSs is better illustrated in time domain
response with 1% step change in the voltage as illustrated in figure 7.1 and 7.2,
corresponding respectively to nominal and maximum conditions
The responses of all three stabilizers, CPSS, GAPSS and DEPSS are plotted
simultaneously for the weak transmission with heavy power transfer. System is more
stable in this case, following any disturbance. All three controllers are able to damp
the oscillations thus improving the system dynamic stability significantly. DEPSS with
its remarkable performance shows it superiority over the GAPSS and the CPSS by
settling within 3 – 3.5 seconds. The GAPSS settles in 3.5- 4 seconds while the CPSS
between 6 -7 seconds. DEPSS also presents some overshoot slightly higher than
CPSS and more or less equal to the GAPSS.
Figure 7.2 : Step response at Nominal condition
51
DEPSS and GAPSS are able to damp the oscillations associated with the
electromechanical mode, and stabilize the system. On the other hand, the CPSS is
less effective in damping these oscillations. DEPSS displays its efficacy by settling
within 3-4 seconds. GAPSS settles within 5 to 6 seconds. The overshoots are also
observed in the response where both DE-PSS and GAPSS present the biggest.
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=1 and Q=0.62222 at Xe=0.9
time(s)
chan
ge in
Vt
(p.u
)
DEPSS
GAPSSCPSS
Figure 7.3: Step response at Maximum condition
52
7.4.3 Robustness tests of PSSs
The PSSs are tested over the selected range of the power system operating
conditions in equations 7.1 and 7.2. The time domain responses for the robust test
are provided in appendix E.2
7.4.3.1 Eigenvalues and Damping ratio
The damping ratio associated with their eigenvalues were compared over a range of
fifteen operating conditions by varying the reactance, Xe = 0.7, 0.8, 0.9 and the
generator power, P = 0.6 – 1.0 p.u.
The tables below summarize each PSS’ performances. The values in brackets are
the damping ratio.
Table 7.14 CPSS eigenvalues and Damping ration under robust test
Xe
0.7 0.8 0.9
P
0.6 -0.71 ±4.20 (0.168)
-0.62 ±3.93 (0.156)
-0.54 ± 3.68 (0.146)
0.7 -0.72 ±4.11 (0.173)
-0.61 ±3.81 (0.159)
-0.52 ±3.53 (0.147)
0.8 -0.71 ±4.00 (0.176)
-0.59 ±3.66 (0.159)
-0.48 ±3.33 (0.144)
0.9 -0.68 ±3.86 (0.175)
-0.54 ± 3.47 (0.155)
-0.42 ±3.05 (0.136)
1 -0.64 ± 3.69 (0.17)
-0.48± 3.22 (0.146)
-0.32 ± 2.63 (0.121)
53
Table 7.15 GAPSS eigenvalues and damping ratio under robust test
Xe
0.7 0.8 0.9
P
0.6 -1.2 ± j4.36 (0.27)
-1.09 ±j4.06 (0.25)
-1.00± j3.79 (0.25)
0.7 -1.29 ± j4.27 (0.29)
-1.16 ± j3.94 (0.28)
-1.05 ±j3.63 (0.27)
0.8 -1.35 ± j4.16 (0.31)
-1.20± j3.78 (0.3)
-1.08 ± j3.40 (0.31)
0.9 -1.39 ±4.01 (0.33)
-1.22 ± j3.56 (0.33)
-1.07± j 3.07 (0.34)
1 -1.42 ±j3.81 (0.34)
-1.21 ± j3.25 (0.35)
-1.06 ± j2.53 (0.39)
Table 7.16 DEPSS eigenvalues and damping ratio under robust test
Xe
0.7 0.8 0.9
P
0.6 -1.27 ± j4.44 (0.275)
-1.16 ± j4.13 (0.271)
-1.08 ±j3.85 (0.27)
0.7 -1.37 ± j4.36 (0.300)
-1.25 ± j4.02 (0.297)
-1.15± j3.69 (0.298)
0.8 -1.46± j4.26 (0.323)
-1.31 ± j3.86i (0.323)
-1.20 ±j 3.47 (0.328)
0.9 -1.52 ± j4.12 (0.350)
-1.36± j3.65 (0.350)
-1.23 ± j3.12 (0.367)
1 -1.56± j3.93 (0.374)
-1.39± j3.33 (0.385)
-1.24 ± j2.51 (0.444)
54
DEPSS and GAPSS display better damping performance than CPSS for every
condition. DEPSS outperforms GAPSS in all selected conditions.
The average damping performance over the entire range is as follows:
Table 7.17 Average PSSs damping ratio
CPSS GAPSS DEPSS
Average damping 0.155 0.318 0.330
DEPSS is 6.1% more robust than GAPSS and 53% than CPSS.
7.4.3.2 Eigenvalues plot under robust test
The system’ eigenvalues were plotted on the s-plane. Figure 8.6 – 8.9 illustrate the
effects on the system when the operating conditions change. Notice that, the fast
non-oscillatory poles associated with the control mode, are not included in the s –
plane’ plots, since they are barely affected by the changes.
Figure 7.3 shows the open – loop poles of the system for all conditions. The system
is poorly damped and unstable for some of the conditions. The circled eigenvalues
causing the instability are associated with the electromechanical modes. The relative
stability line is also included in the plot.
55
-8 -7 -6 -5 -4 -3 -2 -1 0 1-20
-15
-10
-5
0
5
10
15
20
Real axis
Imag
inar
y ax
is
Figure 7.4, shows the CPSS’ closed – loop poles. The CPSS achieve stability for all
conditions. But it also presents poor damping for some operating conditions. Hence,
CPSS does not guarantee robust performance for the selected set of conditions.
-8 -7 -6 -5 -4 -3 -2 -1 0 1-20
-15
-10
-5
0
5
10
15
20
Real axis
Imag
inar
y ax
is
Figure 7.4: Open - loop eigenvalues
Figure 7.5: CPSS closed - loop eigenvalues
56
Figure 7.5 shows the GAPSS closed – loop poles for the selected operating
conditions. GAPSS has managed to restrict some eigenvalues to the left of the
relative stability margin, at certain operating conditions. Hence, GAPSS is robust and
guarantees stability for all conditions.
-8 -7 -6 -5 -4 -3 -2 -1 0 1-20
-15
-10
-5
0
5
10
15
20
Real axis
Imag
inar
y ax
is
Figure 7.6: GAPSS closed-loop eigenvalues
57
-8 -7 -6 -5 -4 -3 -2 -1 0 1-20
-15
-10
-5
0
5
10
15
20
Real axis
Imag
inar
y ax
is
Figure 7.6 shows the DEPSS closed –loop poles. DEPSS is able to retrain most of
the system’s eigenvalues to the left of the chosen stability margin. Therefore, DEPSS
is the most robust stabilizer of all three.
Figure 7.7 : DEPSS closed-loop eigenvalues
58
8 CONCLUSION
The application of the Differential Evolution (DE) to optimally tune a Power System
Stabilizer (PSS) capable of stabilizing a system over a wide range of operating
conditions has been successfully investigated. An eigenvalue based objective
function was implemented for the DE to find the optimum PSS’ parameters at a
particular operating condition.
The objective function consists, shifting and assigning the eigenvalues associated
with the electromechanical modes, to the left side of the s – plane, at a specific
location with damping ratio and frequency of oscillation.
The DEPSS’ s performance was tested and simulated on a single machine infinite
bus (SMIB) system. Results have been presented, in chapter 8 and appendix E, for a
wide range of operating conditions, to establish the efficacy of the DEPSS.
Therefore, the following conclusions are drawn:
• DEPSS is able to provide robust stabilization over the specified range of
operating conditions.
• The performance evaluation of the DEPSS compared to the GAPSS and the
CPSS, revealed that both DE and GA based PSS outperform the CPSS in every
system’s conditions. DEPSS surpasses GAPSS all selected conditions.
Therefore DE is better and 6 % more robust than GA in terms of damping ratio
and 53 % better than CPSS.
• DEPSS is able to restrain the eigenvalues to the specific region, denoted by the
relative stability, for most of the operating conditions.
• The attractive feature of the eigenvalue based objective function is that it allows
the relocation of the electromechanical modes to a desired region with a desired
damping, unlike the phase compensation, eigenvalue shifting method gives
considerable insights on the system performances.
DEPSS was implemented in MATLAB using DeMat package. As a result, DE’s
simple, yet effective method offers much potential for a practical implementation.
59
9 RECOMMENDATIONS
Although considerable number of issues have been successfully investigated in the
area of tuning PSSs, several problems remain unresolved. Based on the research
carried out for this thesis, further issues need to be investigated. Therefore, the
following recommendations are made:
• Investigate the application of DE to tune the PSS in a multimachine system.
• Investigate the application of DE in phase compensation method. The method
consists of tuning the PSS parameters to compensate the phase lag through the
generator, excitation system and the power system. Then, conduct a
comparative study between DE based eigenvalue shifting method and DE based
phase compensation.
• Investigate the application of other Evolutionary Algorithms (EAs) to tune a PSS
with faster convergence than DE.
60
REFERENCES
[1] Jesus Fraile - Ardanuy, P.J. Zufiria: “Design and comparison of adaptive power
system stabilizer based on neural fuzzy networks and genetic algorithms”,
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[2] Adrian Andreou: “On power system stabilizers: Genetic algorithms based tuning
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66
APPENDIX A: SYSTEM EQUATIONS
A.1 Equations of state square matrix variables
The variables of the State matrix A without PSS from equation 3.12 is given as
follows:
The variables in equation 3.13 which include the PSS are given as follow:
1 2
4 1
3
5 6
0
2 0 0 0
10
' ' '
10
do do do
A A
A A A
K KD
M M Mf
K KAT T K T
K K K K
T T T
π
− − − = − − − − −
1 2
4 1
3
5 6
1 1 2 1
2 2 2 2
3 1 1 3 2 1 3 3
2 4 2 4 2 4 4 2 4 4
3 1 1 3 2 1 3
2 4 2 4 2 4
0 0 0 0
2 0 0 0 0 0 0
10 0 0 0
' ' '
10 0 0
10 0 0
1 10 0
10
do do do
A A A
A A A A
C C C
C C C
C C C
K KD
M M Mf
K K
T T K T
K K K K K
T T T TA
K K K T K K T
T T M T M T
K T K K TT K K TT T
T T T T M T T M T T T T
K T K K TT K K TT
T T T T M T T M T
π
− − −
− −
− − −=
− − −
− − − − −
− − − 3
4 2 4 4
1 1
W
T
T T T T
− − −
67
A.2 Heffron – Phillips diagram and constants
The block diagram in Figure A.1 describes the dynamics of the SMIB system. The
constants 1 6K K− in the block diagram describe the dynamic characteristic of the
system [7] known as the Heffron – Phillips K constants. K1 and K2 are derived from
the electric torque, while (K3 and K4) from the field winding circuit equations and (K5,
K6) from the terminal voltage. The relating equations are indexed in Appendix A2.
The constants K2, K3, K4 and K6 are usually positive and they affect the system
differently. K2 – K4 influence the electric torque in different manner depending on the
oscillation frequency. When K4 is positive, a positive damping torque component is
introduced [7]. However, for negative value of K4, the damping will be negative.
K5 on the other hand is commonly negative in practice [7]. In the case where K5 is
positive, the AVR decreases the synchronizing torque and increases the damping
Figure A.9.1 Heffron – Philips 3rd model of SMIB system with PSS included
68
torque. For negative values of K5, the AVR introduces a positive value of
synchronizing torque and negative damping torque component [7].
The expression of the constants below are for a lossless network [16] where 0RZ =
and I eZ x= . On the stator 0aR = .
Where,
,20
1 0 0 0 0, ,
0 02 ,
,
3
,
4 0 0,
,0 0 0 00
5 ,0
06 ,
0
( )cos sin
( ) ( )
sin
( )
( )
( )
( )sin
( )
cos sin
( ) ( )
( )
e d
e d e d
e d
e d
e d
e d
e d
q d d q
t e q e d
q e
t e d
E x xK E E
x x x x
EK
x x
x xK
x x
x xK E
x x
x v x vEK
v x x x x
v xK
v x x
δ δ
δ
δ
δ δ
−= + + +
=+
+=+
−=+
= + + +
=+
( )
( ) ( )
0 00 22 2
0 0
0 0
2 20 0 0
20
00
0 0 0
2 2
0 0 0 0 0
0 010
0 0
( )
tan
q
tq
e q t q
d q q
q t d
q
q q d q
d q e q d e
d q e
q d e
P vi
P x v Q x
v i x
v v v
Q i xi
v
E v i x
E v i x v i x
v i x
v i xδ −
=+ +
=
= −
+=
= +
= + + −
+=
−
69
APPENDIX B: TUNING GUIDELINES FOR A CPSS
The objective of the PSS is to compensate for the phase lag of a signal introduced by
the generator, exciter, and power system GEP(s). The phase lag strongly depends
on the frequency of the signal ranging from 0.2 to 1.5 Hz, thus covering the inter-area
and local mode as described in chapter two.
In designing the CPSS, two main criteria critical for proper operation have to be
satisfied; the time constants T1-T4 for the phase compensation and the gain to
provide adequate damping [52]. Over compensation reduces synchronizing torque.
Thus, the PSS(s) phase must balance the GEP(s) phase. The phase lead to be
provided to compensate for the phase lag is different for different frequency [52].
The following tuning guidelines are recommended by [30, 52]:
1. Since the signal from the GEP(s) is passed through the washout stage before the
phase lead, to eliminate the steady – state bias, it important to choose
appropriate value for Tw. According to [30, 52], it would be adequate to choose
the time constant Tw between 1 – 2 sec if the damping of the local mode is the
only concern. But Tw = 10 s when inter – area is considered.
2. As for the phase lag compensation, ( ) ( ) ( )P s GEP s PSS s= × should pass through
-90o at around 3.5 Hz.
3. The compensated phase lag at local mode should be below 45o, most preferably
at 20o.
4. The compensator gain at high frequencies, which is proportional to 1 3
2 4
TT
T T should
be minimized to reduce noise amplification through the PSS.
70
APPENDIX C: SYSTEM OPERATION
C.1 Single Machine Infinite Bus data
C.1 Generator parameters
MVA H dX qX 'dX '
qX '0dT
'0qT
225 3.53 1.81 1.76 0.3 0.6 7.8 0.9
C.2 AVR parameters
Ka aT maxfdE minfdE
200 0.05 5.0 -5.0
71
C.2 Load flow reports
At minimum operating condition where P = 0.1 & X e = 0.7,
LOAD-FLOW STUDY REPORT OF POWER FLOW CALCULATIONS 10-May-2008 SWING BUS : BUS 2 NUMBER OF ITERATIONS : 4 SOLUTION TIME : 0.047 sec. TOTAL TIME : 0.141 sec. TOTAL REAL POWER LOSSES : 0.
TOTAL REACTIVE POWER LOSSES: 0.785345.
GENERATION LOAD BUS VOLTS ANGLE REAL REACTIVE REAL REACTIVE 1.0000 1.0300 42.8133 1.00 0.4362 0 0 2.0000 1.0000 0 -1.00 0.3492 0 0 LINE FLOWS LINE FROM BUS TO BUS REAL REACTIVE 1.0000 1.0000 2.0000 1.0000 0.436 2
1.0000 2.0000 1.0000 -1.000 0.3492
72
At maximum operating condition where P = 0.1 & X e = 0.9,
LOAD-FLOW STUDY REPORT OF POWER FLOW CALCULATIONS 10-May-2008 SWING BUS : BUS 2 NUMBER OF ITERATIONS : 5 SOLUTION TIME : 0 sec. TOTAL TIME : 0.032 sec. TOTAL REAL POWER LOSSES : -1.11022e-016. TOTAL REACTIVE POWER LOSSES: 1.17678. GENERATION LOAD BUS VOLTS ANGLE REAL REACTIVE REAL REACTIVE 1.000 1.0300 60.9017 1.00 0.6222 0 0 2.000 1.0000 0 -1.00 0.5546 0 0 LINE FLOWS LINE FROM BUS TO BUS REAL REACTIVE 1.0000 1.0000 2.0000 1.0000 0.6222 1.0000 2.0000 1.0000 -1.0000 0.5546
73
APPENDIX D: MATLAB CODES & SIMULATION
TOOLS
D.1 simulation tools
The tuning process described in chapter six was applied to design the PSS of a
single machine infinite bus (SMIB) system represented in figure 8.1. The SMIB was
designed and modelled using the power system toolbox (PST), depicted in the
following section. The system’s parameters are given in Appendix C1
Power System Toolbox
PST is a MATLAB based software that provides models of machines and control
systems to perform transient stability simulations of a power system, and to build
state variable models in small signal analysis and damping controller design [51].
PST is constituted of m-files representing models that can be assembled to tailor an
application by following a set of rules described in [51]. Furthermore, PST offers the
possibility to perform the load flow calculation in addition to system state space
matrices needed for this thesis.
The driver m-file is provided for small signal stability analysis (svm_mgen).This
function provides an environment that requires only the system data to be specified
and act much like stand-alone small signal stability programs [51]. The svm_mgen
driver calculates the system’s state matrices, damping ratios, eigenvalues,
eigenvectors and participation factors. In addition, the driver can perform the load
flow.
The svm_mgen was modified to accommodate the DEPSS and GAPSS as detailed
in the following section.
Operations
As in [38], the approach taken in programming the DEPSS and GAPSS was to have
a driver program that would call the PST functions that provide linearized models of
the power system, implement the DE and SGA to obtain the PSSs’ parameters and
then perform step test responses.
74
Therefore, the structure used in [38] to perform the curve fitting exercise, was
modified and adapted to achieve the eigenvalues’ shifting method, described in
chapter six, by following the tuning procedure in figure 6.2.
The mains m-files used to perform the simulation are described in subsequent
sections and the codes are indexed in Appendix D.
smibDriver.m
Purpose
Main file and driver for the power system analysis.
Description
smibDriver (Pmin, Pstep, Pmax, Xemin, Xestep, Xemax) calls other files to linearize
the system and to tune the PSS.
The function’ inputs are the system operating conditions where ‘P’ is the output
power from the generator and ‘Xe’ the reactance of the line. Pmin, Pstep, Pmax,
Xemin, Xestep and, Xemax are respectively the minimum, the increment and the
maximum power and line’s reactance of the system.
smibDriver outputs numerous graphs such as the open-loop system’ eigenvalues
positions on the s-plane and the step response.
This function is a modified version of model_developer used in [38].
Model_maker & Step_testerMod.m
purpose
These m-files are used to form the state matrices of a power system model,
linearized about an operating point that is set by a load flow and then, perform modal
analysis.
Description
These MATLAB script files calls the models of the PST to select data files, perform a
load flow, form a linearized model by perturbing each state in turn and to do a modal
analysis of the given system.
These two m-files were developed in [38] by modifying the svm_mgen in the PST
toolbox by removing the user prompt and setting the following parameters to default:
75
System frequency: 50Hz
Datafiles: Model_maker: smib.m
Step_testerMod : smib2.m
Perform load flow: yes
Further modifications were appended to these functions such as the participation
matrix and the corresponding eigenvalues.
Model_maker.m
The function is called as followed,
[A_mat, B_mat, C_mat, P_mat, L_vec] = Model_maker (P,Q),
The outputs are the state space matrices (A_mat, B_mat, C_mat), which are used to
analyse the open – loop eigenvalues positions in the s-plane together with the
damping ratio and the oscillation frequencies. P_mat, L_vec are the participation
matrix and vectors of eigenvalues to identify the most participating poles in the
electromechanical modes.
Step_testerMod.m
The function is called as follows:
[A_mat, B_mat, C_mat, P_mat, L_vec] = Step_testerMod (P,Q,Tw, k,T1, T2, T3, T4)
The inputs to this function are the operating conditions and the PSS parameters.
The outputs are the state space matrices used to analyse the system closed – loop’
eigenvalues. P_mat and L_vec are also used for the same purpose as in
Model_maker.
pssOptimizer.m
purpose
The purpose of this file is to call dePSS and gaPSS functions.
Description
76
This function calls the dePSS and gaPSS to find the optimal PSS’ parameters to be
returned to smibDriver for step testing.
pssOtimizer further, plots the closed – loop eigenvalues on the s-plane.
Invocation: [tw,k,t1,t2,t3,t4] = pssOptimizer (P,Xe).
dePSS.m
Purpose
Find the optimum parameters to tune the PSS that provide adequate performance
over a wide range of operating conditions.
Description
This function implements DE. The population size and maximum number of
generations, the mutation rate and the mutation scale factor are set in this file.
This MATLAB script calls the DE toolbox to perform the optimization by evaluating
the objective function’s file ‘eigenShiftDE’.
Invocation: [tw, k1, t1, t2, t3, t4] = dePSS (Po, X)
Where Po and X are the respectively the Power and the line reactance.
gaPSS.m
Purpose
Find the optimum parameters to tune the PSS to provide a minimum performance
over a wide range of operating conditions.
Description
This function implements SGA. The population size and maximum number of
generations, the crossover probability and the mutation scale factor are set in this
file.
This function uses the GAOT to optimize the objective function ‘eigenShiftGA’ by
minimizing the fitness value.
Invocation: [tw, k2, t12, t22, t32, t42] = gaPSS (Po, X).
77
eigenShiftDE.m and eigenShiftGA.m
These MATLAB scripts are the objective function of DE and GA respectively. They
are called to evaluate individuals by their ability to restrict the closed – loop
eigenvalues to the left of the relative stability chosen by the user.
These m-files call the Step_tester, described in section 7.2, to check for the load flow
convergence and calculate the state matrices. The system matrices are then
checked for controllability which enables the shifting of the unstable modes and
achieve the desired damping.
Program flow chart
The interaction between the cited m-files is summarized in figure 7.1 below.
Figure D.9.2 Program flow chart
78
D.2 MATLAB codes
pssOptimizer.m
function [tw,k,t1,t2,t3,t4,k2,t12,t22,t32,t42] = pssOptimize r(P,Xe) %syntax:[tw,k,t1,t2,t3,t4,k2,t12,t22,t32,t42] = pssDesign(Power,Reactance) %pssDesign %calls the functions dePSS and gaPSS to tune the pa rameters of two PSSs %using the phase compensation technique.Returns PSS parameters %k,t1-t4(from dePSS) and k2,t12-t24(from gaPSS). tw is set as 1 %(a typical value for local mode oscillations). % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %T. MULUMBA (MLMTSH005) %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stab ilizers Based on % Genetic Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %Using structure from A.A Phiri," Optimal tuning of Power system stabilizer % using Evolution Al gorithms" Departement of % Electrical Enginee ring UCT, Thesis 2007. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% global Xemin Xestep Xemax global Pmin Pstep Pmax P = P Xe = Xe % pause %Obtain parameters from DE [tw,k,t1,t2,t3,t4]=dePSS(P,Xe) %Obtain parameters from GA [tw,k2,t12,t22,t32,t42]= gaPSS(P,Xe) % pause %Calculate matrices [a,b,c,P_mat,L,L_idx]= Step_testerMod(1,0.5,k,tw,t1 ,t2,t3,t4); L_e = L; P_e = real(P_mat); damp(a); New_eigvalues = []; New_eigvalues1 = []; % pause figure hold for Xe=Xemin:Xestep:Xemax
79
for P=Pmin:Pstep:Pmax P Xe %******************** DIFFEENTIAL EVOLUTION******** *************** % %run Step_tester to obtain closed - loop eigenvalues for DE [a_mat,b_mat,c_mat,bus2]=Step_tester2(P,Xe, k,tw,t1,t2,t3,t4); %Store the open loop eigenvalues damp(a_mat); % pause evalue1=eig(a_mat); New_eigvalues=[New_eigvalues;evalue1']; % % %Plot DE C/L eigenvalues subplot(2,1,1) % y = -20:20; % x = -1*ones(y); % plot(x,y,'--r') drawnow plot(real(New_eigvalues),imag(New_eigvalues ), 'bx' ) title( 'System Closed-loop poles with DE' ) xlabel( 'Real axis' ); ylabel( 'Imaginary axis' ); %************************************************** ************** %*********************GENETIC ALGORITHM************ ************** %run Step_tester to obtain closed - loop eigenvalue s for SGA P Xe [a_mat1,b_mat1,c_mat1,bus21]=Step_tester(P,Xe,k2,tw ,t12,t22,t32,t42); %Store the Closed-loop eigenvalues damp(a_mat1); evalue2=eig(a_mat1); New_eigvalues1=[New_eigvalues1;evalue2']; %Plot DE C/L eigenvalues subplot(2,1,2) % plot(x,y,'--r') drawnow plot(real(New_eigvalues1),imag(New_eigvalue s1), 'kx' ) title( 'System Closed-loop poles with GA' ) xlabel( 'Real axis' ); ylabel( 'Imaginary axis' ); end end hold off return
80
dePSS.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % T. MULUMBA MLMTSH005 %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stab ilizers Based on % Evolution Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [tw,k,t1,t2,t3,t4] = dePSS(Power,Reactance) %************************************************** ********************** %syntax:[tw,k,t1,t2,t3,t4]=dePSS(Power,reactance) %dePSS: Differential Evolution for tuning power sys tem stabilizer %Uses eigenvalues shifting methods % %Implemented using DeMat package by J.V. Zandt and A. Neumaier %K.V. Price, R.M. Storm and J.A. Lampinen,"Differen tial Evolution: % A practical approach to global optimi zation,"Springer 2005. %************************************************** ********************** % Operating condition at which the controller is op timally tuned F_P = Power; F_Xe= Reactance; %************************************************** ****************** % Script file for the initialization and run of the differential % evolution optimizer. %************************************************** ****************** % F_VTR "Value To Reach" (stop when ofunc < F_V TR) F_VTR = -1; % I_D number of parameters of the objective f unction I_D = 5; % FVr_minbound,FVr_maxbound vector of lower and b ounds of initial population % the algorithm seems to work especially well if [FVr_minbound,FVr_maxbound] % covers the region where the global mini mum is expected % *** note: these are no bound constr aints!! *** FVr_minbound = [16 0.6 0.02 0.02 0.02]; FVr_maxbound = [20 4 2 2 2]; I_bnd_constr = 1; %1: use bounds as bound constraints, 0: no bound constraints % I_NP number of population members I_NP = 100; % I_itermax maximum number of iterations (gen erations) I_itermax = 100;
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% F_weight DE-stepsize F_weight ex [0, 2] F_weight = 0.8; % F_CR crossover probabililty constant e x [0, 1] F_CR = 1.0; % I_strategy 1 --> DE/rand/1: % the classical version of DE. % 2 --> DE/local-to-best/1: % a version which has been use d by quite a number % of scientists. Attempts a ba lance between robustness % and fast convergence. % 3 --> DE/best/1 with jitter: % taylored for small populatio n sizes and fast convergence. % Dimensionality should not be too high. % 4 --> DE/rand/1 with per-vector-di ther: % Classical DE with dither to become even more robust. % 5 --> DE/rand/1 with per-generatio n-dither: % Classical DE with dither to become even more robust. % Choosing F_weight = 0.3 is a good start here. % 6 --> DE/rand/1 either-or-algorith m: % Alternates between different ial mutation and three-point-recombination. I_strategy = 1; % I_refresh intermediate output will be produce d after "I_refresh" % iterations. No intermediate output will be produced % if I_refresh is < 1 I_refresh = 5; % I_plotting Will use plotting if set to 1. Will skip plotting otherwise. I_plotting = 0; %************************************************** ************************* % Problem dependent but constant values. For speed reasons these values are % defined here. Otherwise we have to redefine them again and again in the % cost function or pass a large amount of parameter s values. %************************************************** ************************* %------------------------------------------------- S_struct.F_P = F_P; S_struct.F_Xe = F_Xe; S_struct.I_NP = I_NP; S_struct.F_weight = F_weight; S_struct.F_CR = F_CR; S_struct.I_D = I_D; S_struct.FVr_minbound = FVr_minbound;
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S_struct.FVr_maxbound = FVr_maxbound; S_struct.I_bnd_constr = I_bnd_constr; S_struct.I_itermax = I_itermax; S_struct.F_VTR = F_VTR; S_struct.I_strategy = I_strategy; S_struct.I_refresh = I_refresh; S_struct.I_plotting = I_plotting; %************************************************** ****************** % Start of optimization %************************************************** ****************** [FVr_x,S_y,I_nf] = deopt( 'eigenShiftDE' ,S_struct); tw = 2; k = FVr_x(1); t1 = FVr_x(2); t2 = FVr_x(3); t3 = FVr_x(4); t4 = FVr_x(5);
eigenshiftDE.m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Function: S_MSE= eigenShift(FVr_temp, S_s truct) % Author: Rainer Storn % Description: Implements the cost function to be minimized. % Parameters: FVr_temp (I) Paramter ve ctor % S_Struct (I) Contains a variety of parameters. % For details see Rundeopt.m % Return value: S_MSE.I_nc (O) Number of c onstraints % S_MSE.FVr_ca (O) Constraint values. 0 means % the constra ints % are met. Va lues > 0 measure the % distance % to a partic ular constraint. % S_MSE.I_no (O) Number of o bjectives. % S_MSE.FVr_oa (O) Objective f unction values. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % T. MULUMBA MLMTSH005 %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stab ilizers Based on % Evolution Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function S_MSE= eigenShiftDE(FVr_temp, S_struct)
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%---Assign temporary values to parameters---------- -------------------------- tw=2.0; k=FVr_temp(1); t1=FVr_temp(2); t2=FVr_temp(3); t3=FVr_temp(4); t4=FVr_temp(5); F_P = S_struct.F_P; F_Xe= S_struct.F_Xe; % %---State space matrices--------------------------- ------------ [a,b,c,P_mat,L,L_idx]= Step_testerMod2(F_P,0.7,k,tw ,t1,t2,t3,t4); a_mat = a; b_mat = b; d_mat = 0; L_vec = L; P_real = real(P_mat); L_idxVec = L_idx; %----Check for controllability--------------------- ------------ mat_contr = ctrb(a_mat,b_mat); det_check = det(mat_contr);
%Condition for controllability if det_check ~= 0 %find the electromechanical modes P_row = P_real(1,:); spd_mode = P_row - max(P_row); dummy = find(spd_mode == 0); lamnda = L_vec(dummy(1)); %Set conditions d = L_vec > 0; %objective function J = real(lamnda) + 2.0; sigma = -real(lamnda)/sqrt(real(lamnda)^2+ ... imag(lamnda)^2); %Check if individuals meet the conditions. Penalize if otherwise if sum(d)>0 J = J+100; end if (sigma < 0.2 || sigma > 0.4) J = J+100; end if (imag(lamnda)> 4.10 || imag(lamnda)< 3.85) J = J+100; end F_val =J+10; else disp( 'THE SYSTEM IS NOT CONTROLLABLE THUS CANNOT SETTLE' ) disp( 'THE EIGENVALUE SHIFT CANNOT BE PERFORMED' )
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disp( 'Change the system operating conditions' ) end %----strategy to put everything into a cost functio n------------ S_MSE.I_nc = 1; %no constraints S_MSE.FVr_ca = 0; %no constraint array S_MSE.I_no = 1; %number of objectives (costs) S_MSE.FVr_oa(1) = F_val;
gaPSS.m
function [tw,k2,t12,t22,t32,t42]=gaPSS(Po,X) %syntax:[tw,k2,t12,t22,t32,t42]=gaPSS(Power,reactan ce) %gaPSS: genetic algorithm for tuning power system s tabilizer %Uses phase compensation technique % %Implemented using Genetic Algorithm for Optimisati on Toolbox(GAOT) by %C. Houck,J. Joines and M. Kay %http://www.ise.ncsu.edu/mirage/GAToolBox/gaot/ % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %T. MULUMBA (MLMTSH005) %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stab ilizers Based on % Genetic Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %Modified structure from A.A Phiri," Optimal tuning of Power system stabilizer using Evol ution Algorithms" Departement of Electrical Engineering UCT, Thesis 2 007. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% global Pw global Xr Pw = Po; Xr = X; % Setting the seed back to the beginning for compar ison sake rand( 'seed' ,0) % Crossover Operators xFns = 'simpleXover' ; xOpts = [0.65]; % Mutation Operators mFns = 'binaryMutation' ; mOpts = [0.1]; % Termination Operators termFns = 'maxGenTerm' ;
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termOps = [100]; % 200 Generations % Selection Function selectFn = 'roulette' ; selectOps = []; % Evaluation Function evalFn = 'eigenShiftGA' ; evalOps = []; % Bounds on the variables bounds = [16 20;0.6 4;0.02 2;0.02 2;0.02 2]; % GA Options [epsilon float1/binary0 display] gaOpts=[1e-6 0 1]; % Generate an intialize population of size 20 startPop = initializega(100,bounds, 'eigenShiftGA' ,[],[1e-6 0]); % Run the GA [x endPop bestPop trace]=ga(bounds,evalFn,evalOps,s tartPop,gaOpts, ... termFns,termOps,selectFn,selectOps,xFns,xOpts,m Fns,mOpts); % Plot the best over time clf plot(trace(:,1),trace(:,2)); %Return variables tw=2; k2=x(1); t12=x(2); t22=x(3); t32=x(4); t42=x(5); return
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eigenshiftGA.m
function [sol,val] = eigenShiftGA(sol,options) %syntax:[sol,val] = eigenShiftEval(sol,options) %Fitness function for Gentic Algorithm %Calculates the fitness of the population,T. %The performance of each individual is calculated a ccording to its ability %to restrain all eigenvalues to the left of the rel ative stability, thus %minimizing the objective function J. % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %T. MULUMBA (MLMTSH005) %University of Cape Town %Thesis Project:Optimal Tuning of Power System Stab ilizers Based on % Genetic Algorithms %Supervisor: Assoc. Prof. K.A. Folly %13 May 2008 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% global Pw global Xr tw=2; k=sol(1); t1=sol(2); t2=sol(3); t3=sol(4); t4=sol(5); %Check for load flow and get the system state space matrices [a,b,c,P_mat,L,L_idx]= Step_testerMod(1,0.7,k,tw,t1 ,t2,t3,t4); a_mat = a; b_mat = b; L_vec = L; P_real = real(P_mat); %real values of participation factor matrix L_idxVec = L_idx; [wn,z,e_v]=damp(a_mat); %Check for the controllability of the system p = ctrb(a_mat,b_mat); det_check = det(p); %Condition for controllability if det_check ~= 0 %get natural frequency, damping ratio and system po les %[Wn,z,P] = damp(a_mat); P_row = P_real(1,:); spd_mode = P_row - max(P_row); dummy = find(spd_mode == 0); lamnda = L_vec(dummy(1)); d = L_vec > 0;
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%objective function J = real(lamnda) + 2.0; sigma = -real(lamnda)/sqrt(real(lamnda)^2+ ... imag(lamnda)^2); %A = 0;B = 0; C = 0; if sum(d)>0 J = J+100; end if (sigma < 0.2 || sigma > 0.4) J = J+100; end if (imag(lamnda)> 4.10 || imag(lamnda)< 3.85) J = J+100; end val =J+10; else disp( 'System is not controllable' ) end
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APPENDIX E: SIMULATION RESULTS DATA &
GRAPHS
E.1 Data
The participation matrices used to track the eigenvalues associated with the
electromechanical modes are provided below relative to the operating condition.
Each columns of the matrix represents an eigenvalue. Each row is associated with a
state.
Minimal condition
The participation matrix of the open loop system is given below:
No PSS: P_mat minimum condition
-0.0393 0.5197 0.5197 0 0 0 ω∆ -0.0393 0.5197 0.5197 0 0 0 δ∆
0.0003 -0.0002 -0.0002 0.4441 0.4441 0.1119 'qE∆
1.0782 -0.0391 -0.0391 0 0 0 'dE∆
0 0 0 0.1401 0.1401 0.7198 Avr∆ 0 -0.0001 -0.0001 0.4159 0.4159 0.1683 Avr∆
eigen 1 eigen 2 eigen 3 eigen 4 eigen 5 eigen 6
Eigenvalues -1.205 eigen 1 -0.4222 - 5.0371i eigen 2 -0.4222 + 5.0371i eigen 3 -5.5423 - 18.1786i eigen 4 -5.5423 + 18.1786i eigen 5 -59.1579 + 0i eigen 6
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Nominal system condition
At this operating condition, the open loop exhibit undamped oscillatory behaviour.
Therefore, the PSSs are included to improve stability.
No PSS: P_mat
-0.0119 0.50702 0.50702 -0.0011 -0.0011 9.8E-05 ω∆
-0.0119 0.50702 0.50702 -0.0011 -0.0011 9.8E-05 δ∆
0.00379 0.00134 0.00134 0.4431 0.4431 0.10733 'qE∆
1.02051 -0.0101 -0.0101 -0.0002 -0.0002 -7E-07 'dE∆
-0.0001 -0.0016 -0.0016 0.1366 0.1366 0.73011 Avr∆
-0.0003 -0.0037 -0.0037 0.42271 0.42271 0.16236 Avr∆
eigen 1 eigen 2 eigen 3 eigen 4 eigen 5 eigen 6
Eigenvalues
0.0212 + 3.9827i eigen 1 0.0212 - 3.9827i eigen 2 -1.728 eigen 3 -5.9046 +17.2474i eigen 4 -5.9046 -17.2474i eigen 5 -58.5794 eigen 6
CPSS: P_mat
Eigenvalues
-0.447 eigen 1 -1.844 eigen 2 -1.306 -1.644i eigen 3 -1.306+1.644i eigen 4 -2.775 -4.282i eigen 5 -2.775 +4.282i eigen 6 -4.284 -16.78i eigen 7 -4.284+16.78i eigen 8 -58.62 eigen 9
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GAPSS: P_mat
0.090173 -0.0707 -0.0707 0.084897 0.489728 0.489728 -0.00661 -0.00661 9.48E-05 ω∆
-0.00127 -0.05016 -0.05016 -0.0089 0.591216 0.591216 -0.03641 -0.03641 0.000866 δ∆
0.001821 0.01401 0.01401 0.004821 -0.0936 -0.0936 0.522185 0.522185 0.108168
'qE∆
0.003369 0.109517 0.109517 0.772733 0.002621 0.002621 -0.00019 -0.00019 -7.5E-07 'dE∆
-0.00084 -0.0061 -0.0061 -0.00405 0.040243 0.040243 0.104655 0.104655 0.727305 Avr∆
-3.6E-05 -0.00199 -0.00199 -0.0005 0.002612 0.002612 0.417858 0.417858 0.163571 Avr∆
0.826683 0.105967 0.105967 -0.02777 -0.00542 -0.00542 -5.8E-06 -5.8E-06 -2.6E-08 1v∆
0.067885 0.435924 0.435924 0.533344 -0.24651 -0.24651 0.009959 0.009959 3E-05 2v∆
0.012214 0.463533 0.463533 -0.35458 0.219111 0.219111 -0.01144 -0.01144 -3.4E-05 3v∆
eigen 1 eigen 2 eigen 3 eigen 4 eigen 5 eigen 6 eigen 7 eigen 8 eigen 9
Eigenvalues
-0.4449 eigen 1
-1.2115 + 0.5666i eigen 2
-1.2115 - 0.5666i eigen 3
-1.4862 + 3.7973i eigen 4
-1.4862 - 3.7973i eigen 5
-1.772 eigen 6
-4.4269 +16.9785i eigen 7
-4.4269 -16.9785i eigen 8
-58.6272 eigen 9
DEPSS: P_matrix
0.09017 -0.0707 -0.0707 0.0849 0.48973 0.48973 -0.0066 -0.00661 9.5E-05 ω∆
-0.0013 -0.0502 -0.0502 -0.0089 0.59122 0.59122 -0.0364 -0.03641 0.00087 δ∆
0.00182 0.01401 0.01401 0.00482 -0.0936 -0.0936 0.52218 0.522185 0.10817 'qE∆
0.00337 0.10952 0.10952 0.77273 0.00262 0.00262 -0.0002 -0.00019 -7E-07 'dE∆
-0.0008 -0.0061 -0.0061 -0.004 0.04024 0.04024 0.10465 0.104655 0.7273 Avr∆
-4E-05 -0.002 -0.002 -0.0005 0.00261 0.00261 0.41786 0.417858 0.16357 Avr∆
0.82668 0.10597 0.10597 -0.0278 -0.0054 -0.0054 -6E-06 -5.8E-06 -3E-08 1v∆
0.06788 0.43592 0.43592 0.53334 -0.2465 -0.2465 0.00996 0.009959 3E-05 2v∆
0.01221 0.46353 0.46353 -0.3546 0.21911 0.21911 -0.0114 -0.01144 -3E-05 3v∆
eigen 1 eigen 2 eigen 3 eigen 4 eigen 5 eigen 6 eigen 7 eigen 8 eigen 9
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Eigenvalues
-4.49E-01 eigen 1
-1.448-0.652i eigen 2
-1.448-0.652i eigen 3
-1.98E+00 eigen 4
-1.581 -3.93i eigen 5
-1.581 -3.93i eigen 6
-4.507-16.94i eigen 7
-4.507-16.94i eigen 8
-58.62 eigen 9
Maximum system condition
No PSS: P_matrix
0.0061895 0.4978948 0.497895 -0.00103 -0.00103 8.26E-05
0.0061895 0.4978948 0.497895 -0.00103 -0.00103 8.26E-05
-0.001758 0.0031671 0.003167 0.442851 0.442851 0.109723
0.9891524 0.0053913 0.005391 3.32E-05 3.32E-05 -1.38E-06
6.214E-05 -0.001294 -0.00129 0.138998 0.138998 0.724529
0.0001644 -0.003054 -0.00305 0.420179 0.420179 0.165585
eigenval1 eigenval2 eigenval3 eigenval4 eigenval5 eigenval6
Eigenvalues
-1.77 eigenval1
0.0885 - 2.8462i eigenval2
0.0885 + 2.8462i eigenval3
-5.7253 -17.7221i eigenval4
-5.7253 +17.7221i eigenval5
-58.8874 eigenval6
ω∆δ∆
'qE∆'dE∆
Avr∆Avr∆
92
CPSS: P_matrix
0.0286773 0.0071147 0.522018 0.522018 -0.0748 -2.74E-16 -0.00256 -0.00256 8.18E-05 ω∆
-0.000202 0.0048303 0.4379 0.4379 0.150617 3.54E-15 -0.01581 -0.01581 0.000579 δ∆
0.0002217 -0.001409 -0.01612 -0.01612 -0.03432 2.24E-13 0.478746 0.478746 0.110254
'qE∆
-0.000161 0.9831655 0.006745 0.006745 0.003437 6.75E-14 3.48E-05 3.48E-05 -1.39E-06 'dE∆
-0.000142 -3.4E-05 0.006648 0.006648 0.023019 -1E-13 0.12057 0.12057 0.722722 Avr∆
-3.45E-06 0.0001312 -0.0014 -0.0014 0.008521 -2.15E-13 0.413902 0.413902 0.166345 Avr∆
0.9694355 -0.001065 0.010284 0.010284 0.010964 6.86E-17 4.84E-05 4.84E-05 7.69E-08 1v∆
0.002893 0.005811 0.110177 0.110177 -0.48848 1.216204 0.021562 0.021562 9.13E-05 2v∆
-0.00071 0.0014554 -0.07626 -0.07626 1.401041 -0.21624 -0.01649 -0.01649 -7.22E-05 3v∆
eigenval1 eigenval2 eigenval3 eigenval4 eigenval5 eigenval6 eigenval7 eigenval8 eigenval9
Eigenvalues
-0.2429 eigenval1
-1.7654 eigenval2
-0.233 - 2.6355i eigenval3
-0.233 + 2.6355i eigenval4
-4.0869 eigenval5
-10 eigenval6
-4.9280 -17.6448i eigenval7
-4.928 +17.6448i eigenval8
-58.9166 eigenval9
GAPSS: P_mat
0.121986 -0.18468 -0.1847 0.00826 0.6229 0.6229 -0.0034 -0.0034 8.1E-05 ω∆
-0.00263 -0.0267 -0.0267 0.00216 0.55176 0.55176 -0.0252 -0.0252 0.0008 δ∆
0.00168 0.001176 0.00118 -0.0007 -0.0575 -0.05754 0.50064 0.50064 0.11049 'qE∆
-0.00158 -0.0382 -0.0382 1.0654 0.00625 0.00625 3.6E-05 3.6E-05 -1.4E-06 'dE∆
-0.00109 -0.0013 -0.0013 -0.0002 0.0283 0.0283 0.11266 0.11266 0.72195 Avr∆
-4.2E-05 -0.00076 -0.0008 6.5E-05 0.00304 0.00304 0.41437 0.41437 0.16668 Avr∆
0.781547 0.122326 0.12233 -0.0036 -0.0113 -0.01135 2.4E-05 2.4E-05 3E-08 1v∆
0.093033 0.67354 0.67354 -0.0115 -0.2177 -0.21771 0.00341 0.00341 1.2E-05 2v∆
0.007139 0.454596 0.4546 -0.0599 0.07432 0.07432 -0.0025 -0.0025 -9E-06 3v∆
eigenval1 eigenval2 eigenval3 eigenval4 eigenval5 eigenval6 eigenval7 eigenval8 eigenval9
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Eigenvalues
-0.4288 eigenval1
-1.301 - 0.685i eigenval2
-1.301 + 0.685i eigenval3
-1.7526 eigenval4
-1.068 - 2.5i eigenval5
-1.068 + 2.5i eigenval6
-4.55 -17.52i eigenval7
-4.55 +17.52i eigenval8
-58.9294 eigenval9
DEPSS: P_mat
0.114404 -0.02554 -0.1949 -0.1949 0.65373 0.65373 -0.0033 -0.0033 8.1E-05 ω∆
-0.00257 0.003092 -0.1882 -0.1882 0.7143 0.7143 -0.0268 -0.0268 0.00075 δ∆
0.001595 -0.00045 0.02849 0.02849 -0.0834 -0.08345 0.49917 0.49917 0.11044 'qE∆
-0.00151 1.000717 -0.0023 -0.0023 0.00262 0.00262 3.6E-05 3.6E-05 -1.4E-06 'dE∆
-0.00104 0.001025 -0.0189 -0.0189 0.04272 0.04272 0.11511 0.11511 0.72213 Avr∆
-4.0E-05 4.38E-05 -0.0047 -0.0047 0.00454 0.00454 0.41684 0.41684 0.16661 Avr∆
0.794598 0.010367 0.13178 0.13178 -0.0343 -0.03426 -5E-06 -5E-06 -2.2E-08 1v∆
0.081483 -0.10575 1.04665 1.04665 -0.5417 -0.54171 0.00717 0.00717 2.6E-05 2v∆
0.013073 0.116504 0.20198 0.20198 0.2415 0.2415 -0.0082 -0.0082 -2.9E-05 3v∆
eigenval1 eigenval2 eigenval3 eigenval4 eigenval5 eigenval6 eigenval7 eigenval8 eigenval9
Eigenvalues
-0.2429 eigenval1
-1.7654 eigenval2
-0.3233 - 2.6355i eigenval3
-0.3233 + 2.6355i eigenval4
-4.0869 eigenval5
-10 eigenval6
-4.9280 -17.6448i eigenval7
-4.9280 +17.6448i eigenval8
-58.9166 eigenval9
94
E.2 Graphs
This section includes the graphs tested for robustness. A step change of 0.001 in
input voltage was applied for the all range of operating condition.
Figure E.2 : Xe = 0.7 and P = 0.7
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.6 and Q=0.17203 at Xe=0.7
time(s)
chan
ge in
Vt (p
.u)
DEPSS
GAPSS
CPSS
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.7 and Q=0.22131 at Xe=0.7
time(s)
chan
ge in
Vt
(p.u
)
DEPSS
GAPSS
CPSS
Figure E.1: Xe = 0.7 and P =0.6
95
Figure E.4 : X = 0.7 and P = 0.9
Figure E.3 : X = 0.7 and P = 0.8
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.8 and Q=0.28062 at Xe=0.7
time(s)
chan
ge in
Vt (p
.u)
DEPSS
GAPSS
CPSS
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.9 and Q=0.35148 at Xe=0.7
time(s)
chan
ge in
Vt
(p.u
)
DEPSS
GAPSS
CPSS
96
Figure E.5 : Xe = 0.8 and P = 0.6
Figure E.6 : Xe = 0.8 and P = 0.7
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.6 and Q=0.18698 at Xe=0.8
time(s)
chan
ge in
Vt (p
.u)
DEPSS
GAPSS
CPSS
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.7 and Q=0.24554 at Xe=0.8
time(s)
chan
ge in
Vt (p
.u)
DEPSS
GAPSS
CPSS
97
Figure E.7: X =0.8 and P = 0.8
Figure E.8 : X =0.8 and P = 0.9
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.8 and Q=0.31734 at Xe=0.8
time(s)
chan
ge in
Vt
(p.u
)
DEPSS
GAPSS
CPSS
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.9 and Q=0.40544 at Xe=0.8
time(s)
chan
ge in
Vt
(p.u
)
DEPSS
GAPSS
CPSS
98
Figure E.9 Xe = 0.8 and P = 1.0
Figure E.10 Xe = 0.9 and P = 0.6
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=1 and Q=0.51516 at Xe=0.8
time(s)
chan
ge in
Vt (p
.u)
DEPSS
GAPSS
CPSS
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.6 and Q=0.20423 at Xe=0.9
time(s)
chan
ge in
Vt
(p.u
)
DEPSS
GAPSS
CPSS
99
Figure E.11: X = 0.9 and P = 0.7
Figure E.12 : X = 0.9 and P = 0.8
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.7 and Q=0.27338 at Xe=0.9
time(s)
chan
ge in
Vt (p
.u)
DEPSS
GAPSS
CPSS
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.8 and Q=0.36039 at Xe=0.9
time(s)
chan
ge in
Vt
(p.u
)
DEPSS
GAPSS
CPSS
100
Figure E.13: X = 0.9 and P = 0.9
0 1 2 3 4 5 6 7 8 9 103
4
5
6
7
8
9
10
11
12
13
x 10-3 Step response for op cond. P=0.9 and Q=0.47185 at Xe=0.9
time(s)
chan
ge in
Vt (p
.u)
DEPSS
GAPSS
CPSS
101
APPENDIX F: SOFTWARE CD
The software CD contains the MATLAB applications developed to find the optimal
parameters used to tune the PSS. The CD also includes the Genetic Algorithm
Toolbox (GAOT) version 5 and the Differential Evolution package (DeMAt).
GAOT and DeMAT are free toolbox available for download from
“http://www.ise.ncsu.edu/mirage/GAToolBox/gaot/”and
http://www.icsi.berkeley.edu/~storn/code.html.
The “*.MAT” files are in the folder “PSS Optimizer”. The driver program is
“smibDriver.m”. The following instruction must be followed in order to use properly:
i. Open MATLAB.
ii. In the MATLAB “current Directory”, set the path to the “PSS Optimizer” folder
iii. From the command window, enter:
“ smibDriver(Pmin, Pstep, Pmax, Xemin, Xestep, Xemax)”
iv. Or Opening the file “smibDriver” in the current directory window and execute.
v. If the error: “sim_fle could not be append” occurs, close the MATLAB
application and then restart. It will work.
The simulation’ data results are contained in the doc file “Results.doc” and can be
access after simulation.
![Design of Multi-stage Fuzzy PID Bundled Artificial Bee ...€¦ · field of tuning methodologies are all part of the PSS topic [1]. The action of a PSS is to extend the angular stability](https://img.dokumen.tips/doc/110x75/5eacf956c599975f0f2fa2dd/design-of-multi-stage-fuzzy-pid-bundled-artificial-bee-field-of-tuning-methodologies.jpg)
