A FUZZY LOGIC BASED AUTOMATIC VOLTAGE
REGULATOR FOR ALTERNATOR TERMINAL
VOLTAGE AND REACTIVE POWER CONTROL
A PROJECT WORK PRESENTED IN PARTIAL FULFILMENT
FOR THE AWARD OF MASTER OF ENGINEERING (M. Eng)
DEGREE IN ELECTRICAL ENGINEERING
BY
OKOZI, SAMUEL OKECHUKWU
B. Eng (Hons), FUTO
PG/M.ENG/06/40693
DEPARTMENT OF ELECTRICAL ENGINEERING
UNIVERSITY OF NIGERIA, NSUKKA.
MARCH, 2009
A FUZZY LOGIC BASED AUTOMATIC VOLTAGE
REGULATOR FOR ALTERNATOR TERMINAL
VOLTAGE AND REACTIVE POWER CONTROL
A PROJECT WORK PRESENTED IN PARTIAL FULFILMENT
FOR THE AWARD OF MASTER OF ENGINEERING (M. Eng)
DEGREE IN ELECTRICAL ENGINEERING
BY
OKOZI, SAMUEL OKECHUKWU
B. Eng (Hons), FUTO
PG/M.ENG/06/40693
DEPARTMENT OF ELECTRICAL ENGINEERING
UNIVERSITY OF NIGERIA, NSUKKA.
SUPERVISOR: VEN. ENGR. PROF. T.C MADUEME
MARCH, 2009
A FUZZY LOGIC BASED AUTOMATIC VOLTAGE REGULATOR FOR ALTERNATOR TERMINAL VOLTAGE AND REACTIVE POWER CONTROL
A PROJECT WORK PRESENTED IN PARTIAL FULFILMENT FOR THE
AWARD OF MASTER OF ENGINEERING DEGREE IN ELECTRICAL
ENGINEERING
BY: OKOZI, SAMUEL OKECHUKWU
B. Eng (Hons), FUTO
PG/M.ENG/06/40693
DEPARTMENT OF ELECTRICAL ENGINEERING UNIVERSITY OF NIGERIA, NSUKKA.
MARCH, 2009
AUTHOR: --------------------------------------------- OKOZI, SAMUEL OKECHUKWU SUPERVISOR: --------------------------------------------- VEN. ENGR. PROF. T.C. MADUEME HEAD OF DEPARTMENT: ------------------------------------------ ENGR. PROF. M.U. AGU EXTERNAL EXAMINER: ------------------------------------------- ENGR. PROF. J.C. EKE
DECLARATION I, Okozi, Samuel Okechukwu, a postgraduate student in the Department of
Electrical Engineering, with Registration number PG/M.ENG/06/40693
humbly declare that this is my work and it has not been submitted in part or full
for any other Diploma or Degree of this university to the best of my knowledge.
------------------------------------ Okozi, Samuel Okechukwu
CERTIFICATION
OKOZI, Samuel Okechukwu, a postgraduate student in the Department
of Electrical Engineering, with Registration number PG/M.ENG/06/40693, has
satisfactorily completed the requirements for course and research work for the
Degree of Master of Engineering (M. Eng) in the department of Electrical
Engineering University of Nigeria, Nsukka.
The work embodied in the Dissertation is original and has not been
submitted in part or full for any other Diploma or Degree of this university to
the best of my knowledge.
--------------------------------------- ------------------------------ Ven. Engr. Prof. T.C. Madueme Engr. Prof. M.U. Agu SUPERVISOR HEAD OF DEPT.
DEDICATION
This work is dedicated to the Almighty God who made it possible for my dream to come true, and to the memory of my dear mother, Late Mrs. Rosaline Okozi.
ACKNOWLEDGEMENT I wish to express my profound gratitude to my supervisor , Ven. Engr. Prof T.C.
Madueme who has been like a father throughout the cause of this work. I thank
the Head of the Department, Engr. Prof. M.U. Agu for his immense help and
open door policy even on a short notice.
I am also grateful to Engr. Dr. L.U. Anih whose encouraging words I shall
always remember and also very grateful to Engr. Dr. E. S. Obe for providing me
with lots of journals on fuzzy logic from far away Germany.
In a special way, I thank Engr. Dr. S.N. Ndubisi of Electrical/Electronic
Engineering Department, Enugu State University of Technology (ESUT) for
giving me the foundation of what fuzzy logic is all about.
. I also remain grateful to the following lecturers in this Department for their
encouragements Engr. B.O Nnadi, Engr. C. Odeh and Engr.S.O. Oti.
Also not left out in my train of appreciation were my colleagues, Mbadiwe,
Cosmas, Umor, Nelson, Ben and N.D for our cooperation cannot be forgotten in
a hurry. Also included are my friends, Cornelius, Jude, Naze and Eniola of
Power Holding Company of Nigeria (PHCN), New Haven work centre, Enugu.
Finally, I must not forget my family members for the support and understanding
throughout this programme.
To all of you and the Almighty God, I remain grateful.
Samuel Okechukwu Okozi
ABSTRACT
The frequency of a power system is affected by changes in real power while the
terminal voltage magnitude is affected by changes in reactive power. The load
frequency control loop (LFC) controls the real power and frequency while the
Automatic voltage regulator (AVR) controls the reactive power and voltage
magnitude of an alternator.
The role of the AVR is to hold the terminal voltage magnitude of a synchronous
generator at a specified level. As the reactive load of the consumers increases
beyond the rated value of the generator, the result will be a decrease in the
terminal voltage of the generator. Most of the appliances therefore cannot be
powered hence the need to bring the voltage level back to the specified level.
The conventional control approach to the terminal voltage and reactive power
regulation involves the development of the linear equations of the AVR models
and then using the appropriate control theory to develop the controller.
This conventional modeling and control approach has been seen to perform
poorly when the system operating conditions change (or vary). This is due to the
fixed parameters and the linearized models of the system. This and other
reasons have led to the use of fuzzy logic control and programming in the
control of the terminal voltage and reactive power of AVR of an alternator.
Conventional modeling and control approach is highlighted, fuzzy dynamic
programming approach is used extensively and the results were simulated using
MATLAB software packages.
DEFINITION OF TERMS Rise time: The time for a system to respond to a step input and attains a
response equal to the magnitude of the input. Peak time: The time for a system to respond to a step input and rise to a
peak response. Overshoot: The amount the system output response proceeds beyond the
desired response. Settling time: The time required for the system output to settle within a
certain percentage of the input amplitude. PID controller: A controller with three terms in which the output of the
controller is the sum of a proportional term, an integrating term, and a differentiating term, with an adjustable gain for each term.
Steady-state error: The error when the time period is large and when the transient response has decayed, leaving the continuous response.
LIST OF SYMBOLS AND ABBREVATIONS
KR Sensor gain KA Amplifier gain KE Exciter gain KG Generator gain
Rτ Sensor time constant Aτ Amplifier time constant Eτ Exciter time constant Gτ Generator time constant
Vtss Steady-state response of the AVR Vs(s) Output voltage of the sensor Vref Reference voltage Vt Generator terminal voltage VR Input voltage to the exciter VF Field voltage PID Proportional Integral Derivative PI Proportional Integral
refV Reference/specified voltage to generator
tV Terminal voltage of the generator 1V Voltage error (Vref - Vt )
eV∫ Integral of the error, ∫ Ve
)(te Error Int. e Integral of the error e U Output of the fuzzy rules (input to the plant)
1g Scaling gain for tuning of the membership functions for e (t)
2g Scaling gain for tuning of the membership functions for “int. e” 0g Scaling gain for tuning of the membership functions for u
d(t) Power angle of the generator in radian w(t) Rotor speed of the generator in rad/sec. D Damping coefficient of the generator H Inertia coefficient of the generator w0 Speed of the generator at the operating point
mP Motor mechanical power in p.u. )(tPe Active electrical power delivered by the generator in p.u. )(tE f Equivalent EMF in the excitation coil in p.u.
)(, tfE Transient EMF in the generator coil in p.u.
)(, tdE Transient direct axis EMF of the generator
)(, tqE Transient EMF of the generator in the quadrature axis
dX Direct axis reactance of the generator
dX , Transient direct axis reactance of the generator )(, tX q Transient reactance in the quadrature axis in the generator in p.u.
dI Direct axis current of the generator qI Quadrature axis current of the generator
fI Excitation field current
sV Infinite bus voltage in p.u. fU Output of the PI fuzzy controller
dsX Stator reactance in the direct axis
SMIB Single Machine Infinite Bus LFC Load frequency control AVR Automatic Voltage Regulator MATLAB Matrix Laboratory PSS Power system stabilization VAR Volt Ampere Reactive FLC Fuzzy Logic Control PT Potential Transformer NV Negative Very NL Negative Large NB Negative Big NM Negative Medium NS Negative Small Z Zero PS Positive Small PM Positive Medium PB Positive Big PL Positive Large PV Positive Very FIS Fuzzy inference System D/A Digital/Analogue FLPSS Fuzzy Logic Power System Stabilizer FPI-AVR Fuzzy Proportional Integral-Automatic Voltage Regulator CAVR Conventional Automatic Voltage Regulator G Generator CB Circuit Breaker
LIST OF TABLES Table 3.1 34
LIST OF FIGURES AND DIAGRAMS Figure 1.1 Conventional control design 5 Figure 1.2 Fuzzy control design 6 Figure2.1 Schematic diagram of LFC and AVR of a
synchronous generator 9 Figure 2.2 A simple diagram of an AVR 10 Figure 2.3 Block diagram of a simple AVR 12 Figure 2.4 Root locus plot for equation 2.13 15 Figure 2.5 Terminal voltage response 17 Figure 2.6 Simulink model for figure 2.5 18 Figure 2.7 Block diagram of a simple AVR
compensated with a stabilizer 19 Figure 2.8 Terminal voltage step response of CAVR
with stabilizer 20 Figure 2.9 Simulink model for figure 2.7 21 Figure 2.10 An AVR compensated with PID controller 22 Figure 2.11 Simulink model for figure 2.10 23 Figure 2.12 Terminal voltage step response with PID 24 Figure 3.1 Fuzzy controller architecture 27 Figure 3.2 Fuzzy control system 28 Figure 3.3a Membership functions for the error 32 Figure 3.3b Membership functions for eV∫ 32 Figure 3.3c Membership function for the terminal voltage 33 Figure 3.4 FIS Editor 36 Figure 3.5 The Rule Editor 37 Figure 3.6 The Rule Viewer 38 Figure 3.7 Schematic diagram for fuzzy AVR operation 40 Figure 3.8 Fuzzy-AVR control loop 40 Figure 3.9 Terminal and desired voltage step response
with FPI-AVR 42 Figure 3.10 Fuzzy input voltage to the generator 42 Figure 3.11 A Simple machine infinite bus power system 44 Figure 3.12a Step response of FPI-AVR to three phase fault 47 Figure 3.12b Fuzzy input voltage to the generator 47 Figure 4.1 Terminal voltage response 49 Figure 4.2 Terminal voltage step response of CAVR
with stabilizer 50 Figure 4.3 Terminal voltage step response with PID 51 Figure 4.4a Terminal and desired voltage step response
with FPI-AVR 52 Figure 4.4b Fuzzy input voltage to the generator 52 Figure 4.5a Step response of FPI-AVR to three phase fault 53 Figure 4.5b Fuzzy input voltage to the generator 53
CONTENTS Title page
Declaration
Certification
Dedication:
Acknowledgement:
Abstract: i
Definition of terms iii
List of symbols and abbreviation iv
List of tables vi
List of figures and Diagrams vii
Contents: viii
Chapter 1 Introduction: 1
Chapter 2 Conventional Control Approach 9
2.1 Sensor Model 11
2.2 Amplifier Model 11
2.3 Exciter Model 11
2.4 Generator Model 12
2.5 Test for stability 14
2.6 Test for Controllability 16
2.7 AVR Excitation with stabilizer 18
2.8 AVR Excitation with PID controller 22
2.9 Overview of fuzzy logic 24
Chapter 3 Fuzzy Logic Control Approach 27
3.1 Fuzzy Logic Controller Design 27
3.2 Fuzzification 29
3.3 Formulation of the rule base and the membership
function definition 29
3.4 The inference mechanism 33
3.5 Defuzzification 38
3.6 Mechanical equations of a SMIB 44
3.7 Electrical generator dynamics 44
3.8 Electrical equations of a SMIB 44
3.9 Linear model of SMIB 45
Chapter 4 Simulation Results 48
Chapter 5 Conclusions 54
References 55
Appendices 57
CHAPTER ONE
INTRODUCTION
The main objectives of the control strategy in power system is to generate and
deliver power in an interconnected system as economically and reliably as
possible while maintaining the voltage and frequency in the steady-state within
permissible limit [1].
A reliable, continuous supply of electric energy is essential for the functioning
of today’s complex societies. Due to a combination of increasing energy
consumption and impediments of various kinds concerning the extension of
existing electric transmission networks, these power systems are operated closer
and closer to their limits.
Deregulatory efforts will tighten the economical constraints under which
utilities have to operate their own network or allow or prevent competitors from
using it. This in turn will require more precise power flow control which is
made possible by phase angle controllers being developed using new power
electronic equipment. However, it is to be expected that these highly non-linear
components will introduce harmonics and require non-linear control in order to
prevent system destabilization.
This situation requires a significantly less conservative power system
operation regime which, in turn, is possible only by monitoring and controlling
the system state in much more detail than was necessary previously.
In electric power systems, [2], there are three different control levels:
Generating Unit Controls which consist of prime mover control and excitation
control with automatic voltage Regulator (AVR) and power system stabilization
(PSS). The first controls generator speed deviation and energy supply system
variable like boiler pressure or water flow. Excitation control aims at
maintaining the generator terminal voltage and reactive power output within its
machine-dependent limits.
System Generation Control which determines active power output such that
the overall system generation meets the system load. It further controls the
frequency and the tie line flows between different power system areas.
Transmission Control monitors power and voltage control devices like tap-
changing transformers, synchronous condensers and static VAR compensators.
From the view point of system automation, [3], Generating Unit Control
is a complete closed-loop system and in the past a lot of effort has been
dedicated to improve the performance of the controllers. The main problem for
example with excitation control is that the control law is based on a linearized
machine model and the control parameters are tuned to some nominal operating
conditions. In case of a large disturbance, the system conditions will change in a
highly non-linear manner and the controller parameters are no longer valid. In
this case the controller may even add a destabilizing effect to the disturbance by
for example adding negative damping.
These problems provide an important motivation to explore other modern
control techniques like fuzzy logic control.
The system frequency is affected by changes in real power while changes in
reactive power affect the voltage magnitude. While the Load Frequency Control
(LFC) controls the real power and frequency, the Automatic Voltage Regulator
(AVR) controls the reactive power and terminal voltage magnitude. Because the
excitation time constant is much smaller than the prime mover time constant, its
transient decays much faster. For this reason, the cross-coupling between the
LFC loop and AVR loop is negligible and hence the load frequency and
excitation voltage control can be analyzed independently.
In modern large power system interconnected networks, manual regulation is
not feasible and therefore automatic generation and voltage regulation
equipment are installed on each generator.
The increasing technological demands and performance requirements call for
complex production and manufacturing systems that in turn requires
sophisticated control systems. To satisfy the product quality requirement in a
flexible production environment, an advanced control techniques that can deal
with uncertainty and non-linear ties need to be introduced. Hence the need for
fuzzy logic control which can handle both linear and non-linear system. Fuzzy
logic is a paradigm for an alternative design methodology which can be applied
in developing both linear and non-linear systems [4].
Ndubisi and Agu [5] puts it thus, Fuzzy logic concept incorporates an
alternative way which allows one to design a controller using a higher level of
abstraction without knowing the plant model.
Conventional modeling and control approaches based on differential equations
are often insufficient, mainly due to the lack of precise formal knowledge about
the process to be controlled. Unlike the conventional control, if the
mathematical model of the process is unknown we can design fuzzy controllers
in a manner that guarantees certain key performance criteria.
Lee et al [6] states that while the conventional control starts with a mathematical
model of the process and controllers are designed based on the model, fuzzy
logic control on the other hand starts with heuristics and human expertise
knowledge (in terms of fuzzy IF-THEN rules) and controllers are designed by
synthesizing these rules. An important source of information to the fuzzy design
is the knowledge of the plant operators, control engineers and process designers.
Fuzzy logic control (FLC) reduces the time and complexity in analyzing the
differential equations involved in the conventional control and hence in the
overall design development cycle as depicted in Figures 1.1 and 1.2
respectively.
Conventional Design Methodology
Fig 1.1: Conventional control design
Understand physical system and control requirements
Develop a Linear Model of Plant, Sensors and Actuators
Develop an algorithm for the controller
Determine a simplified controller from control theory
Simulate, Debug, and Implement the Design
Fuzzy based Design Methodology
Fig.1.2: Fuzzy control design
Using the conventional approach, our first step is to understand the physical
system and its control requirements. Based on this understanding, our second
step is to develop a model which includes the plant, sensors and actuators. The
third step is to use linear control theory in order to determine a simplified
version of the controller, such as the parameters of a PID, PI controllers. The
Understand physical system and control requirements
Design the controller using fuzzy rules
Simulate, Debug and implement the design
fourth step is to develop an algorithm for the simplified controller. The last step
is to simulate the design including the effects of non-linearity, noise, and
parameter variations. If the performance is not satisfactory we need to modify
our system modeling, re-design the controller, re-write the algorithm and re-try.
With fuzzy logic the first step is to understand and characterize the system
behavior by using our knowledge and experience. The second step is to directly
design the control algorithm using fuzzy rules, which describe the principles of
the controller’s regulation in terms of the relationship between its inputs and
outputs. The last step is to simulate and debug the design. If the performance is
not satisfactory we only need to modify some fuzzy rules and re-try.
Although the two design methodologies are similar, the fuzzy based
methodology substantially simplifies the design loop. This results in some
significant benefits, such as reduced development time and simpler design.
As reported in [7], seven fuzzy subsets were employed to develop software
written in C++ in the design of a fuzzy AVR of a controller for a synchronous
generator.
Fuzzy experts like Lofti Zadeh proved that the greater the number of fuzzy
subsets, the better the performance of the controller. The cumbersome nature of
the C++ language was reduced by the use of the MATLAB software.
In this work, a rule based fuzzy logic controller is developed for controlling the
terminal voltage and reactive power of a synchronous generator. Eleven fuzzy
subsets were utilized to enhance the performance of the fuzzy controller. The
work is arranged in this order; first the conventional control approach is
discussed, followed by the fuzzy logic control approach. A comparison is made
between the results of the two control approaches. All the simulations are
carried out using MATLAB software package. The results show reduction in
percent overshoot, rise time, peak time, settling time and overall responses
when fuzzy logic approach is applied.
CHAPTER TWO
CONVENTIONAL CONTROL APPROACH
A typical arrangement of a simple AVR block with the different models is as
shown in figure 2.1.
Gen. field
Steam
∆PG, ∆QG
tiePΔ
CPΔ
Fig. 2.1: The schematic diagram of LFC and AVR of a synchronous
generator
As mentioned earlier, a change in reactive power affects mainly the magnitude
of the terminal voltage. The excitation system maintains generator voltage and
controls reactive power flow. The role of the Automatic Voltage Regulator
(AVR) is to hold the terminal voltage magnitude of a synchronous generator at a
Frequency Sensor
Automatic voltage Regulator (AVR)
Excitation System
Voltage sensor
Valve control Mechanism
Load frequency Control (LFC)
G Turbine
∆PV
specified level. An increase in the reactive power load of the generator leads to
a drop in the magnitude of the terminal voltage. This drop in the magnitude of
the terminal voltage is sensed through a potential transformer (PT), this voltage
is rectified and compared to the d.c set point signal. The compared voltage
(error signal) is then amplified and sent to the exciter which controls the exciter
field, and increases the exciter terminal voltage which results in the increase of
the generator e.m.f.
The simple schematic diagram of a simple AVR only can be then drawn from
the figure 2.1 as shown in figure 2.2
Exciter + + VR fV _ _ _ Vs P.T
Fig. 2.2: A simple diagram of an AVR
The transfer function of each model of the AVR is developed as follows;
2.1 SENSOR MODEL
In this model, the potential transformer here senses the terminal voltage of the
generator and then rectifies the output at this end through a bridge rectifier. This
Σ
Rectifier
Amplifier
Stabilizer
G EV refV
_
model has a gain of KR and a time constant of Rτ . The transfer function of this
model is therefore given by;
)()(
sVsV
t
s = s
K
R
R
τ+1 2.1
2.2 AMPLIFIER MODEL
The terminal voltage of the generator at any time is compared with the reference
voltage and the error, EV = (Vref – Vs) is then amplified in this model. This model
has a gain KA and a time constant of Eτ . The transfer function of this model is
represented by;
)()(
sVsV
E
R = s
K
A
A
τ+1 2.2
2.3 EXCITER MODEL
This model which can be a solid state device has a single time constant Eτ and a
gain EK . The voltage of an exciter has no simple relationship with the terminal
voltage because of the saturation effect in the magnetic circuit. Modern exciter
ignores the saturation effects and other nonlinearities. The transfer function of
this model is as written below;
)()(
sVsV
R
F = s
K
E
E
τ+1 2.3
2.4 GENERATOR MODEL
The terminal voltage of the synchronous generator is dependent on the generator
load. The e.m.f generated by the synchronous generator is a function of the
magnetization curve. The Generator model has a time constant and a gain
constant of Gτ and GK respectively. The transfer function relating the terminal
voltage of the generator to the generator field voltage is given by;
)()(
sVsV
F
t =s
K
G
G
τ+1 2.4
The block diagram of the AVR can now be drawn in the figure 2.3
by utilizing the model equations 2.1, 2.2, 2.3.
)(sRV )(sFV VF(s)
_ Vs(s) Amplifier Exciter Generator
Sensor Fig 2.3: Block diagram of a simple AVR.
The block diagram of an isolated generator with a dc automatic voltage
regulator is shown in figure 2.3. The first block represents the gain and time
constant of an electronic amplifier, the second block is the time constant of the
dc generator field winding, and the third block represents the a.c generator field
time constant. The voltage transducer (Sensor) time constant is in the negative
feedback path. From the above block diagram, the transfer function relating the
Σ sK
A
A
τ+1
sK
R
Rτ+1
sK
G
G
τ+1
sK
E
E
τ+1
)(stV )(sEV refV
generator terminal voltage Vt (s) to the reference voltage Vref (s) using Mason’s
gain rule [8] is;
)(
)(sV
sV
ref
t =RGEARGEA
RRGEA
KKKKsssssKKKK+++++
+)1)(1)(1)(1(
)1(τττττ 2.5
The open-loop transfer function of the block diagram is written as
KG(s) H(s) =)1)(1)(1)(1( ssss
KKKK
RGEA
RGEA
ττττ ++++ 2.6
Also using Ackermann’s formula [9], the state equations of this excitation
system in equation 2.5 are
x1 Aτ1− 0
A
Kτ− 0 x1
A
Kτ
x2 Eτ1
Eτ1− 0 0 x2 0
dtd x3 = 0 0
Rτ1−
Rτ1 x3 + 0 Vref
Vt 0 Gτ1 0
Gτ1− Vt 0
tV = [0 0 0 1]⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
tvxxx
3
2
1
2.9
With the AVR system of generator having the following parameters,
KE=KG=KR=1, Aτ =0.1, Eτ =0.4, Gτ =1.0 and Rτ =0.05, 2.10
The characteristic equation then becomes
1+ KG(s) H(s) = 1+0.05s)s)(10.4s)(10.1s)(1(1
)1)(1)(1(AK++++
= 1+)20)(1)(5.2)(10(
500++++ ssss
AK
= 50077525.30735.334
500
++++ ssssAK 2.11
Also the steady-state response of the system is
Vtss= 0
lim→s
sVt(s) = A
A
KK+1
2.12
The characteristic polynomial equation becomes
s4+33.5s3+307.5s2+775s+500+500KA = 0 2.13
2.5 TEST FOR STABILITY
For control stability, the range of KA is found using the Routh-Hurwitz array;
s4 1 307.5 500+500KA
s3 33.5 775 0
s2 284.365 500+500KA 0
s1 58.9KA – 716.1 0 0
s0 500+500KA
From the table for absolute stability,
58.9KA – 716. 1≤ 0
∴KA<12.16
The root-locus plot as K varies from 0 to 12.16 is shown in figure 2.4 and
obtained using the MATLAB commands shown in appendix I.
Fig. 2.4: Root locus plot for equation 2.13
The result shows that the loci intercept the imaginary axis ( ωj ) at s = ±j4.81 for
KA = 12.16 which shows that the system is marginally stable for KA = 12.16.
If the amplifier gain is set at KA = 10, the state-space equation becomes;
-1.0 1.0 0
A = 0 -2.5 2.5 2.15
0 0 -10
0
B = 0 2.16
0
2.6 TESTING FOR CONTROLLABILITY
0 0 2.5
Pc = [B AB A2B] = 0 25 -312.5 2.17
100 -100 1000
The determinant of Pc = -6250 and ≠ 0.
Therefore the system is controllable.
The steady-state response is
101
10+
=tssV = 0.909
The steady-state error, essV = 1.0 – 0.909
= 0.091
Using equation 2.7 with KA= 10, the terminal voltage response of the system is
obtained using the following MATLAB commands shown in appendix II or the
Simulink model shown in figure 2.6. The result is as shown in figure 2.5
Fig. 2.5: Terminal Voltage Response
From the terminal voltage step response, it is seen that the system has a large
percent overshoot and a long settling time with an oscillatory response, and a
steady-state error of more than 9 percent.
The time-domain performances specifications of the system are as follows;
Rise time = 0.247
Peak time = 0.791
Settling time = 19.04
Percent overshoot = 82.46
The Simulink model for the block diagram in figure 2.3 is shown in
figure 2.6. The simulation result shows that the same voltage response as in
figure 2.5 was obtained.
Fig. 2.6: Simulink model for figure 2.5
2.7 AVR EXCITATION WITH STABILIZER
The basic function of a power system stabilizer is to add damping to the
generator power oscillations by controlling its excitation using an
auxiliary stabilizing signal [10]. From the analysis above, the AVR step
response was not satisfactory even with a small amplifier gain of
Vref
Vt
Step
1
0.05s+1
Sensor
Scope
1
s+1
Generator
1
0.4s+1
Exciter
Clock
0.1s+1
10
Amplifier
KA = 10. It is therefore necessary to increase the relative stability by
introducing a controller that can add a zero to the AVR open-loop transfer
function. A possible way of doing this is by adding a stabilizer feedback
to the control as shown in figure 2.7.
Vref(s) Ve(s) VR(s) VF(s) Vt(s) _
SV _ Amplifier Exciter Generator
Stabilizer
Sensor Fig 2.7: Block diagram of a simple AVR Compensated with a Stabilizer
An improved or satisfactory response of the AVR can be obtained by proper
adjustment of the stabilizer gain KF and time constant Fτ . For a stabilizer with
gain KF =2 and Fτ =0.04sec, the closed-loop transfer function becomes
1375002748755.270962136455.58)50045(250
)()(
2345
2
+++++++
=sssss
sssV
sV
ref
t
The steady-state response of the system becomes
Vtss= 0
lim→s
sVt(s) = 909.0137500
)500)(250(=
And the steady-state error becomes
sK
A
A
τ+1
sK
R
R
τ+1
sK
G
G
τ+1
sK
E
E
τ+1
sK
F
F
τ+1
∑
Vess= 1-0.909
=0.091
By adjusting KF and Fτ properly, the terminal voltage step response can
be obtained using the MATLAB commands shown in appendix III.
Fig. 2.8: Terminal Voltage step Response of CAVR with stabilizer
From the voltage step response, the time-domain specifications are as follows;
Rise time = 2.95
Peak time = 7.08
Settling time = 8.08
Percent overshoot = 4.13
The Simulink model for the block diagram in figure 2.7 is shown in figure 2.9.
Figure 2.9: Simulink model for fig. 2.7
2.8 AVR EXCITATION WITH PID CONTROLLER
In order to reduce the steady-state error and improve the dynamic response, a
proportional integral derivative (PID) controller is added. The transfer function
of the PID controller is
cG (s) = PK + S
K I + sKD 2.18
With the addition of the PID controller, the block diagram of the AVR is then
drawn as shown in figure 2.10.
Vref(s) EV RV FV
)(sVt _ PID Amplifier Exciter Gen.
Sensor Fig. 2.10: An AVR compensated with Proportional Integral Derivative
(PID) controller.
Also applying Mason’s gain rule [8], to the block diagram of the figure 2.10, the
closed loop transfer function becomes; )(
)(sV
sV
ref
t =
))()(1)(1)(1)(1())((2
2
GERADIPRGEA
GEADIP
KKKKKsKsKsTsTsTsTsKKKKsKsK++++++
++ 2.19
The steady-state response is
Vtss = 0
lim→s
sVt(s) =RGEAI
GEAI
KKKKKKKKK 2.20
When the parameters of the AVR in equation 2.7 with KP=1.00, KI=0.25,
KD=0.28 are substituted in equations 2.12 and 2.13 respectively;
)()(sV
sV
ref
t =1250550021755.3075.33
)25.6257(2002345
2
+++++++
sssssss 2.21
And the steady-state response of the system becomes
sK
R
R
τ+1
sK
A
A
τ+1
Ks
KK DI
P ++ sK
E
E
τ+1 sK
G
G
τ+1
Σ tV
Vtss = 0
lim→s
sVt(s) =1250
)25.6)(200(
= 1.00 2.22
The steady-state error = 1- Vtss
=1-1.00
=0.00
The results show that the addition of the PID controller brings the steady-state
error to zero. This is common for all integral control. The simulink model of the
block diagram in figure 2.10 is as shown in figure 2.11
Fig.2.11: Simulink model for PID control.
Vref
Vt
Step
1
0.05s+1
Sensor
Scope
1
s+1
Generator
1
0.4s+1
Exciter
PID
DiscretePID Controller
Clock
10
0.1s+1
Amplifier
Using the simulink model shown in figure 2.11, the step response obtained is
shown in figure 2.12.
Fig. 2.12: Terminal voltage step response with PID
The result shows that the system has a very negligible overshoot with a settling
time of about 1.5 seconds.
2.9 OVERVIEW OF FUZZY LOGIC
In control engineering, the primary objective is to distill and apply knowledge
about how to control a process so that the resulting control system will reliably
and safely achieve high-performance operations. When the operating conditions
of the generator changes like the presence of a fault, the conventional control
approach becomes ineffective. This therefore requires an alternative control
strategy. Fuzzy logic control is a better alternative. Fuzzy logic provides a
methodology for representing and implementing our knowledge about how best
to control a process.
Fuzzy logic is a system conceived by Lofti Zadeh in 1965 in his paper titled
“none other than fuzzy” as a method of dealing with the imprecision of practical
systems [2]. Fuzzy logic implements human experiences and preferences via
membership functions and fuzzy rules. Fuzzy control provides a formal
methodology for representing, manipulating, and implementing a human’s
heuristic knowledge about how to control a system.
Fuzzy logic is basically a multivalued logic that allows intermediate values to
be defined between conventional evaluations like yes/no, true/false, black/white,
etc.
In fuzzy systems, values are indicated by a number (called a truth value) in the
range from 0 to 1, where 0.0 represents absolute falseness and 1.0 represents
absolute truth. The ordinary Boolean operators that are used to combine sets
will no longer apply, we know that in Boolean operations, 1 and 1 is 1, but what
is 0.7 and 0.3? This will be covered in the fuzzy operations. As was seen in the
conventional control approach, if the mathematical equations of say the PID
Controller or the machine models were unknown, it would have been very
difficult to adjust their parameters in order to obtain the voltage step response. It
is therefore said that while differential equations are the languages of the
conventional control, rules are the languages of the fuzzy logic. Fuzzy Control
Systems are model-free estimators [11]. In other words the designers do not
need to state how the outputs depend mathematically on the inputs i.e. a well
defined mathematical model of the system is not required for the fuzzy
controller design.
When fuzzy logic or systems are used to model a process and controllers are
designed based on this model, then the resulting controllers are called Fuzzy
Logic Controllers [12].
According to [5], fuzzy logic controller (FLC) is a special kind of a state
variable controller governed by a family of rules and a fuzzy inference
mechanism.
The Fuzzy Logic Controller (FLC) algorithm uses heuristic strategies, defined
by linguistically described statements. The fuzzy logic control algorithm reflects
the mechanism of control implemented by people, without using any formalized
knowledge about the controlled plant in the form of mathematical models, and
without an analytical description of the control algorithm.
CHAPTER THREE
FUZZY LOGIC CONTROL APPROACH
3.1 FUZZY LOGIC CONTROLLER DESIGN
Fuzzy Logic Controller design is a three-stage process [13]. It comprises of
fuzzification, inference mechanism and defuzzification stages. To design the
controller, firstly, membership functions for the input variables (error, and
integral of error must be specified). Secondly, the fuzzy inference system must
be defined which consists of a series of “If…..then…..” linguistic rules. Then
finally, the membership functions for the output must be selected. A structure of
a fuzzy logic controller is shown in figure 3.1.
Ref. signal e _ y(t)
u (t) -
To plant Feedback Signal
Fig. 3.1: Fuzzy controller architecture
From the diagram, the plants outputs are denoted by y (t), its input is denoted by
u (t), and the reference input to the fuzzy controller is denoted by r (t). The
fuzzy logic controller has three main components:
1. The inference mechanism incorporates the rule base which holds the
knowledge in the form of a set of rules of how best to control the system,
Fuzzifier
Rule Base
Inference Engine
Defuzzifier ∑
e∫
and evaluates which of the control rules are relevant at the current time
and then decides what the input to the plant should be.
2. The fuzzification interface that simply modifies the inputs so that they
can be interpreted and compared to the rules in the rule base.
3. The defuzzification interface that converts the conclusion reached by the
interference mechanism into the inputs to the plant.
For the terminal voltage and reactive power control for the automatic voltage
regulator (AVR), the plant output is the terminal voltage Vt(s) while the input to
the plant is the Vf(s), and the reference input to the fuzzy controller is the
Vref(s).
With these nomenclatures, the fuzzy control system can be shown in figure 3.2.
r + e u y
-
Fig 3.2: Fuzzy Control System
3.2 CHOOSING THE FUZZY LOGIC AVR CONTROLLER
INPUTS AND OUTPUTS
In designing a fuzzy controller, variables which can represent the dynamic
performance of the plant to be controlled are chosen as the inputs to the
Σ Fuzzy Controller
Plant
controller [6]. The error (e) and the integral of the error are the inputs to the
controller.
For the fuzzy logic AVR control, the error between the reference voltage Vref
and the terminal voltage Vt i.e. Ve and the integral of the error V1 which is the
difference between the immediate and previous voltage error values are
considered as the inputs to the fuzzy controller while the output is the specified
output voltage of the alternator Vt
3.3 FUZZIFICATION
In fuzzy logic, linguistic variables are used as opposed to numeric variables
[14]. The input variables above are measured based on the expert knowledge or
the knowledge of the plant operator and quantified linguistically. In this
controller, eleven fuzzy subsets are chosen. These are:
“Negative Very large” (NV)
“Negative Large” (NL)
“Negative Big” (NB)
“Negative Medium” (NM)
“Negative Small” (NS)
“Zero” (Z)
“Positive Small” (PS)
“Positive Medium” (PM)
“Positive Big” (PB)
“Positive Large” (PL)
“Positive Very large” (PV)
These input variables are assigned numerical values as (-1, -0.8, -0.6, -0.4, -0.2,
0, 0.2, 0.4, 0.6, 0.8 and 1) which stands for Negative Very large, Negative
Large, Negative Big, Negative Medium, Negative Small, Zero, Positive Small,
Positive Medium, Positive Big, Positive Large and Positive Very large
respectively. The use of numbers for linguistic descriptions simply quantifies
the sign of the error and indicates the size in relation to the other linguistic
values. However, it does not represent any particular value of voltage error. In
the real world, measured quantities are real numbers (crisp). The process of
converting a numerical variable (real number) into a linguistic variable (fuzzy
number) is called fuzzification. Figure 3.3 shows the membership functions for
the input and output functions. Membership function is a plot of a function μ
versus e (t) that takes on special meaning. The function μ quantifies the
certainty that e (t) can be classified linguistically as “positive small” or
“Positive medium” and so on as the case may be.
As [14] puts it, the membership function quantifies in a continuous manner,
whether values of e (t) belong to (are members of) the set of values that are
“positive small”, and hence it quantifies the meaning of the linguistic statement
“error is positive small”.
There are several definitions of membership function which includes; bell-
shaped function, trapezoidal function, B-spline function, triangular function,
Gaussian function. The choice of any function however, depends on the type of
fuzzy approach and the designer’s choice. For our Fuzzy-AVR control,
triangular membership function is suitable and therefore used.
The inputs are mapped into membership functions, and a degree of membership
shows to what degree each input belongs to a particular linguistic label. The
membership can take on values between 0 and 1 for each of the linguistic labels.
Once the membership functions for the inputs are made, an intelligent decision
rules are then made on what the output will be. This process of decision making
is called the inference mechanism.
Figure 3.3a: Membership functions for the error
Figure 3.3b: Membership functions for the Integral of error
Figure 3.3c: Membership function for the terminal voltage
3.4 The inference mechanism
There are control laws that govern the operation of the controller in the design
of a conventional controller. In the design of fuzzy logic controller, there are
linguistic rules that allow the operator to develop a control decision in a more
familiar human environment.
With the two inputs for this FLC, an (11x11) decision table is constructed as
shown in Table 3.1. Every entity in the table represents a rule. The antecedent of
each rule conjuncts Ve and V1 fuzzy set values. A combination of experience
and common sense is used in obtaining the entries for the matrix [12]. For
instance, if the voltage does not change, then Int_error = Zero. An example of
the thi rule is: If Ve is PM and V1 is NS then U is PS. This means that if the
voltage error is positive medium and the integral of the voltage error is negative
small then the output of the controller should be positive small. If the error is
positive small and the integral of the error is Negative Small, then the output of
the controller should be Zero.
Table 3.1: Decision table of 121 rules for the Fuzzy-AVR controller
Ve U NV NL NB NM NS Z PS PM PB PL PV NV NV NV NV NV NV NV NL NB NM NS Z NL NV NV NV NV NV NL NB NM NS Z PS NB NV NV NV NV NL NB NM NS Z PS PM NM NV NV NV NL NB NM NS Z PS PM PB
eV∫ NS NV NV NL NB NM NS Z PS PM PB PL Z NV NL NB NM NS Z PS PM PB PL PV PS NL NB NM NS Z PS PM PB PL PV PV PM NB NM NS Z PS PM PB PL PV PV PV PB NM NS Z PS PM PB PL PV PV PV PV PL NS Z PS PM PB PL PV PV PV PV PV PV Z PS PM PB PL PV PV PV PV PV PV
The inference mechanism evaluates which of the control rules are relevant at the
current time and then decides what the input to the plant or the output of the
fuzzy should be.
As [14] puts it, the inference mechanism is a way of determining the
applicability of each rule called “matching”. Hence, the inference mechanism
seeks to determine which rules are on to find out which rules are relevant to the
current situation. The inference mechanism will seek to combine the
recommendations of all the rules to come up with a single conclusion. There are
two types of fuzzy inference systems in the Fuzzy MATLAB toolbox:
Mamdani-type and Sugeno-type. The most commonly used method for inferring
the rule output is the Mamdani method because of its wide acceptance. The
processes involved in the Mamdani method includes; Fuzzy Inference
System(FIS), Membership Function Editor, Rule Editor, Rule Viewer, Surface
Viewer. The FIS handles the high-level issues for the system which includes the
number and names of the inputs and outputs. The defuzzification technique is
also set at the FIS.
The membership function editor is used to define the shapes of all the
membership functions associated with each variable as shown in figure 3.4.
Fig. 3.4: FIS Editor
The Rule Editor is for editing the list of rules that defines the behavior of the
system. The 121 rules in the rule table of table 3.1 are encoded in the fuzzy
controller through the Rule Editor.
Figure 3.5: The Rule Editor
The Rule Viewer and the Surface Viewer are used for looking at, as opposed to
editing, the FIS. They are strictly read-only tools. The Rule Viewer is a
MATLAB technical computing environment based display of the fuzzy
inference diagram as shown in figure 3.6. They are used as a diagnostic; it can
show (for example) which rules are active, or how individual membership
function shapes are influencing the results. The Surface Viewer is used to
display the dependency of one of the outputs on any one or two of the inputs—
that is, it generates and plots an output surface map for the system.
Figure 3.6: The Rule Viewer
3.6 Defuzzification
As mentioned earlier, defuzzification converts the conclusion reached by the
interference mechanism into the inputs to the plant. Defuzzification operates on
the implied fuzzy sets produced by the inference mechanism and combines their
effects to provide the most certain controller output which is the output of the
plant. According to some experts, defuzzification is referred to as the
“decoding” the fuzzy set information produced by the inference mechanism or
process into numeric fuzzy controller outputs. In this work, centre of area
method (centroid) is used for defuzzification according to the membership
function of the output shown in figure 3.3c.
In a real-life analysis, the output of the designed controller is fed into a
compatible computer as shown in Fig. 3.7. The computer is connected to the
field winding of the alternator through a digital/analogue (D/A) converter and a
field exciter circuit. For further research, disturbances will be added to a rated
generator system by changing the length of the transmission line between the
generator and the commercial point to see the effectiveness of the FUZZY
LOGIC POWER SYSTEM STABILIZER (FLPSS). A real-time monitoring
system to check the effectiveness of the FLPSS will be available.
For this work, the performance of the fuzzy controller using proportional
integral-Automatic Voltage Controller (FPI-AVR) is examined through the
simulation developed based on the exciter and generator models of figure 3.8
shown below using MATLAB software packages.
Driving Motor
Mechanical Coupling
A
B
C
Fig. 3.7: General Schematic diagram for Fuzzy AVR operation
eV
u Vf
b
Fig.3.8: Fuzzy-AVR control loop
From figure 3.8,
Fuzzy Controler φ
Exciter
SK
E
E
τ+1
Gen.
SK
E
G
τ+1
g1
g2
go
∫Ve
Load
Computer
Filter/ Divider
A/D
D/A Field Exciter
Voltage sensor
2F 2F
1F
125 V DC
Let ),()( 1 tXtV = ),()( 2 tXtV f = )(3 tXb = 3.1
F
t
VV =
SK
G
G
τ+1 or
dtdV = -
G
tVτ
)( + FG
G VKτ
or dtdV = -
G
Xτ
1 + 2XK
G
G
τ 3.2
also u
VF = S
K
E
F
τ+1
or dt
dVF = -E
FVτ
+ uK
E
E
τ=
E
Xτ
2 + uK
E
E
τ 3.3
eV∫ = b = )( 1VVd −∫ = 1V
or 1XVVVdtdb
dtd ==−= 3.4
When equations 3.1, 3.2, 3.3 and 3.4 are combined together, equation 3.5 is
obtained as follows;
Gτ1
− G
GKτ
0
F = ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)(tbVV
dtd
F
t
= ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
3
2
1
XXX
dtd = 0 u
K
E
E
E ττ+−
1 0 ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
3
2
1
XXX
+ ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
100
dV 3.5
1− 0 0
F = 1221 ;1;1 XVuK
XXK
X dE
E
EG
G
G
−+−+−ττττ
3.6
The value of the voltage response is tuned using Runge-Kutta method as in the
program of Appendix IV. The results are shown in figure 3.10.
Fig. 3.9: Terminal and desired voltage step response with FPI-AVR
Fig. 3.10: Fuzzy input voltage to the generator.
It is observed that the overshoot, settling time and rise time are highly reduced
which is an improvement to the CAVR.
From the rule base in table 3.1, the voltage error, Ve = Vref – Vt is negative large
depicts a situation where the terminal voltage is much higher than the reference
or the specified voltage. In power systems, this is termed over voltage. A
decrease in the value of the reactive power of a generator cannot result to an
increase of the terminal voltage beyond the rated value. Instead the presence of
fault can result to an over-voltage, it is therefore necessary to analyze the
performance of the AVR for a simple machine infinite bus power system when
there is a fault and it was cleared.
In this work, simplified dynamic model of a simple machine infinite bus power
system is considered [1]. The model consists of a single synchronous machine
connected through a parallel transmission line to a very large network
approximated by an infinite bus. The classical third order single-axis dynamic
model of the simple machine infinite bus power system is shown in figure 3.1
CB1 CB2 Vs
Generator
Transformer CB3 CB4
Fault
Fig 3.11: A simple machine infinite bus power system
G
3.5 Mechanical equations
0)( ωωδ−= t
dtd 3.7
))((2
))(( 00 me PtP
Ht
HD
dtd
−−−=ω
ωωω 3.8
In this analysis, the mechanical input power Pm is assumed to be constant in the
excitation of the controller design. This implies that the governor action is slow
enough not to have any significant impact on the machine dynamics.
3.6 Electrical generator dynamics
))()((1)( '' tEtE
TtE qf
doq −= 3.9
3.7 Electrical equations (assumed X'd = Xq)
)()()( )( '' tIXXtEtE dddqq −+= 3.10
ds
e Xt
tP 'sd )(sin(t)VE'
)(δ
= 3.11
ds
sqd X
tVtEtI '
' )(cos)()(
δ−= 3.12
ds
q Xt
tI 's )(sinV
)(δ
= 3.13
ds
s
ds XV
Xt
tQ '
2
'sq )(cos(t)VE'
)( −=δ
3.14
)()(' tIKtE fad= 3.15
21
22'' ]))(())()([()( tIXtIXtEtV ddddqt +−= 3.16
3.8 Linear model of Single Machine Infinite Bus (SMIB)
By linearizing the above equations about the operating point, the variable model
of a single machine to infinite bus (SMIB) is obtained as follows;
BuAxX +=.
3.17
Cxy = 3.18
Where the state variable x is defined by
],,[ 'qEX ΔΔΔ= ωδ 3.19
From the matrix above, A, B, C are represented by
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−=
''
'
'
'0
sin).(
cos.
2
0
ds
s
do
dd
ds
qs
XV
TXX
XEV
HA
δ
δω
02
1
HD
−
0''
'
'
'0
1.)(
.1
sin.
2
0
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
−−
−
dodo
dd
do
ds
s
XTXX
T
XV
Hδω
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
'
00
do
c
TK
B
⎢⎢⎣
⎡+
−−= δ
δcos).(
)(sin)(.
)('
2'
'
''
ds
s
t
dd
ds
sd
t
ddq
XV
VIX
XVX
VIXE
C 0 0
'
'''
)1()(
⎥⎥⎦
⎤−
−
ds
d
t
ddq
XX
VIXE
The sub index 0 shows that matrices are operated at operating point.
When a delay of 2 seconds was applied to the system and cleared in 4 seconds,
it represents a three phase fault which was cleared within 4 seconds. The
response was as shown in figure 3.12.
Fig. 3.1
12: (a) Step (b) Fuz
p responsezzy input v
e of FPI-Avoltage to
AVR to thrthe gener
ree phase rator
fault.
CHAPTER FOUR
SIMULATION RESULTS
Simulations were carried out using MATLAB software package to examine the
performance of the fuzzy proportional-AVR (FPI-AVR). Seven results were
obtained under two set of tests as follows;
A. Sudden terminal voltage variation.
The terminal voltages of the generator under sudden variation of load were
examined at four different conditions, namely
1. With Conventional AVR (CAVR)
The parameters defined on page iv,
KE=KG=KR=1, Aτ =0.1, Eτ =0.4, Gτ =1.0 and Rτ =0.05, the response is shown in
figure 4.1
Fig. 4.1: Terminal Voltage Response
The response has the following time-domain performance specifications; rise
time = 0.247, peak time = 0.791, settling time = 19.04 and percent overshoot =
82.46.
The system was not satisfactory due to high settling time and percent overshoot
hence the need for an improvement.
2. With a stabilizer connected between the exciter output and the input, the
transfer function of the stabilizer is 104.0
2+s
s . The response is as shown in
figure 4.2
Fig. 4.2: Terminal Voltage step Response of CAVR with stabilizer.
The time-domain performance specifications are as follows;
Rise time = 2.95, Peak time = 6.08, Settling time = 8.08 and percent overshoot
= 4.13. Since the steady-state error, settling time and percent overshoot is still
high, there is need for improvement in the terminal voltage step response hence
the use of PID to improve the dynamic response and lower or remove the
steady-state error.
3. With a PID in series with the amplifier
The parameters for the PID are KP = 1.00, KI = 0.25, KD = 0.28, the
response is as shown in Figure 4.3.
Fig. 4.3: Terminal voltage step response with PI
The response has a negligible over-shoot and settling time of about 1.28
seconds.
4. With a FPI-AVR
The system percent overshoot was reduced to minimal. This tends to
smooth out operations and improve overall efficiency. The terminal
voltage step response is shown in figure 4.4.
Fig.4.4a: Terminal and desired voltage step response with FPI-AVR
Fig. 4.4b: Fuzzy input voltage to the generator.
B. A th
Fig. 4.5
hree phase
the
5: (a) Step (b) Fuz
e fault was
response o
response zzy input v
applied af
of the FPI-A
of FPI-AVvoltage to
fter 2 secon
AVR is sh
VR to threthe gener
nds and cle
hown in fig
ee phase farator
eared after
gure 4.5.
ault.
4 seconds;
CHAPTER FIVE
CONCLUSIONS
It has been shown that fuzzy logic control can effectively be applied to the
regulation of voltage for a synchronous generator. The results show that the
FPI-AVR performance is very high when compared to conventional AVR
(CAVR). The resulting system was both computationally efficient and had
settling time, overshoot and rise time reduced. The arrangement of the structure
results in a computationally less intensive control algorithm. Another significant
advantage of fuzzy logic is that it can easily accommodate additional input
signals. Therefore integrating controllers becomes much simpler with resultant
performance enhancement.
Fuzzy Logic Control provides a convenient means to develop the controller
which can accommodate the non-linear nature of the exciter-generator system.
Fuzzy logic on the other hand is not without any set back, the set back is
observed on the steady-state error which can be cleared by tuning the gains g0,
g1 and g2 to achieve the required result.
References [1] H.Sadaat, Power systems analysis, Tata McGraw-Hill series publishing
company Limited New Delhi, India 2002. [2] M.Y. Chow and K.Tomsovic, “Tutorial on fuzzy logic applications in
power systems”. IEES-PES winter meeting in Singapore, January, 2000 [3] O.P Malik, D.Niebur and T.Hiyama, “Tutorial on fuzzy logic applications
in power systems”. IEES-PES winter meeting in Singapore, January, 2000
[4] FIDE: why use fuzzy logic? Aptronix Inc, 1996-2000 [5] S.N Ndubisi and M.U.Agu, “A rule-based fuzzy proportional integral
type automatic voltage regulator for turbo generators”. European Journal of Scientific research, Vol. 20, No 4 May 2008 pp 924-933.
[6] K.Y. Lee, “fuzzy logic applications in power systems”. IEES-PES winter
meeting in Singapore, vol.4 January, 2000 [7] A.R.Hassan, T.R. Martis and A.H.M Sadrul Ula “Design and
Implementation of a Fuzzy Controller Based Automatic Voltage Regulator for a Synchronous Generator” IEEE Transactions on Energy Conversion, Vol. 9, No 3, September 1994 pp 550-557.
[8] M.Gopal, Modern Control System theory, Wiley Eastern Limited 1989 [9] R.C. Dorf and R.H.Bishop, Modern Control Systems, Pearson Education
(Singapore) ptc Ltd Pataparganj, India 2004 [10] G. Rogers, Power system oscillations, kluwer Academic publishers,
Nurwell USA 2000 [11] M.G.McArdle, D.J Morrow, P.A.J.Calvert and O.Cadel “A Hybrid PI and
PID Type Fuzzy Logic Controller for Automatic Voltage Regulation of the Small Alternator” IEEE Transactions on Energy Conversion, Vol 9, No 17, September 2001.pp 103-108.
[12] F. Lu and Y.Hsu, “Fuzzy Dynamic programming approach to reactive
/voltage control in a distribution substation”. IEE transactions on power systems vol. 12 No 2, May, 1997.
pp 681-688.
[13] B. J.LaMeres, M.H.Nehrir “Fuzzy Logic Based Voltage Controller for a
synchronous Generator” IEEE Transactions on Computer Applications in Power System, vol 9, No 4, April 1999
[14] K.M. Passino and S. Yurkovich., Fuzzy Control Addison Wesley
Longman, Inc 1998
APPENDICES
APPENDIX 1: (Program for the Root-locus plot of fig. 2.3) %program for the root locus plot of figure 2.3 numc=500; denuc = [1 33.5 307.5 775 500]; grid on; xlabel (real axis) ylabel (imaginary axis) title ('root locus plot for equation 10.') figure (23), rlocus (numc, denuc); APPENDIX II: (Program for the terminal voltage response for figure 2.4) %program for the terminal voltage plot of figure 2.4 numc= [250 5000]; denuc= [1 33.5 307.5 775 5500]; t=0:0.05:20; c=step (numc, denuc, t); xlabel ('t [sec]'); ylabel (‘c [volts]’) grid on; title ('Terminal voltage response'); plot(t, c); grid on; end
APPENDIX III: (Program for the terminal voltage response for fig. 2.6)
%program for the terminal voltage plot of figure 2.6
numc=[250 11250 125000];
denuc= [1 58.5 13645 274875 137500];
t=0:0.05:20;
c=step(numc, denuc, t);
xlabel ('t [sec]');
ylabel('c[volts]');
grid on;
title ('Terminal voltage step response for CAVR with stabilizer');
plot(t, c);
grid on;
end
APPENDIX IV: (Program for the terminal voltage response for fig. 2.9)
%program for the terminal voltage plot of figure 2.9
numc=[1400 5000 1250];
denuc= [1 33.5 307.5 2175 5500 1250];
t=0:0.05:20;
c=step(numc, denuc, t);
xlabel ('t [sec]');
ylabel('c[volts]');
grid on;
title ('Terminal voltage step response for CAVR with PID');
plot(t, c);
grid on;
end
APPENDIX IV: (Program for the FPI-AVR) % This program is used to simulate the PI fuzzy controller. % The variable names are chosen accordingly. % Initialize controller parameters KA=10; % Gain of the amplifier KG =1.0; % Gain of the Generator KE =1.0; % Gain of the Exciter KR=1.0; % Gain of the sensor % Initialize parameters for the fuzzy controller nume=11; % Number of input membership functions for the e universe of discourse numie=11; % Number of input membership functions for the integral of e universe of discourse g1=1;,g2=.01;,g0=10; % Scaling gains for tuning membership functions for
% universes of discourse for e, integral of e %(what we sometimes call "int e" below), and u %respectively
we=0.2*(1/g1); % we is half the width of the triangular input %membership function bases. wie=0.2*(1/g2); base=0.4*g0; %Base width of output membership fuctions % of the fuzzy controller e=[-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1]*(1/g1); % Centers of input membership functions for the int e %universe of discourse of fuzzy controller (a vector of %centers) cie=[-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1]*(1/g2);
rules=[-1 -1 -1 -1 -1 -1 -0.8 -0.6 -0.4 -0.2 0; -1 -1 -1 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2; -1 -1 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4; -1 -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6; -1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8; -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1; -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1; -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1 1; -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1 1 1; -0.2 0 0.2 0.4 0.6 0.8 1 1 1 1 1; 0 0.2 0.4 0.6 0.8 1 1 1 1 1 1]*g0; % Next, we initialize the simulation: t=0; % Reset time to zero index=1; % This is time's index (not time, its index). tstop=30; % Stopping time for the simulation (in seconds) %tstop=60; % Stopping time for the simulation (in seconds), step=0.01; % Integration step size x=[1;0.5;0.1]; % Initial condition on state of the % controller % Next, we start the simulation of the system. while t <= tstop v(index)=x(1); % Output of the controller if t<=.2, vd(index)=.2; end % First, define "test input 1" if t>.2, vd(index)=1; end
%if t<=10, vd(index)=18; end % This is "test input 2" (ramp) %if t>10, vd(index)=vd(index-1)+(4/1500); end % Ramp up 4 in 15 sec. %if t>25, vd(index)=22; end % Fuzzy controller calculations: % First, for the given fuzzy controller inputs we determine % the extent at which the error membership functions % of the fuzzy controller are on (this is the fuzzification part). ie_count=0;,e_count=0; % These are used to count the number of non-zero mf certainities of e and int e e(index)=vd(index)-v(index); % Calculates the error input for the fuzzy controller b(index)=x(3); % Sets the value of the integral of e if e(index)<=ce(1) % Takes care of saturation of the left-most membership function mfe=[1 0 0 0 0 0 0 0 0 0 0]; % i.e., the only one on is the left-most one e_count=e_count+1;,e_int=1; % One mf on, it is the left-most one. elseif e(index)>=ce(nume) % Takes care of saturation % of the right-most mf mfe=[0 0 0 0 0 0 0 0 0 0 1]; e_count=e_count+1;,e_int=nume; % One mf on, it is the % right-most one else % In this case the input is on the middle part of the % universe of discourse for e % Next, we are going to cycle through the mfs to % find all that are on for i=1:nume
if e(index)<=ce(i) mfe(i)=max([0 1+(e(index)-ce(i))/we]); % In this case the input is to the % left of the center ce(i) and we compute % the value of the mf centered at ce(i) % for this input e if mfe(i)~=0 % If the certainty is not equal to zero then say % that have one mf on by incrementing our count e_count=e_count+1; e_int=i; % This term holds the index last entry % with a non-zero term end else mfe(i)=max([0,1+(ce(i)-e(index))/we]); % In this case the input is to the % right of the center ce(i) if mfe(i)~=0 e_count=e_count+1; e_int=i; % This term holds the index of the % last entry with a non-zero term end end end end % The following if-then structure fills the vector mfie %with the % certainty of each membership function of the integral %of e % for its current value (to understand this part of the code see the above % similar code for computing mfe). Clearly, it could %be more efficient to
% make a subroutine that performs these computations %for each of the fuzzy system inputs. if b(index)<=cie(1) % Takes care of saturation of %left-most mf mfie=[1 0 0 0 0 0 0 0 0 0 0]; ie_count=ie_count+1; ie_int=1; elseif b(index)>=cie(numie) % Takes care of saturation of the %right-most mf mfie=[0 0 0 0 0 0 0 0 0 0 1]; ie_count=ie_count+1; ie_int=numie; else for i=1:numie if b(index)<=cie(i) mfie(i)=max([0,1+(b(index)-cie(i))/wie]); if mfie(i)~=0 ie_count=ie_count+1; ie_int=i; % This term holds last entry % with a non-zero term end else mfie(i)=max([0,1+(cie(i)-b(index))/wie]); if mfie(i)~=0 ie_count=ie_count+1; ie_int=i; % This term holds last entry % with a non-zero term end end end end % The next loops calculate the crisp output using only %the non-zero premise of error,e, and integral of the %error e. %This cuts down computation time since we will only %compute the contribution from the rules that are on %(i.e., a maximum of four rules for our case). Minimum %is used for the premise and implication.
num=0; den=0; for k=(e_int-e_count+1):e_int % Scan over e indices of mfs that are on for l=(ie_int-ie_count+1):ie_int % Scan over int e indices of mfs that are on prem=min([mfe(k) mfie(l)]); % Value of premise membership function % This next calculation of num adds up the numerator %for the defuzzification formula. rules(k,l) is the %output center for the rule. base*(prem-(prem)^2/2 is the area of a symmetric % triangle with base width "base" and that is chopped %off at a height of prem (since we use minimum to %represent %the implication). Computation of den is %similar but %without rules(k,l) num=num+rules(k,l)*base*(prem-(prem)^2/2); den=den+base*(prem-(prem)^2/2); end end u(index)=num/den; % Crisp output of fuzzy controller that is %the input to the controller. % Next, the Runge-Kutta equations are used to find the next state. % Clearly, it would be better to use a Matlab "function" for % F (but here we do not so we can have only one program). % This does not implement the exact equations in the %work but if you think about the problem a bit you will %see that these really should be accurate enough. Here, %we are not calculating several intermediate values of %the controller output, only % one per integration step; we do this simply to save % some computations.
time(index)=t; F=[(-x(1) + x(2)) ; (-2.5*x(2) +2.5*u(index)) ; (vd(index)-x(1)) ]; k1=step*F; xnew=x+k1/2; F=[(-xnew(1) + xnew(2)) ; (-2.5*xnew(2) + 2.5*u(index)) ; (vd(index)-xnew(1)) ]; k2=step*F; xnew=x+k2/2; F=[ (-xnew(1) + xnew(2)) ; (-2.5*xnew(2) + 2.5*u(index)) ; (vd(index)-xnew(1))]; k3=step*F; xnew=x+k3; F=[(-xnew(1) + xnew(2)) ; (-2.5*xnew(2) + 2.5*u(index)) ; (vd(index)-xnew(1))]; k4=step*F; x=x+(1/6)*(k1+2*k2+2*k3+k4); % Calculated next state t=t+step; % Increments time index=index+1; % Increments the indexing term % index=1 corresponds to time t=0. end % This end statement goes with the first "while" %statement in the program % Next, we provide plots of the input and output of the % controller along with the reference voltage that we %want to maintain. % It is also easy to plot the two inputs to the fuzzy % controller, % the integral of e (b(t)) and e(t) if you would like %since these values are also saved by the program. subplot(211) plot(time,v,'-',time,vd,'--')
grid on xlabel('Time (sec)') title('Terminal voltage (solid) and desired voltage %(dashed)') subplot(212) plot(time,u) grid on xlabel('Time (sec)') title('Output of fuzzy controller (input to the %Generator)') % End of program
