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Fuzzy sets and fuzzy logic in system control and prediction

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Title:
Fuzzy sets and fuzzy logic in system control and prediction
Creator:
Al-Agtash, Salem Y
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Denver, CO
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University of Colorado Denver
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English
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ix, 65 leaves : ill. ; 29 cm.

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Subjects / Keywords:
Fuzzy sets ( lcsh )
Fuzzy logic ( lcsh )
Electric power -- Mathematical models ( lcsh )
Electric power -- Mathematical models ( fast )
Fuzzy logic ( fast )
Fuzzy sets ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (M.S.)--University of Colorado at Denver, 1995. Electrical engineering
Bibliography:
Includes bibliographical references (leaves 62-65).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Electrical Engineering.
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Salem Y. Al-Agtash.

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Full Text
FUZZY SETS AND FUZZY LOGIC IN
SYSTEM CONTROL AND PREDICTION
by
Salem Y. Al-Agtash
B.Sc., Bogazici University, 1988
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment of
the requirements for the degree of
Master of Science
Electrical Engineering
1995


This thesis for the Master of Science
degree by
Salem Y. Al-Agtash
has been approved
by
Miloje S. Radenkovic
Marv Anderson
Tom Altman
g?// p/for
Date
11


To Yahya
in


Al-Agtash, Salem Y. (M.S., Electrical Engineering)
Fuzzy Sets and Fuzzy Logic in System Control and Prediction
Thesis directed by Associate Professor Miloje S. Radenkovic
ABSTRACT
The thesis discusses fuzzy logic rule-based systems as qualitative design methodol-
ogy for models of diverse physical nature to emulate human-decisionmaking in the
framewark of approximate reasoning. Fuzzy system design procedures are explained
based on different fuzzification and defuzzifcation criteria, system knowledge base,
and decision-making logic. Utilization of fuzzy relation and fuzzy compositional
operators in different types of fuzzy reasoning are also discussed in this manuscript.
An application example concerning the implementation of a fuzzy-logic approach
to provide a structural framework for the representation, manipulation and utiliza-
tion of data and information concerning the prediction of power demand is also
provided. An algorithm has been implemented and fuzzy rules were set to predict
the power demand at each load point on an hourly basis. However, the fuzzy rea-
soning used in the application is based on Mamdanis minimum operation rule as
a fuzzy implication function. The parameters taken into consideration cover envi-
IV


ronmental and weather-related conditions. Prediction of the power demand at each
geographical load point, and hence the country-wide demand, has been tested in
Jordan. Results concerning the daily prediction have been obtained with a standard
deviation error of 4.4%. Meanwhile, the designed fuzzy rule-based model could al-
low the system to replace the requirement for skilled dispatchers in predicting daily
load curves in a fuzzy environment.
This abstract accurately represents the content of the candidates thesis. I rec-
ommend its publication.
Miloje S. Radenkovic
v


Contents
1 Introduction 1
1.1 Background and Motivation......................................... 1
1.2 Literature Survey................................................. 4
1.3 Problem Formulation............................................... 5
1.4 Thesis Outline.................................................... 7
2 Fuzzy Logic, Sets, and Systems 8
2.1 Introduction...................................................... 8
2.2 Notation and Terminology.......................................... 9
2.3 Set Theoretic Operations......................................... 10
2.4 Linguistic and Fuzzy Variables................................... 12
2.5 Fuzzy Systems.................................................... 13
2.6 Fuzzy System Design Procedure.................................... 16
2.6.1 Model Functional and Operational Characteristics........... 16
2.6.2 System Control Variables................................... 18
2.6.3 Fuzzification Strategies................................... 18
2.6.4 Behavior of the Control Surfaces .......................... 20
2.6.5 Decision-making Logic...................................... 22
2.6.6 Defuzzification Strategies................................. 32
3 Application Example 34
3.1 Introduction..................................................... 34
3.2 System Development............................................... 35
vi


3.3 Power Demand Prediction Model................................... 36
3.3.1 System Analysis........................................... 36
3.3.2 Methodology .............................................. 38
3.4 System Application.............................................. 51
3.5 Discussion and System Evaluation................................ 57
4 Conclusions and Future Directions 60
Vll


List of Figures
2.1 Fuzzy sets are membership functions................................ 9
2.2 Diagrammatic representation of cold and hot......................... 11
2.3 Basic configuration of a fuzzy system............................... 15
2.4 Fuzzy system design procedure...................................... 17
2.5 Fuzzy system archeticture............................................ 29
3.1 System fuzzy model................................................... 36
3.2 Fuzzification of the location variable............. 41
3.3 Fuzzification of the population variable........... 42
3.4 Fuzzification of the industrialization variable.... 43
3.5 Fuzzification of the temperature variable............................ 44
3.6 Fuzzification of the climate variable................................ 45
3.7 Fuzzification of the time variable................................... 46
3.8 Fuzzification of system output variable: Area Power Demand. ... 47
3.9 Flowchart of the algorithm.......................................... 54
3.10 An example of power prediction..................................... 56
3.11 The actual load diagram (solid line) and the predicted load diagram
(dotted line) for Tuesday, April 6, 1993........................... 59
vm


ACKNOWLEDGMENT
I would like first of all to express my gratitude to those who have contributed
to the development of this thesis. I am greatly indebted to Dr. Miloje Radenkovic
for his guidance and valuable comments on each part of the thesis.
I also wish to thank the members of the committee, Dr. Gari Leininger, Dr.
Marv Anderson, and Dr. Tom Altman for their helpful comments and suggestions.
The financial support of both the Fulbright Program in the United States Infor-
mation Agency (USIA) through the American-Mideast Educational and Training
Services, Inc. ( AMIDEAST ) and Yarmouk University in Jordan is gratefully
acknowledged.
Finally, I am thankful to my parents and wife for their love and patience.
IX


1 Introduction
Artificial intelligence (AI) has been an active area of research in recent years. It has
been successfully used for various applications due to its rapidity, universality, and
feasibility. Currently, fuzzy systems and neural networks are popular AI method-
ologies. They are dynamic systems that can learn from experience like humans.
For instance, fuzzy expert systems are means of representing, manipulating, and
utilizing data and information that posses non-statistical uncertainty. They also
deal with issues such as reasoning at higher semantic or linguistic levels. Fuzzy
systems provide a structural framework that defines each imprecise factor by fuzzy
set and when several fuzzy factors exist simultaneously, an action is taken accord-
ing to some rules. In this project, the framework of fuzzy logic for representing
uncertainty and imprecision in system control and prediction is used.
This chapter is organized as follows: section one gives a background and mo-
tivations to the research. Section two provides the reader with a literature survey
taking into consideration some practical applications concerning fuzzy set theory.
Section three represents formulation of the problem. Finally, the thesis outline is
given in section four.
1.1 Background and Motivation
In the advent of this century, uncertainty and vagueness were considered to be
disturbing terms in models, theories, and statements. The only theory that dealt
first with these terms was probability theory, [11, 32]. They were treated on basis
of frequentistic interpretation of randomness.
1


The conclusion of statements about events being neither true nor false led to
the formulation of multi-valued propositional calculus by the Polish logician Jan
Lukasiewicz in 1920. He chopped the intermediate ground into multiple pieces and
came out with multi-valence logic. He then defined the indeterminacy as a con-
tinuum, a spectrum between falsehood and truth, [15, 32]. As a matter of fact,
multi-valence logic was worked out in the 1920s and 1930s to deal with Hiesen-
bergs uncertainty principle in quantum mechanics. Then the term vagueness
was used by the logician Bertrand Russel to describe multi-valence. In 1937, a
quantum philosopher Max Black published a paper on vague sets. In 1965, Lotfi
Zadeh published a paper called fuzzy sets which applied Lukasiewiczs multi-
valued logic to sets and groups of objects, [29]. The fuzzy principle states that
everything is a matter of degree, and fuzziness, generally, corresponds to truth,
falsity and indeterminacy, or to presence, absence, and ambiguity.
As the demand for solving optimization problems to complex systems became
of great importance to many system designers, the need to include uncertainty in
system models rather than neglecting its existence became desirable and sometimes
essential, [24, 26, 31]. This, however, has led to numerous attempts to modify
existing formal methods to reflect human mental behavior and be applicable to
reality. In this contest, various uncertainty theories were suggested in the realm of
fuzziness and vagueness. In this direction, fuzzy-based methods are considered to
be the most established theory which is based on fuzzy rules relating fuzzy concepts
in the form of conditional statements: If X is A, then Y is B, [14, 31, 32].
Fuzzy systems store a number of common-sense fuzzy rules some of which are
activated due to input data to some degree, then the fuzzy system produces fuzzy
outputs which will then be interpreted to an output crisp value. Meanwhile, fuzzy
reasoning is similar to common sense reasoning which takes into account uncertainty
2


in the interpretation of events. On the contrary, a mathematical way of reasoning
is based on the principle of two-valued logic (classical logic) where, a truth-table
method is used for the interpretation of formulas of classical propositional calculus
as functions in the field of sets, [32].
In system theory, modeling the physical structure of large and non-linear sys-
tems is extremely diverse and complicated, hence, difficult to control. Traditionally,
controllers are designed on basis of a mathematical description and on the systems
linearized model. Design of such systems faces the problem of choosing the ap-
propriate technique to achieve a particular goal. In literature, there have been a
number of studies for the control and management of such system models such
as state-space and frequency domain techniques, software package-based computa-
tional procedures, Lagrange multiplier, performance index analysis and mathemat-
ical programming analysis. These methods are considered to be mathematically
rigorous and need considerable computational efforts to execute the iteration pro-
cedures and it was found that they have some divergent expectations in real time
analysis [3, 4, 15, 21, 24, 26]. In addition, they often require modifications by
human operators to meet compatible analysis in the real world need.
However, the most complicated group of systems are those involving control that
links a human operator or a group of people coordinating, matching, or predicting
control actions. Management of such systems involves methods to describe human
intellectual activities. In this direction, expert systems are used to reflect the exper-
tise of an expert. Several techniques has been exploited in the literature concerning
the application of expert systems such as fuzzy systems and artificial neural net-
works on various control systems and industrial processes, [4,14, 18, 19, 20, 22, 23].
In this thesis, we exploit fuzzy-based methods for modeling human mental behavior
in complex systems. An application example is provided concerning the power de-
3


mand prediction in the country of Jordan, [3, 4]. The recorded data in the Jordan
Supervisory Control Center (JSCC) were utilized to develop a fuzzy model capable
of providing the system users with continuous predictions of system loads. Mean-
while, the main motivation to this research is to provide suitable measures of the
sophisticated parameters influencing power demand, generation and distribution.
Such measures are, indeed, the reflection of the skilled dispatchers intuition to
schedule the power system (expertise). This expertise is, actually, available in a
form of data-base concerning power demand and generation at each load point and
at every hour during an elapsed period of time which, in turn, would be of great
use for the reflection of past knowledge on future processing.
1.2 Literature Survey
Since the inception of fuzzy set theory, there have been numerous fuzzy-based indus-
trial applications all over the world, including building computer chips and systems
that intelligently control subways, automobile systems and numerous consumer elec-
tronic devices, [32]. Today in Japan, for instance, high-speed fuzzy systems control
subways, stabilize helicopters, and regulate air conditioners. In addition, the fuzzy
logic control method introduced by Mamdani, [20, 21], has been successfully applied
to various complex and nonlinear control problems. Such control methods are used
in cam corders, washing machines, vacuum sweepers, and televisions, [32]. A fuzzy
washing machine, for example, uses load sensors to measure the size and texture of
the wash load and uses a pulsing light sensor to measure the dirt in the wash water.
Fuzzy rules turn these measurements into patterns of water agitation for different
lengths of time.
Furthermore, fuzzy systems have been used for system control and prediction
based on the expert knowledge involved. They estimate functions and control sys-
4


terns with partial descriptions of a system behavior. However, in the area of power
system applications, some research efforts have been devoted to load forecasting.
G.Lambert, for example, [16] has proposed a load shape model which obtains the
load forecasting through operation of base-load and deviation load. Then, fuzzy
conditional statements (rules) were used for the decision making concerning the
power prediction. The system variables taken into consideration were the dry bulb
temperature, wet bulb temperature, relative humidity, wind direction and wind
speed. These variables were related to the country Canada taken as a case study.
The average error found in this approach was 2.01 %. Second, Young II Park, [23]
has used two different methods for short-term load forecasting: a simple moving
average method is used for ordinary days and in the case of holidays and special
days, a fuzzy estimation is made. In ordinary days, he assumed that the change of
load pattern is not so abrupt and a simple moving average method which requires
small calculations can be used. In special days, he assumed that there are impre-
cise fuzzy factors affecting the shape of load curves and these factors have to be
combined with rules for load estimations. The fuzzy factors considered were hot
weather, rainy weather, next to a national day and next to New Years Day. A
utility matrix was then constructed using fuzzy decisions. The degree of influence
on load data was calculated for the computation of the load prediction. Percentage
error was found to be about 1.4 for ordinary days and about 2 for special days.
1.3 Problem Formulation
In this work, fuzzy logic, sets, and theory are exploited in the qualitative approach
of system management and control design procedures. The research implies the
following: First, definition of fuzziness in the terminology of system theory. Sec-
ond, presenting fuzzy set notations, terminologies, and theoretic operations. Third,
5


design procedures of rule-based fuzzy systems for modeling human-like decision-
making within the conceptual framework of fuzzy logic and approximate reasoning.
Fourth, various methods of fuzzy reasoning and different types of implication func-
tions, sentence connectives, and compositional operators are discussed. Finally,
fuzzification and defuzzification criteria necessary for the system design interface
are also discussed.
An application example is also presented concerning the power demand predic-
tions in the country of Jordan. Fuzzy logic concepts are exploited, in this example,
to predict the power demand in an efficient way to help in optimizing the power
generation based on expert knowledge. The designed fuzzy system is used to mimic
the action of a skilled dispatcher in predicting the power demand, [16, 17], [25].
Figure 3.1 shows the power demand predictor model. The input variables to the
fuzzy model are the geographical location, population, climate, temperature, time
and the industrial factor. The output of the fuzzy model is the current predicted
area power demands (PAPDs) at each of the n load points. Computation of the
predicted power demand is done using an algorithm based on fuzzy logic. The algo-
rithm computes the power demand at each load point on an hourly basis concerning
a target period of interest. Knowledge-base involves the use of system data-base
that stores the data concerning the process for the years back. Jordan power sys-
tem is believed to be a good application example for the control and prediction
of system complexity and uncertainty. This is due to the following reasons: First,
the population of Jordan being unevenly distributed. Second, the countrys steady
growth and modernization process. Third, the region political situation having
resulted in mass people movement which, in turn, greatly disturbed the daily life
resources and requirements.
6


1.4 Thesis Outline
In this thesis, fuzzy system design procedures are explained through providing an
application example regarding the power demand prediction in the country of Jor-
dan. Chapter two discusses fuzzy logic, sets, and theory an a substantial basis.
It discusses the various types of fuzzification and defuzzification interface criteria,
knowledge base, engine inference, fuzzy reasoning, and implication functions. Chap-
ter three discusses the implementation of fuzzy logic approach for the prediction
of power demand. The chapter also presents the detailed analytical method used
in the design procedure. Chapter four gives conclusions and direction for future
research.
7


2 Fuzzy Logic, Sets, and Systems
2.1 Introduction
Fuzzy set theory has been an active area of research in recent years. It has been suc-
cessfully used for various applications, especially in the realm of industrial processes,
[14]. Fuzzy set theory describes the mathematical representation of a multi-level
logic known as fuzzy logic. It is a logic which is closer in spirit to human think-
ing and natural language than a traditional logic. Fuzzy logic was first introduced
by Zadeh, [29], in 1965 as a mathematical way to describe vagueness and uncer-
tainty in everyday life. Since then, fuzzy set theory has played an important role in
control systems, system structure identification, state prediction, and in processes
that lack quantitative data regarding the behavioral input-output relations. For
instance, fuzzy logic-based controllers provide means of converting a linguistic con-
trol strategy into an automatic control strategy based on expert knowledge. On the
other hand, rule-based fuzzy predictions reflect human experts thought process in
determining system future behavior. Meanwhile, the literature in fuzzy set theory
and its application has grown beyond scientists expectations to intervene almost
in every aspect of human life.
This chapter is organized as follows: Section two gives some important notations
and terminologies on fuzzy sets. Theoretic operations are also provided in section
three. Section four presents the definition of linguistic and fuzzy variables. Section
five gives an overview on fuzzy systems. Finally, fuzzy system design procedure is
given in section six.
8


Domain=X
Range=/^-P Figure 2.1: Fuzzy sets are membership functions.
2.2 Notation and Terminology
A fuzzy set was defined by Zadeh in his pioneering research on fuzzy set theory, [29],
as a linguistic approach for representing, manipulating, and utilizing data and in-
formation that posses non-statistical uncertainty.
Definition 2.1 Fuzzy Set, [29]: A class or space of objects X with the generic
elements x X characterized by a membership function Ha{x) is called a fuzzy set
A which associates with each point in X a real number in the interval [0,1].
Unlike conventional sets in set theory, fuzzy sets are always functions from a
universe of objects X into [0,1]. This is depicted in Figure 2.1 which illustrates
that the fuzzy set is the function pp that carries X into [0,1].
The support of A is the set of points in X at which pa{x) is positive. The
crossover point in A is an element of X whose grade of membership in A is 0.5. A
fuzzy singleton is a fuzzy set A whose support is point x defined as:
x
where \i is the grade of membership of x in A.
(2.1)
9


Corollary: A fuzzy set A in X is represented as a set of ordered pairs of a
generic element x and its grade of membership:
A = {(x,(jla(x)) ; xX} (2.2)
When X is continuous, a fuzzy set A can be written concisely as:
WxH^ (2-3)
Where the integral sign stands for the union of the fuzzy singletons When X
is discrete, however, the fuzzy set A has a finite support {xi, X2,..., xn} which can
then be represented by the summation.
^ V'' ^A^i)
As an illustration, suppose that the domain of objects X = {x,}; i = 1... oo
is the interval [0,120], with x ^ temperature, then the fuzzy sets of X labeled cold
and hot may be represented as
Figure 2.2 illustrates the diagrammatic representation of cold and hot.
2.3 Set Theoretic Operations
Definition 2.2 Emptiness: A fuzzy set is empty if and only if its membership
function is identically zero on X, i,e
{A = {; VxeX}} (2.7)
10


Membership function
Figure 2.2: Diagrammatic representation of cold and hot.
Definition 2.3 Equivalence: The fuzzy sets A and B are said to be equal
written as A = B if and only if
Ha(x) = Pb(x) for all x in X (2.8)
A
Definition 2.4 Complement: The complement of a fuzzy set A denoted by A
defines the membership function p^(x) for all x £ X by
Pa(x) = 1 ~ M*) (2-9)
Definition 2.5 Containment: A is said to be contained in B or equivalently A
is a subset of B, if and only if Pa(x) < Pb(x) for all x X. In symbols
Ac B <=> (Ia{x) ^ ftB^) (2.10)
Definition 2.6 Union: The union of two fuzzy sets A and B ( A U B) is also a
fuzzy set whose membership function is defined as
fiAuB(x) = max{pA(x),IJ-B(x)} (2.11)
11


Definition 2.7 Intersection: The intersection of two fuzzy sets A and B ( Af\B)
is also a fuzzy set whose membership function is defined as
^nfl(z) = Tnin{{j,A(x),fiB(x)} (2.12)
Proposition Given a fuzzy set A and its complement A then
AC\A ^ 0 (2.13)
AUA ^ X (2.14)
Definition 2.8 Cartesian Product: The Cartesian product of the fuzzy sets
Ai... An in Xi,.. .Xn is also a fuzzy set in the product space X\ x ... x Xn with
the membership function
f^AiAn(^l? j n) min{pAl(zj),... pAn(a'7i)} (2.15)
Definition 2.9 Fuzzy Relation: An n-ary fuzzy relation in X is a fuzzy set R
in the product space X x X x ... X, or symbolically
R = {(xi,x2,...xn),fj,R(xi,x2,...xn)} ; *j6X, i = l...n (2.16)
where fiR is a membership function defined on the universe of discourse.
Definition 2.10 Composition: Composition of a two binary fuzzy relations R\
and R2 denoted by R\ o R2 is also a fuzzy relation in X whose membership function
is defined by
^H!ofl2(a;,y) =Sup-star {p,A(x,v),fJ,B(v,y)} (2.17)
V
2.4 Linguistic and Fuzzy Variables
A physical variable in fuzzy systems is identified as a linguistic or fuzzy variable
whose values are labels or fuzzy subsets of the universe of objects. A fuzzy variable
12


is usually characterized by a quintuple (y, T(x), X) in which y is the system variable;
T(x) is the term set of labels of y defined on the universe of discourse X. As an
illustration consider the temperature variable in a temperature-controlled system
as a fuzzy variable with a term set T(temp) where
T(temp) = {cold, warm, hot}
or
T(temp) = {very cold, cold, warm, hot, very hot}
where each term in T(temp) is characterized by a fuzzy set in X = (0,120)F to
a degree of membership in [0,1]. In this contest, the first case, cold may be
interpreted as a temperature below about 40, warm may be interpreted as a
temperature over 30 and below 70, hot may be interpreted as a temperature
over 60 and below 120. In the second case, very cold may be interpreted as a
temperature below about 30, cold may be interpreted as a temperature over 20
and below 50, warm may be interpreted as a temperature over 40 and below
70, hot may be interpreted as a temperature over 60 and below 90, and finally,
very hot may be interpreted as a temperature over 80 and below 120.
2.5 Fuzzy Systems
Fuzzy systems are defined as systems of modeling human decision-making within
the conceptual framework of fuzzy logic and approximate reasoning. They are rule-
based dynamic systems that can recognize complex input-output relations without
definition. In conventional physical systems, the output value is often used to ad-
just the settings of the system states through some kind of process compensators to
provide best performance. In contrast, fuzzy systems work on the basis of associa-
tive memories. They consist of rules that combine the system input and output. A
typical fuzzy system consists of four principal components, [18, 19], a fuzzification
13


interface, a knowledge base, decision-making logic, and a defuzzification interface.
Figure 2.3 shows the basic configuration of a fuzzy system.
Fuzzification Interface: At this stage, values of the input variables are mea-
sured and transferred into fuzzy representations or linguistic variables viewed
as labels of fuzzy sets. Symbolically:
x=fuzzifier (x0)
where x0 is a crisp input value from a process; a: is a fuzzy set; and fuzzifier
is a fuzzification process.
Knowledge Base: The knowledge base, mainly, represents a classification
skill of an intelligent system. It consists of expert domain knowledge and
attendant control goals. In the knowledge base, the system variables are
associated with a linguistic certainty value forming a set of linguistic control
rules, and they are characterized by the control policy of the expert domain
of the process. Symbolically:
K = (U,C)
where K is referred to as a knowledge base; U is a finite set representing the
universe of objects. C = {ci, C2,... Cm} represents a family of classification
patterns in which c,- is a disjoint subset in U.
Decision-making: Decision-making is the area of command and control to
select the most suitable guidance algorithm based on the system rules of
inference and fuzzy control implications.
14


Figure 2.3: Basic configuration of a fuzzy system.
15


Defuzzification Interface: This stage implies converting the inferred fuzzy
system output to a non-fuzzy output. Symbolically:
y0=defuzzifier (y)
where y is an output fuzzy set value; j/o is a crisp output value; and defuzzifier
is a defuzzification process.
2.6 Fuzzy System Design Procedure
An essential part in the design of fuzzy system is claimed to be understanding the
systems behavior mechanism and identifying the system dynamics in terms of a
conventional input-output model. Second, a conceptual design to formalize and
structure a procedure which would then be modeled and simulated iteratively on a
computer to obtain a desired system behavior. In fact, there are six steps considered
in the design as illustrated in figure 2.4. First, define the model functional and
operational characteristics. Second, define the system control variables. Third,
select a fuzzification strategy. Fourth, define the behavior of the control surfaces.
Fifth, synthesize the structure of the system model (decision-making logic). Finally,
select a method of defuzzification.
Each of the above steps is investigated separately in the following sections.
2.6.1 Model Functional and Operational Characteristics
At this stage, the architectural characteristics of the system are identified in terms
of an input-output process model. This basically, deals with three main points in
the fuzzy system design:
1. What information and data points flow into the system.
16


Figure 2.4: Fuzzy system design procedure.
17


2. What basic transformations are performed on the data.
3. What data elements are output from the system.
2.6.2 System Control Variables
This step implies identifying the space of process state variables {V} and the space
of output variables {T}. However, the system outputs Y = {yi,... ym} are related
to the system states X = {x\,...xm} by some type of transformation function
which takes into account the system unobserved disturbances and the control ac-
tions applied.
Definition 2.11 : Let X designate the space of observed process states, U the
space of control actions with which a set of objectives {Z*} is satisfied, E the space
of unobserved disturbances, and Y the space of process outputs, then
Y = F(X, U, E)
where F is the operator transforming the input X into the output Y. In fuzzy sys-
tems the transformation operator F is the rule-base which presents the realization
of the system strategies, symbolically
F = (I,Z*)
where = <£(.,.) to process the information on the states of the
input and output I(X, Y) and the systems desired objectives {Z*}
2.6.3 Fuzzification Strategies
In a fuzzy model, a fuzzification strategy implies decomposing each process state
(input) variable and output variable into a set of fuzzy representations known as
fuzzy sets. A fuzzy set linguistically approximates uncertain or imprecise sampled
18


values to a degree of membership in the interval [0,1]. However, the fuzzy sets
representing the system state variables form a fuzzy input space, while those repre-
senting the system output variables form a fuzzy output space. Noticeably, in the
fuzzification process, the uncertain information of the system universe of discourse
is either continuous or discrete. In the discrete universe, the fuzzification process
is carried out based on labeling segments of intervals with fuzzy sets. Each generic
element in the segment is identified with a grade of membership value in [0,1].
On the other hand, if the universe is continuous, a discrete universe is formed by
a discretization of the continuous universe, [18]. This is important in the system
data quantification and representation with fuzzy sets. Quantification discretizes
a universe into a certain number of segments known as quantization levels. Each
segment forms a discrete universe labeled with a fuzzy set that identifies a grade of
membership value in [0,1].
Finding the number of fuzzy sets or linguistic terms associated with each variable
defined on the same universe of discourse is believed to be a complex issue. Some
rules of thumb have been suggested by [6], [8]. First, the number of fuzzy sets
associated with a variable should generally be an odd number. Second, each set
should overlap between 10 and 50 percent of the neighboring space. Finally, the
density of the fuzzy sets should be highest around the optimal control point of the
system.
The grade of membership function defining a fuzzy set may be represented by
a vector of numbers (denoted by [15] as a fit vector) or by a functional form (e.g.
bell-shaped function, triangular-shaped function ...etc.). A fit vector represents a
discrete universe of discourse X whose entries are the membership values at each
sample datum in X. In this case, the fuzzy set A; is written as in Eq. 2.4. On the
other hand, a functional form represents a continuous universe of discourse X, and
19


the fuzzy set A; is of the form in Eq. 2.3.
As an illustration, the fuzzy sets, in fuzzy logic controllers, usually have a mean-
ing such as NB: negative big; NM: negative medium; NS: negative small; ZE: zero;
PS: positive small; PM: positive medium; PB: positive big; or any other common
sense linguistic terms that best describe a physical variable in a control system.
It is also important to mention that in some applications, the observed data
are disturbed by a random noise which is measurable only in a statistical sense.
The fuzzification strategies, in this case, should involve both uncertainty and ran-
domness. In this sense, Bharathi [7] and Dubois [10] have a broad explanation on
the fuzzification of statistical data. Two methods were suggested. First, bijective
transformation which transforms a probability measure into a possibility measure
by using the concept of the degree of necessity. Second, hybrid number arithmetics
regarding both probabilistic and possibilistic modes of characterization.
2.6.4 Behavior of the Control Surfaces
A control surface is a hypothetical surface created by the interaction of rules with
fuzzy sets. Defining the behavior of the control surfaces implies writing fuzzy con-
ditional statements of the form, If-Then rules which reflects or expresses system
experts domain knowledge. Fuzzy rules are usually formulated in linguistic terms
simulating human decision-making behavior. Such a group of rules is deduced from
the observation of human experts actions in terms of the input-output data. A
fuzzy rule is mainly defined by the relation between observation (antecedent) and
action (conclusion or consequent). The maximum number of fuzzy rules is deter-
mined by the cardinality Z), or the number of representations, of a term set in
the fuzzy input space. For example, if the cardinality of two input fuzzy variables
£i and xi in a fuzzy system are D{x\) = 5 and D(x2) = 7 respectively, then the
20


maximum number of rules will be 5 x 7 which represent the total possible input
combinations. In some cases, fewer rules may be used but in a compromise of
knowledge.
Meanwhile, based on expert knowledge and the fuzzy model of the process, the
linguistic description of the process dynamic characteristics formulates a fuzzy set
of control rules for optimizing the overall system performance. Therefore, expert
experience and system engineering knowledge have a crucial role in the characteri-
zation of the operational rules in a fuzzy system.
In this reasoning model, for instance, a multi-input single-output fuzzy model
with an observation space [/, and action space V, would be characterized by the
fuzzy rules of the type
Rk : If xi is Aly and x2 is A22 ... and xn is A2 then y is B4
where Rk is the kth rule (1 < k < nl); nl:the cardinality of the observation space;
xi, (1 < l < n2) are fuzzy input variables; y is the fuzzy output variable; A} and
are the fuzzy sets of x\ and y in the universe of discourses U and V respectively.
Rk can be rewritten as
Rk : If X is A*' then Y is Bi
where A1 = (AJ1, A22... AJJ^2), (1 < i < ml \ ml : number of fuzzy sets in xi) are
the fuzzy sets of the linguistic variables X = {x\,x2,.. .xn2) in the universe of
discourse U = (Ui,U2... Un2). In fuzzy systems, a fuzzy rule is implemented by a
fuzzy implication function in which R1 is written symbolically
Rk = (A*1 and AJ,2 and ... A^22) B* in Ui x U2... Un2 x V (2.18)
PR} = /^(a*1 and Aij2 and ...a|^2)(Uiu2wn2,u) (2.19)
= [vA?Ma,ndfiAj2(u2)...(iAx2(ub2)] > (2.20)
where (Ay- and A22 ... AjJJj2; Bl) is the fuzzy association in Ui x U2 ... Un x V, and
21


> denotes a fuzzy implication function.
Example : A fuzzy rule in an anti-lock braking system might be:
Ri= If brake temperature is Warm and speed is Slow then brake pressure is
Slightly Decreased.
Here, the observation space U is characterized by the set F(X) = { degree of
brake temperature} x { term for speed } and the action space V is characterized by
the set F(Y) = { terms of brake pressure }. For instance, x may have a term set as
{ Hot, Warm, Cold } of which all the elements represent a common sense linguistic
representation over the universe of discourse of U\. x2 may have a term set as {
Slow, Medium, Fast } of which all the elements represent a common sense linguistic
representation over the universe of discourse of C/2. Finally, y may have a term set
as { Slightly Decreased, Decreased, Constant, Slightly Increased, Increased} of
which all the elements represent a common sense linguistic representation over the
universe of discourse of V. Hence, the cardinality representing the number of all
possible system rules is equal to 3 x 3 = 9. Meanwhile, the fuzzy rule in the above
example can be written symbolically as:
R1 = {A{ and A\) B1 (2.21)
PR1 = and a>)(12,u) (2.22)
= [va* (i)and/iAi (u2)] > i (u) (2.23)
2.6.5 Decision-making Logic
The decision-making logic is an interface between the human model and the formal
model used to represent a particular judgment situation. This interface relates the
human information processing activities with the imperfect knowledge of system
information and organization. Decision-making logic involves concepts of system
22


fuzzy relation, sentence connectives, Compositional operators, and interface mech-
anisms.
Fuzzy Relation
The concept of a fuzzy relation is a generalization of the concept of a function.
In fuzzy logic, fuzzy relations are fuzzy system rules expressed with some type of
fuzzy implication function. Moreover, there are many fuzzy implication functions
proposed in literature with the following basic properties, [18, 19] : fundamen-
tal property, smoothness property, unrestricted inference, symmetry of generalized
modus ponens and generalized modus tollens. Lee, [19] classified them into three
categories: the fuzzy conjunction, the fuzzy disjunction, and the fuzzy implication.
The former two are closely related to the fuzzy Cartesian product, and the latter is
a generalization of implication in propositional calculus, modus ponens, and modus
tollens.
Considering the fuzzy rule If x is A then y is 5 which can be represented
by a fuzzy implication function A ) B, where A is the antecedent and B is the
consequent in the universes U and V with the membership functions ha and fis
respectively. Then
Definition 2.12 Fuzzy conjunction, [19]: The fuzzy conjunction for all u U
and v V is defined by
B = Ax B
-L
Ha{u)*iib{v)
(2.24)
(2.25)
luxV (u,v)
Here is an operator representing intersection, algebraic product, bounded product,
or drastic product.
Definition 2.13 Fuzzy disjunction, [19, 29]: The fuzzy disjunction for all u U
23


and v V is defined by
Ax B
(2.26)
(2.27)
AyB =
_ f Ha(u) Hb(v)
JuxV (tt,u)
Here is an operator representing union, algebraic sum, bounded sum, or drastic
sum. For more details on the operators, refer to [19].
Definition 2.14 Fuzzy implication, [30,18]: The fuzzy implication for all u U
and v V is defined by
1. Material implication:
A y B = A*B
(2.28)
2. Propositional calculus:
A y B = A(A*B)
(2.29)
3. Extended propositional calculus:
A y B = {AxB)*B
(2.30)
4. Generalization of modus ponens:
AyB = sup{c [0,1], A* c < B} (2.31)
5. Generalization of modus tollens:
A y B = inf{t [0,1], B t < A}
(2.32)
where the operators and are defined as in definitions 2.12 and 2.13.
24


Furthermore, here are some examples of fuzzy implications adopted in many
practical applications as mentioned in, [19, 21]:
Min-operation rule of fuzzy implication
Re = Ax B
-L
ha{u) n iiB{v)
UxV U.V
Product operation rule of fuzzy implication
Rp = A x B
= L
Va(u)(ib(v)
UxV u,v
Arithmetic rule of fuzzy implication
Ra = (A x V)(U x B)
in(i fiA(u) + fiB(v))
-l
UxV
U.V
Maximum rule of fuzzy implication
Rm = (AxB)V(AxV)
{ha{u) n Hb{v)) u (1 Ha(u))
= l
UxV
U, V
Sentence Connectives
(2.33)
(2.34)
(2.35)
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)
The sentence connectives and and else are commonly used in system fuzzy
rules, and connective is interpreted as an intersection (min operator) in the
Cartesian product, and else is interpreted as a union (max operator).
25


Definition 2.15 and, [19, 29]: Let U and V be the universes of A and B
respectively, then the antecedent of if A and B then (7 is also a fuzzy set in the
product space U x V with
Vaxb(u,v) = Tnin{fiA(u),fj,B(v)} (2.41)
The fuzzy relation in this context, is defined as
R = (A x B) > C (2.42)
Definition 2.16 else, [19, 29]: Let U and V be the universes of A and B
respectively, then the connective else in the fuzzy rule if A and B then C else
D is interpreted using the following fuzzy relations:
(A x B) - -*C (2.43)
(Ax B) D (2.44)
X 1 D (2.45)
A A (Ax B) -*D (2.46)
Compositional Operator
A Compositional operator is referred to as a Sup-Star composition, where star
denotes an operator ( min, max, product, .. .etc.).
Definition 2.17 Compositional rule of inference, [19, 29]: Let R be a fuzzy
relation from U to V and A is a fuzzy subset of U, then the Compositional rule of
inference asserts that the solution of the fuzzy subset B of V is
B = AoR
P-Aor{v) = Sup star {/iA(u), hr{u, v)}
u&J, vSV
26
(2.47)
(2.48)


where o is a Compositional operator. The main Compositional operators mentioned
in the literature are the Sup-min, Sup-product, Sup-bounded and Sup-drastic
product, [19, 29] Selection of a Compositional operator depends on the nature
of some defined problems. Meanwhile, Sup-min and Sup-product operators are the
most frequently used in fuzzy system applications.
Inference Engine
The inference engine is a mechanism for manipulating rules from the knowledge
base, forming inferences and drawing conclusions. The conclusions can be deduced
in a number of ways which depend on the structure of the engine. An uncertainty
of information in the knowledge base would, however, certainly induce uncertainty
in conclusions. Therefore, the inference has to provide a suitable measure of un-
certainty in the conclusion which can be comprehensive and interpretable to the
system user. Lee, [19], stated four types of fuzzy reasoning. First, a reasoning
based on Mamdanis minimum operator rule. Second, a reasoning based on Larsens
product operation rule. Third, a reasoning based on Tsukamotos method with a
monotonic membership function. Finally, a reasoning which is based on taking the
consequences of a rule as a function the input fuzzy variables.
Definition 2.18, [19], Given an observation A' and B' in a 2-input, 1-output
fuzzy system with A\,..., Am as fuzzy subsets in U and Bi,... ,Bm as fuzzy
subsets of V, and C\,..., Cm as fuzzy subsets of W. Then the action C'k inferred
from the kth rule Rk of a set of N\ x N2 rules is defined as follows
C'k = {A',B')oRk (2.49)
where A' U, B' V, C' 6 W, and Rk G U x V x W
Pcdu) = Sup-star (nA'(u),HB'(v),fiRk(u,v,u)) (2.50)
* ueu.vev
27


= Sup star (fiA'(u) pB'{v),pCk{u))
u&j,v£y
= (*k*'Pck{u)
here, ak : Pa>(u) //b(u), : implication operator, * : fuzzy reasoning resulting
from the implication function used.
When the input data driven from the observation space is read, one or multiple
rules will be activated in parallel but to different degrees, as shown in figure 2.5. The
kth activated rule will produce a non-null output Ck with a non-negative weight ak
depending on the fuzzy reasoning and the implication operator used. The inferred
consequence Ck due to the kth rule is given by equations (2.49, 2.50). Finally,
the resultant inferred consequence C' equals the union of the individual weighted
outputs C'k. Hence,
m
C'={jc'k (2.51)
k=1
and, the membership function of the inferred consequence C' can be written as
771
pc> = U (2.52)
k=1
771
= \J ak r pCk(u) (2.53)
k=l
m:number of activated rules due to the observations A' and B'
In real time processes, the inputs are usually measured by sensors and are crisp
values which will then be converted into fuzzy sets in the fuzzification process. In
turn, these will activate a number of rules parallely in the knowledge base processing
which will be interpreted to give a control action, prediction result or parameter
estimation value. As an illustration, consider the input data Uo> which activate
the following two control rules
28


Figure 2.5: Fuzzy system archeticture.
Ri : if a: is Ai and y is B\ then z is C\
R2 : if a; is A2 and y is B2 then z is C2
Then the weighing factors ai and a2 of the first and second rule may be ex-
pressed as
= HAiM */*b,(vo)
a 2 = fiAtiu o)*^s2(uo)
Here could be any implication operator in the class of (minimum, algebraic prod-
uct, bounded product, or drastic product) or the class of (union, algebraic sum,
bounded sum, drastic sum, or disjoint sum). Then the fuzzy reasoning due to the
first and second rule will lead to the control decisions:
C[ = (A^B'^oR,
C'2 = (A2,B2)oR2
29


with membership functions
A*cj(w) = ai *-pCl(w)
= (*2 *lichfa)
Then the inferred consequence C' is
a = c[uc2
fj,c>{u) = |/c;()U/ic'(w)
= ai * fid (w) U a2 *' nc2 (t*>)
Selection of * depends on the type of fuzzy reasoning
1. Type 1 Mamdanis minimum operation rule: The individual inferred con-
sequence from the kih rule is computed by using the minimum operation rule
(defined by an operator (A)) as an implication function in the fuzzy reasoning
encountered. Hence,
fic> (u) = ak A k = 1,2 (2.54)
Then
Hc>{u) = (2.55)
= [ati A fic, (w)] U [a2 A fic2 (w)]
fJiC1 ' The membership function of the resultant inferred consequence C'.
2. Type 2 Larsens product operation rule: The individual inferred conse-
quence from the kth rule is computed by using the product operation rule
(defined by an operator (.)) as an implication function in the fuzzy reasoning
encountered. Hence,
30


f*ck M = ak-pck (w) k = 1,2
(2.56)
Then
He1 G*>) Vc[ U /^c'
= [i-M<7, (u)] U [aj.fiCj (w)]
(2.57)
To obtain a deterministic control action for the first two types of reasoning a
defuzzification is required, as will be discussed later in this chapter.
3. Type 3 Tsukamotos Method: The results inferred from the first and second
rule are a.\ and a2 such that
ai = C\(wi)
a2 = C2(w2)
And the crisp control action is the weighted combination
atiW! + a2w2
w =
(2.58)
(2.59)
(2.60)
<*1 + <*2
4. Type 4 The consequence of a rule is a function of input variables: In this
mode of reasoning the fuzzy rules are of the form
Ri : if x is A\ and y is Bi then z =
R2 : if x is A2 and y is B2 then z = j2{x, y)
fi : is a function of the process state variables x, y defined in the input
subspaces. The control action inferred from the first rule is ai/i(uo,i>o) and
that is inferred from the second rule is ct2f2(uo, uo). The crisp value is then
defined as
aifi{uo>vo) + a2f2(uo,v0)
oti + a2
(2.61)
31


2.6.6 Defuzzification Strategies
The defuzzification process is a mapping from a fuzzy output space defined over
the universe of discourse U into a space of non fuzzy (crisp) control action. The
mapping function is denoted by a defuzzification strategy. The most commonly
used strategies are: the max criterion, the mean of the max, and the center of
gravity, [12, 15].
1. Max criterion: This is the simplest defuzzification strategy. The output crisp
value w is computed by choosing the element that has a maximal membership
value in the output fuzzy distribution C'. In particular, to find wmax of the
C' distribution where
Hc'(wmax) = sup{y,C'{w)} ; w 6 W (2.62)
If the output space W equals a finite set of values {twi,..., wp}, then the
supremum in Eq. 2.62 is replaced with a maximum
Pc'{wmax) =max {pc'(wi)} (2.63)
2. Mean of the max: The crisp output value w generated from this strategy is
based on computing the mean value of all fuzzy outputs whose membership
value reaches the maximum, i.e.
w = Et (2-64)
;=i K
where tu,- is the support value at which the membership function is maximum,
and k is the number of these values.
3. Center of gravity: In this defuzzification strategy, all the information in the
output fuzzy distribution C' is used to compute the output value as the cen-
32


troid w
a f-o0ufJ'C>(w)dv
W~ J-ooVC>(u)duj
(2.65)
In the case of discrete fuzzy systems, the fuzzy centroid w of the fit-vector C'
is
a YJk=i UkUC'juk)
Efc=1 l*C'(Wk)
where ujk is the centroid of C'k.
(2.66)
33


3 Application Example
3.1 Introduction
This chapter discusses the implementation of a fuzzy-logic approach on the predic-
tion of power commitments. In fact, prediction of power demand plays an important
role in various kinds of power system applications such as unit commitment and
economic dispatch. It can be used for the system operation as an aid for the oper-
ator to know the power demand in the lead-time period. However, there have been
a number of studies in this field during the past few decades. Traditional methods
used are known to be inaccurate for days with special events. This is due to the
fact that power prediction for those days requires the knowledge and experience
of an expert in a control center. Due to this fact, several artificial intelligence
(AI) methodologies have been proposed to replace the expert in the control cen-
ter. Some of these methods have shown encouraging results concerning short-term,
medium-term, and long-term power demand predictions.
In the literature, fuzzy systems and neural networks are important AI method-
ologies used for power demand prediction, [16, 23]. Fuzzy systems simulate human-
like performance. They are composed of rules that relate system inputs to system
outputs using fuzzy logic. Meanwhile, fuzzy logic simulates human thinking that
measures the degree of uncertainty and vagueness in a physical variable. Fuzzy sys-
tems are dynamic systems that recognize an input-output relationship using fuzzy
associations (rules). They deal with issues at a linguistic level by providing a struc-
tural framework that defines each imprecise factor by a fuzzy set and when several
fuzzy factors exist simultaneously, an action is taken accordingly with some rules.
34


The proposed study was aimed at developing a fuzzy system for power demand
predictions on both short-term and medium-term lead-time periods. The imprecise
environmental, industrial, population, and weather related factors were considered.
This chapter is organized as follows: section two gives a general description
of the system development. The power demand prediction model is presented in
the third section. System application is discussed in the fourth section. Finally,
discussion and system evaluation are presented in section five.
3.2 System Development
The computational unit (fuzzy model) depicted in Fig.3.1, is used to predict the
total power demand based on expert knowledge. The input variables to the fuzzy
model are the geographical location, population, climate, temperature (dry bulb
temperature), time and the industrial factor. The outputs of the fuzzy model are
the current predicted area power demands (PAPDs) at each of the n load points.
The presented model is used to predict the total power demand in the days ahead.
The architectural characteristics of the model are designed in such a way to evolve
the information flowing at each load point then executing all the necessary rules
combining the input values to the output model, [3, 4].
Computation of the predicted power demand has been implemented using an
algorithm based on fuzzy logic. The algorithm computes the power demand indi-
vidually at each load point and on hourly basis. The input data fed into the model
concerning each state variable are fuzzified, which, in turn, will activate at least one
of the system rules. Each activated rule will emit a weighted fuzzy output variable.
Summation of all weighted fuzzy outputs is then defuzzified to produce the power
demanded by each load point and at every hour concerning the target period of
interest. The system data taken from the Jordan Supervisory and Control Center
35


Time -
Climate
Temperature
Geo. Location____
Industrialization -
Population_________
Power
Demand
Predictor
PAPD1
PAPDn
Figure 3.1: System fuzzy model.
(JSCC) were utilized in the derivation of fuzzy system rules based on the human
operator experience.
3.3 Power Demand Prediction Model
3.3.1 System Analysis
In this section, the fuzzy model used to predict the total power demand is presented.
As shown in Fig.3.1, the input variables to the fuzzy model are: time of the day
(T), geographical location (L), temperature (K), climate (C), population (P) and
industrialization (I) These variables are treated as state fuzzy variables in the
observation space. The outputs of the fuzzy model are the predicted area power
demands (PAPDs). The amount of power consumption varies according to the
variation of systems state fuzzy variables concerning a target period.
Time of the day factor inherently imbeds human requirements for electricity as
related to the day and night activities, weekdays, weekends, holidays and finally
seasonal requirements. The temperature factor supports energy requirements re-
36


lated to weather conditions. The location factor involves special requirements, and
peoples habits that may influence energy needs. The climate also reflects the reac-
tion of humans to energy needs as a response to rainy, snowy and cloudy weather
conditions. Finally, the population and industrialization factors have a significant
effect on power consumption. For example, it could be expected that a certain
load point which is lightly populated, highly industrial, and under cold and cloudy
weather conditions, will have a medium power demand. Meanwhile, another load
point under the same other conditions would require higher power demand, if it were
highly populated. It should be noted that the models main objective is to predict
the total power demand within considerable bounds of uncertainty. The prediction
of power demand is dependent on the system variables that may vary from time to
time. Fuzzification of variables involves a trade off of precision in prediction and
computation time. Increasing the number of fuzzy regions would improve the preci-
sion of power prediction but, on the other hand, would increase the computational
time and hence the time between two adjustments becomes larger. For example,
the fuzzy variable, temperature, may have three labels: Cold(C), Warm(W) and
Hot(H), and in a more precise identification it may have five labels: Very Cold(VC),
Cold(C), Warm(W), Hot(H), and Very Hot(VH). Then a 0C may be cold (the first
case) or very cold (the second case). As a matter of fact, consumption of power is
considerably dependent on temperature. That is, a decision is taken in favor of low
power demand when the combination includes cold in the first case and in favor
of very low power demand when the combination includes very cold in the second
case. In summary, decomposing a certain fuzzy variable into large number of fuzzy
regions leads to a more precise prediction of the power demand. However, the cost
of better precision is a large processing time, large memory needed to absorb all
data and temporal results, and large step size between two consecutive adjustments.
37


3.3.2 Methodology
The fuzzy system constructed here is a fuzzy associative memory (FAM) system,
[15]. The fuzzy associative memories (FAMs) axe transformations that map fuzzy
sets to fuzzy sets. It encodes a bank of compound FAM rules that associate multiple
outputs with multiple inputs.
Fuzzification of System State Variables
In a fuzzy system, the physical variables are identified as fuzzy variables. A fuzzy
variable describes the discrete values of the physical variable with linguistic common
sense terms. These linguistic terms are called fuzzy-set values or fuzzy regions. For
example, the temperature variable taken in our study is a fuzzy variable that takes H
(hot), W (warm), and C (cold) as fuzzy-set values. The set W covers temperatures
between 10 and 30 degree celsius to a degree of membership between zero and one.
Different fuzzy quantifications may be done depending on the number of fuzzy-
set values (fuzzy regions). In our application, prediction of power demand on a
country-wide basis, the physical variables involved are chosen in a way that they
are highly effective on the amount of power consumption.
In the model, each of the system state fuzzy variables is decomposed into a
reasonable number of fuzzy regions following the rules of thumb,[8], that is by
choosing an odd number of labels associated with a variable and each label should
overlap between 10 and 50 percent of the neighboring space, finally the density
of fuzzy sets should be highest around the optimal control point of the system
and should thin out as the distance from that point increases. The membership
function associated with each of the state variables is designed to have the form of
a triangular-shaped function.
Figure 3.2 shows the fuzzification of the location variable where northern (N),
38


center (C), and southern (S) parts of Jordan are the chosen fuzzy regions, and
this is due to the fact that those regions are the most dynamic regions in Jordan.
Fuzzification of the population and industrialization variables are shown in Fig.
3.3 and Fig. 3.4 respectively. The words slightly, normally and heavily are used
to indicate how populated or how industrialized is the considered area. Figure 3.5
shows the fuzzification of the temperature variable. Since power consumption is
not very responsive to a small variation in temperature in almost all regions, we
chose the usual terms cold (C), warm (W), and hot (H) in representing the regions
temperature (dry bulb temperature). The climate variable is fuzzified as clear (CR),
partially cloudy (PD), and cloudy (CD) as shown in Fig. 3.6. Measures of cloudiness
are indicated with numbers between one and eight (Meteorological Department).
Finally, the fuzzification of time of the day variable being decomposed into the
fuzzy regions: late at night (LN), early morning (EM), morning (M), midday (MD),
afternoon (AF), evening (E), and finally night (N) as shown in Fig. 3.7.
Fuzzification of System Output Variable
The output of the designed fuzzy model is the amount of power demand at the
associated area. In some areas (load points), the range of variation in power con-
sumption is so wide. Therefore, the fuzzy variable PAPD is decomposed into large
number of fuzzy regions to improve the precision of the area power demand pre-
diction. There are seven fuzzy regions denoted as very very low (VVL), very low
(VL), low (L), medium (M), high (H), very high (VH) and finally very very high
(VVH) as shown in Fig. 3.8. Also the figure shows that the whole amount of power
consumption to be covered by the fuzzy representations has been taken as a variable
K that would be given a certain value depending on the area under consideration.
The membership function associated with the output fuzzy variable is designed to
39


have the form of a trapezoid-shaped function as in the state variables.
Knowledge Base
1- Data-Base
In this case, the FAM bank is a 6 dimensional space with a 7 x (3)5 = 1701
possible fuzzy set entries. Let the domain of time (T), geographical location (L),
temperature (K), climate (CL), population (P) and industrialization (I) be quanti-
fied to n points such that:
U\ = ^2j -jin
U2 = /l, ^2,
7/3 ki, &2 , kji
u4 j *"21 I'n
U5=Pl,P2,-,Pn
Ue = *1,*2,
and the domain of PAPD to p points such that
V pd\, pd2, , pdp
Let the subsets A'1, Bt2, C*3, DlA Et5 and Fl6 represent the ilth, i2th, i3th, iAth,
i5th, and iQth fuzzy region in the state variables T,L,K,CL,P and I respectively.
Finally, let Gl7 represent the i7th fuzzy region in the output variable PAPD, il
and i7=1...7 i2, i3, i4, i5, and i6=1...3. The subsets A'1, Bi2, Ci3, DiA, Ei5, FiG
and Gl7 define the membership functions pg>2, pci3, /*£', PEisi Pf>6 and /j,Gir
that map the elements of tj of Ui, lj of U2, kj of U3, rj of U4, pj of U5, ij of Uq
40


Menbership
Geographical Location
a ~
- I u
a ca

LJ
fc-* * v < O' < a iA m h < <
£ o LJ Ul ** < 4 LI z c
< O' (Q a II a z X < < 4 a
L. uJ D < \A V| Ci.3: u. >- < m ae X (4 < X 14 a < a h- 4 a *
T
City
Figure 3.2: Fuzzification of the location variable.
41
ma*an
SHEDtAH
qvera
AQABA


Menbership Populate
City
Figure 3.3: Fuzzification of the population variable.
42


Henbership
Industrialization
City
Figure 3.4: Fuzzification of the industrialization variable.
43


Membership
Temperature
Figure 3.5: Fuzzification of the temperature variable.


Membership
Clinate
Figure 3.6: Fuzzification of the climate variable.
45


A
wbership
Tine OP The Day
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0
Figure 3.7: Fuzzification of the time variable.
Hours
46


M'W
Figure 3.8: Fuzzification of system output variable: Area Power Demand.
47


and pdj of V to degrees of membership in the interval [0,1]. The membership value
indicates how much tj belongs to the subset Axl and how much lj belongs to the
subset Bt2, etc. Each of the subsets described above can be represented by a fit
vector whose entries are the membership values associated with the support values
addressed to that fuzzy region, such that:[15]
A = (fljl . Oj'n)
Bi2 = (bil...bin)
Ci3 = (cil...cin)
Di* = (dil...din) (3.1)
E* (e,i... e,n)
Fi6 = (fil...fin)
Gi7 = (gu...gip)
where ( il and i7=1...7 i2, i3, i4, i5, and i6=1...3 ) indicate the number of fuzzy
regions in a particular fuzzy variable,
As an illustrative example, let the fuzzy sets A3 *, B1, C1, D3, E2, F3 and G5 en-
code the area power demand-prediction association (Morning, North, Cold, Cloudy,
Normally populated, Heavily industrial; High) that can be interpreted linguistically
as
R= IF T is A3 AND L is B1 AND K is C1 AND CL is D3 AND P is
E2 AND I is F3 ; THEN PAPD is G5
2- Rule-Base
The association combining the state variables with the output variable are the FAM
system rules that define the behavior of the system. The number of these rules or
associations is dependent on the system state variables, each of which is divided
into fuzzy regions.
48


Fuzzy system rules have been set in the algorithm and justified by analyzing and
contrasting the behavior of the designed system. As a matter of fact, the rule-base
of the system under study is said to have the form of a multi-input single-output
(MISO) system, and is defined as follows:
Rmiso = {Rmisoi R-misoi Rmiso}
where Rmiso ' (& = 1 : n) represents the kth rule such that:
Rmiso-' if T is Ail AND L is Bi2 AND K is Ci3 AND CL is DiA AND
P is Ei5 AND I is Fi6 ; THEN PAPD is Gi7
The antecedent of Rmiso frms a fuzzy set Aa x Bl2 x Ct3 x Dt4 x El5 x Fi6
in the product space Ui x U2 x U3 x U4 x Us x Uq. The inferred consequence has a
membership function defined by the fuzzy implication representing the kth rule as
Rah so = (Ail and #i2 and Ci3 and Du and Ei5 and Fi6) > Gir
fiRk = (t)and^*2 (Oand/^c*3 (fc)and//Du (r)and//£;is (p)and//^i6 (?)] ) fiGn(j>d)
MI SO
where {Ail and Bt2 ... F*6; G*7) is the fuzzy association in Ui x Ui... Un x V, and
> denotes a fuzzy implication function. Such types of rules, in the literature are
referred to as state evaluation fuzzy rules.
Decision-Making Logic
As has been mentioned, the developed fuzzy system is aimed to emulate a skilled
human operator in the control center. In other words, it is aimed to model human
decision-making in the framework of approximate reasoning.
In this application, fuzzy reasoning of the first type is based on Mamdanis
minimum operation rule as a fuzzy implication function. In this mode of reasoning,
49


the kth rule leads to the output prediction G'k such that
PG'k (pd) = Qk A Pak (pd) (3.2)
and the resultant inferred prediction G' is the union of the individual output pre-
dictions G'k.
When a set of input values is read, one or multiple rules will be activated in
parallel but to a different degree, as shown in Fig. 2.5. The kth activated rule
will produce a non-null output Gk with a nonnegative weight ak The inferred
consequence G'k is then defined such that,
pG'k (pd) = ak *' pGk (pd) (3.3)
= ock A pGk (pd)
where
ak = pa*i (tj) Pb*2(h) ^c*3 (fy) P-D* fo) Pe* (Pj) Pf* (*j)) (3.4)
Tnin(pA>i (tj)) Pbi2 (tj)) Pc*3 (&})) PDxi (Fj)i Pe,s (Pj)) Pf,b (ij))
Then the resultant inferred output consequence equals the individual weighted
outputs Gk, such that:
771
G' = UGi (3.5)
k-1
771 PG'(pd) = (J pG'k(pd) k=l (3.6)
m
= U akApck(pd)
k=l
m:number of activated rules due to the observations A\ B', C\ D', E\ and F'.
50


Defuzzification
In the defuzzification process, to produce a single numerical output corresponding
to the predicted area power demand, a center of gravity defuzzification process is
encountered.
The fuzzy centroid pd of the output G' is:
_ EfcLi pdkPG'jpdk)
P Eti Mp4)
where pdk is the centroid of G'k.
3.4 System Application
(3.7)
Jordan has been taken as a case study for the proposed model. There are 25
main load points, geographically distributed all over the country and connected
together by 132KV and 400KV national grid lines. Each load point experiences
both different weather conditions and distinct variations of power demand. The
implemented algorithm, shown in Fig.3.9, scans over all the load points on an hourly
basis. At each load point the related data describing the system fuzzy variables are
manipulated by calling the associated fuzzy variable function in such a way as to
compose all the possible rules. A weighted output fuzzy label is then searched to
satisfy an activated rule. As was explained in the previous section, different rules
may be activated at the same time, and the combination of their outputs is then
defuzzified to compute the predicted area power demand at that hour. Finally, the
numerical value of the computed area power demand is added successively to the
total power demand. Implementation of the algorithm, as depicted in Fig.3.9, is
based on the following procedure:
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1. Decide on whether the computation is to be carried out for an ordinary day
or a special day.
2. Begin computation at 1 oclock with one hour increment based on a 24 hour
system.
3. Scan over the 25 load points for each associated hour.
4. Call the time function which classifies day time into an associated fuzzy rep-
resentation^) and determines its degree of membership between (0,1) to that
representation(s).
5. Call the temperature function which classifies the weather dry bulb tempera-
ture at that hour and in that region into an associated fuzzy representation(s)
and determines its degree of membership between (0,1) to that representa-
tion^).
6. Call the climate function which classifies the degree of cloudiness at that hour
and in that region into an associated fuzzy representation(s) and determines
its degree of membership between (0,1) to that representation(s).
7. Call the location function which classifies the regions location into an associ-
ated fuzzy representation(s) and determines its degree of membership between
(0,1) to that representation(s).
8. Call the population function which classifies population of the region into an
associated fuzzy representation(s) and determines its degree of membership
between (0,1) to that representation(s).
9. Call the industrialization function which classifies the regions industrializa-
tion into an associated fuzzy representation(s) and determines its degree of
52


membership between (0,1) to that representation.
10. In the rule function, the input fuzzy representations activate at least one rule
which, in turn, produces a weighted output fuzzy representation.
11. Finally, the weighted outputs are then defuzzified to produce a numerical
value for the predicted area power demand using the defuzzification function.
The defuzzification process is done in the same way as it was explained in the
previous section.
12. The procedure above is repeated for each load point and at each hour.
Functions in the items 4 through 9 convert the input values into fuzzy repre-
sentations with a degree of membership in the same way as they are represented in
the Figures 2 through 7.
An example of power prediction
Consider the forecasting for the Bayader load point for an ordinary day (April
11, 1993) and under the following conditions:
1. Degree of cloudiness is five.
2. Dry bulb temperature of 7.8(7.
3. Time of the day is 9 oclock.
Solution
Following the procedure described in sections 3.3 and 3.4, The input data set
(Bayader, 5, 7.8(7, 9) activates two rules:
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Figure 3.9: Flowchart of the algorithm.
54


1. (M, C, C, PC, NP, NI; M) rule which is interpreted as:
IF time T is morning (M) AND location L is center (C) AND temperature
K is cold (C) AND climate CL is partly cloudy (PC) AND population P is
normally populated (NP) AND industrialization / is normally industrial (NI)
THEN area power demand (PAPD) should be medium (M)
The output PAPD is then set to high but to a degree of membership given by :
min(fiM(9),l*c(Bayader), fic( 7.8), //pc(5), fiNp(Bayader), pNi(Bayader))
Wi = mm(0.7,1,1,1,0.334,1) = 0.334
2. (M, C, C, PC, HP, NI; H) rule which is interpreted as:
IF time T is morning (M) AND location L is center (C) AND temperature
K is cold (C) AND climate CL is patly cloudy (PC) AND population P is
highly populated (HP) AND industrialization I is normally industrial (NI)
THEN area power demand (PAPD) should be high (H)
The output PAPD is then set to high but to a degree of membership given by :
mm(/ZAf (9), fjic(Bayader), /jtc(7.8), /ipc(5), fJ,Hp(Bayader), fiNi(Bayader))
tt>2 = mm(0.7,1,1,1,0.667,1) = 0.667
The above two rules describe the area power demand for the observed values of
the system state variables. The resulting outputs axe combined, and fuzzy centroid
is computed. The training parameter K has been assigned a value of 80 MW for
the Bayader load point. Fig.3.10 illustrates the computational procedure for the
above example. In this case the fuzzy centriodal area power demand value equals
47.315 MW.
55


ia
|Ijj!|j!SIiimi(!iilgB
Figure 3.10: An example of power prediction.


3.5 Discussion and System Evaluation
In the preceding sections we have suggested a fuzzy model for power demand predic-
tion based on the fuzzy variables time, temperature, climate, location, population,
and industrialization. Some of these variables have more effect than others on the
amount of power demand. For instance, the time of the day variable has a high
effect, and this is due to the fact that there are fixed working hours in public offices
and some other industrial areas. However, temperature and climate variables have
also strong effects. This can be noticed from the example given in the previous
section, where a change of 2 MW at the Bayader load point is due to a change of
14C7 in temperature and 5 degrees in cloudiness. Location, population, and in-
dustrialization axe also relevant variables, but they are fixed for each load point in
the short-term load forecasting computations. Hence, these variables can be frozen
for short-term power-demand predictions. This, in turn, will facilitate prediction
processing concerning the time, temperature, and climate variables. This has been
attempted in the model, where each load point was denoted with a fixed variable
covering its location, population, and industrialization, and has shown encouraging
results. The system data concerning actual temperatures, actual degree of cloudi-
ness, and actual power demand at each hour and at each load point have been used
to set up the rules. Data cover April 1 to May 15, 1993. Data concerning power
demand have been taken from the JSCC, and weather-related data have been taken
from the Meteorological Department in Jordan.
As an application example, the load curve has been estimated for Tuesday, April
6, 1993, during the 24 hours, and with the actual weather readings as inputs to the
system. Figure 3.11 shows the actual load curve (solid line), and the predicted load
curve (dotted line) given by the model. The two curves have been compared for the
indices: average error (eavg), maximum error (tmax), minimum error (emin), root
57


mean square error (erms), and standard deviation error (estd)- The above indices
were evaluated for the data obtained, and their values are: eavg of 0.5219%, emax
of 9.8799%, emin of 0.1115%, erms of 4.4338%, and finally an estj of 4.4977%.
The above indices indicate that the system is qualified to replace the work of an
operator. This an interesting result, as it opens the door to industrial automation
in a fuzzy environment. It is worth mentioning that one should not view the actual
values (sold line) in Fig. 3.11 as optimal values. On the contrary, the results (dotted
line) are closer to optimal values. The fact that a computer has a greater ability
to compare, remember, and correlate the conditions for different times, faster and
more accurately than humans, reinforces this belief.
58



Figure 3.11: The actual load diagram (solid line) and the predicted load diagram
(dotted line) for Tuesday, April 6, 1993.


4 Conclusions and Future Directions
This thesis presents a fuzzy approach in the qualitative modeling of system control
and management. However, the recent developments in the area of fuzzy systems
include the application of fuzzy logic controllers (FLC) in numerous industrial appli-
cations, [14]. They have been applied in warm water process, [12], traffic junction,
[22], power system applications, [4,16], [23, 33], fuzzy memory devices, [28], etc. In
these applications, the designed FLC hardware system can operate at a speed of 10
mega fuzzy logic inferences per second (FLIPS). Fuzzy chips and fuzzy computers
have been also designed[28]. Fuzzy chips, for instance, can perform approximately
250 000 FLIPS at 16-MHz clock, and fuzzy computers which were first built by
OMRON Tateishi Electric Co.Ltd are capable of processing fuzzy information at
a speed of approximately 10 Mega FLIPS. A fuzzy computer comprises of a fuzzy
memory, a set of inference engines, a MAX block, a defuzzifier, and a control unit
processing data in a fuzzy environment. Further improvement in this direction for
faster processing and production of reliable components may keep this field in pace
with other sciences.
However, the rise in the complication of data and controls with a diverse phys-
ical nature has significantly hampered the formulation of both quantitative and
qualitative models, and consequently, the development of design methodology. As
a consequence, much attention in system design has to be devoted to analytical
fuzzy methods to simplify models of deterministic and stochastic nature.
On the other hand, the concept of fuzzy systems being applied in this work
through an adaptive approach based on the use of fuzzy logic for predicting hourly
60


power demand in the country of Jordan has shown promising results. An initial
evaluation of the approach has reflected reasonable results that are compatible
with the actual needs. Further evaluation may be gained through the implemen-
tation of sufficient teaching of the model. Indeed, the implemented algorithm has
shown little divergence in realizing the system performance and computational time.
Meanwhile, it can still provide the control center operator with indications of load
fluctuations, due to any inevitable conditions. The total predicted power demand,
to help decide on an optimum number of generators to be committed, has also been
investigated in substantial research in Jordan through the application of an arti-
ficial neural network. Training a neural network would help in scheduling weekly,
monthly and yearly maintenance for all the available generators.
Finally, we conclude this thesis by providing the following suggestions for future
studies:
Further investigation to derive knowledge base and fuzzy rules from the skilled
operators decision-making structure in an efficient and systematic methodol-
ogy-
Development and structuring of an adaptive fuzzy system capable of providing
varying fuzzy regions. This, in turn, will improve system precision relying on
narrowing or widening regions.
Another important topic for forthcoming investigation is to provide a sound
theoretical background means in the development of a theory of deterministic
and stochastic fuzzy systems. In response, an attempt is needed to search the
application and extension of fundamental notations of system dynamic theory
such as state space representation, controllability, observability, stability, etc.
to the conditions and problems on which fuzzy systems are applicable.
61


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Full Text

PAGE 1

FUZZY SETS AND FUZZY LOGIC IN SYSTEM CONTROL AND PREDICTION by Salem Y. Al-Agtash B.Sc., Bogazici University, 1988 A thesis submitted to the Faculty of the Graduate School of the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering 1995 1

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This thesis for the Master of Science degree by Salem Y. Al-Agtash has been approved by Miloje S. Radenkovic Gary G. Leinin /( Marv Anderson Tom Altman Date 11

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To Yahya lll

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Al-Agtash, Salem Y. (M.S., Electrical Engineering) Fuzzy Sets and Fuzzy Logic in System Control and Prediction Thesis directed by Associate Professor Miloje S. Radenkovic ABSTRACT The thesis discusses fuzzy logic rule-based systems as qualitative design methodol ogy for models of diverse physical nature to emulate human-decisionmaking in the framewark of approximate reasoning. Fuzzy system design procedures are explained based on different fuzzi:fication and defuzzifcation criteria, system knowledge base, and decision-making logic. Utilization of fuzzy relation and fuzzy compositional operators in different types of fuzzy reasoning are also discussed in this manuscript. An application example concerning the implementation of a fuzzy-logic approach to provide a structural framework for the representation, manipulation and utiliza tion of data and information concerning the prediction of power demand is also provided. An algorithm has been implemented and fuzzy rules were set to predict the power demand at each load point on an hourly basis. However, the fuzzy rea soning used in the application is based on Mamdani's minimum operation rule as a fuzzy implication function. The parameters taken into consideration cover enviiv

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ronmental and weather-related conditions. Prediction of the power demand at each geographical load point, and hence the country-wide demand, has been tested in Jordan. Results concerning the daily prediction have been obtained with a standard deviation error of 4.4%. Meanwhile, the designed fuzzy rule-based model could allow the system to replace the requirement for skilled dispatchers in predicting daily load curves in a fuzzy environment. This abstract accurately represents the content of the candidate's thesis. I rec ommend its publication. Miloje S. Radenkovic v

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Contents 1 Introduction 2 1.1 Background and Motivation 1.2 Literature Survey . 1.3 Problem Formulation 1.4 Thesis Outline .... Fuzzy Logic, Sets, and Systems 2.1 2.2 2.3 2.4 2.5 2.6 Introduction . . . . Notation and Terminology Set Theoretic Operations Linguistic and Fuzzy Variables Fuzzy Systems . . . . . Fuzzy System Design Procedure 2.6.1 Model Functional and Operational Characteristics 2.6.2 System Control Variables 2.6.3 Fuzzification Strategies . 2.6.4 Behavior of the Control Surfaces 2.6.5 Decision-making Logic . 2.6.6 Defuzzification Strategies 3 Application Example 3.1 Introduction .... 3.2 System Development Vl 1 1 4 5 7 8 8 9 10 12 13 16 16 18 18 20 22 32 34 34 35

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3.3 Power Demand Prediction Model 36 3.3.1 System Analysis 36 3.3.2 Methodology 38 3.4 System Application 51 3.5 Discussion and System Evaluation 57 4 Conclusions and Future Directions 60 Vll

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List of Figures 2.1 Fuzzy sets are membership functions. . . 2.2 Diagrammatic representation of cold and hot. 2.3 Basic configuration of a fuzzy system .. 2.4 Fuzzy system design procedure. 2.5 Fuzzy system archeticture. 3.1 System fuzzy model. . 9 11 15 17 29 36 3.2 Fuzzification of the location variable. 41 3.3 Fuzzification of the population variable. 42 3.4 Fuzzification of the industrialization variable. 43 3.5 Fuzzification of the temperature variable. . 44 3.6 Fuzzification of the climate variable. 45 3. 7 Fuzzification of the time variable. . 46 3.8 Fuzzification of system output variable: Area Power Demand. 47 3.9 Flowchart of the algorithm. . . 54 3.10 An example of power prediction.. 56 3.11 The actual load diagram (solid line) and the predicted load diagram (dotted line) for Tuesday, April6, 1993. . . . . . . . . 59 viii

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ACKNOWLEDGMENT I would like first of all to express my gratitude to those who have contributed to the development of this thesis. I am greatly indebted to Dr. Miloje Radenkovic for his guidance and valuable comments on each part of the thesis. I also wish to thank the members of the committee, Dr. Gari Leininger, Dr. Marv Anderson, and Dr. Tom Altman for their helpful comments and suggestions. The financial support of both the Fulbright Program in the United States Infor mation Agency (USIA) through the American-Mideast Educational and Training Services, Inc. ( AMIDEAST ) and Yarmouk University in Jordan is gratefully acknowledged. Finally, I am thankful to my parents and wife for their love and patience. lX

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1 Introduction Artificial intelligence (AI) has been an active area of research in recent years. It has been successfully used for various applications due to its rapidity, universality, and feasibility. Currently, fuzzy systems and neural networks are popular AI method. ologies. They are dynamic systems that can learn from experience like humans. For instance, fuzzy expert systems are means of representing, manipulating, and utilizing data and information that posses non-statistical uncertainty. They also deal with issues such as reasoning at higher semantic or linguistic levels. Fuzzy systems provide a structural framework that defines each imprecise factor by fuzzy set and when several fuzzy factors exist simultaneously, an action is taken accord ing to some rules. In this project, the framework of fuzzy logic for representing uncertainty and imprecision in system control and prediction is used. This chapter is organized as follows: section one gives a background and mo tivations to the research. Section two provides the reader with a literature survey taking into consideration some practical applications concerning fuzzy set theory. Section three represents formulation of the problem. Finally, the thesis outline is given in section four. 1.1 Background and Motivation In the advent of this century, uncertainty and vagueness were considered to be disturbing terms in models, theories, and statements. The only theory that dealt first with these terms was probability theory, [11, 32]. They were treated on basis of frequentistic interpretation of randomness. 1

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The conclusion of statements about events being neither true nor false led to the formulation of multi-valued propositional calculus by the Polish logician Jan Lukasiewicz in 1920. He chopped the intermediate ground into multiple pieces and came out with multi-valence logic. He then defined the indeterminacy as a con tinuum, a spectrum between falsehood and truth, [15, 32]. As a matter of fact, multi-valence logic was worked out in the 1920's and 1930's to deal with Riesen berg's uncertainty principle in quantum mechanics. Then the term "vagueness" was used by the logician Bertrand Russel to describe multi-valence. In 1937, a quantum philosopher Max Black published a paper on vague sets. In 1965, Lotfi Zadeh published a paper called "fuzzy sets" which applied Lukasiewicz's multi valued logic to sets and groups of objects, [29]. The fuzzy principle states that everything is a matter of degree, and fuzziness, generally, corresponds to truth, falsity and indeterminacy, or to presence, absence, and ambiguity. As the demand for solving optimization problems to complex systems became of great importance to many system designers, the need to include uncertainty in system models rather than neglecting its existence became desirable and sometimes essential, [24, 26, 31]. This, however, has led to numerous attempts to modify existing formal methods to reflect human mental behavior and be applicable to reality. In this contest, various uncertainty theories were suggested in the realm of fuzziness and vagueness. In this direction, fuzzy-based methods are considered to be the most established theory which is based on fuzzy rules relating fuzzy concepts in the form of conditional statements: If X is A, then Y is B, [14, 31, 32]. Fuzzy systems store a number of common-sense fuzzy rules some of which are activated due to input data to some degree, then the fuzzy system produces fuzzy outputs which will then be interpreted to an output crisp value. Meanwhile, fuzzy reasoning is similar to common sense reasoning which takes into account uncertainty 2

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in the interpretation of events. On the contrary, a mathematical way of reasoning is based on the principle of two-valued logic (classical logic) where, a truth-table method is used for the interpretation of formulas of classical propositional calculus as functions in the field of sets, [32]. In system theory, modeling the physical structure of large and non-linear sys tems is extremely diverse and complicated, hence, difficult to control. Traditionally, controllers are designed on basis of a mathematical description and on the system's linearized model. Design of such systems faces the problem of choosing the ap propriate technique to achieve a particular goal. In literature, there have been a number of studies for the control and management of such system models such as state-space and frequency domain techniques, software package-based computa tional procedures, Lagrange multiplier, performance index analysis and mathemat ical programming analysis. These methods are considered to be mathematically rigorous and need considerable computational efforts to execute the iteration pro cedures and it was found that they have some divergent expectations in real time analysis [3, 4, 15, 21, 24, 26]. In addition, they often require modifications by human operators to meet compatible analysis in the real world need. However, the most complicated group of systems are those involving control that links a human operator or a group of people coordinating, matching, or predicting control actions. Management of such systems involves methods to describe human intellectual activities. In this direction, expert systems are used to reflect the exper tise of an expert. Several techniques has been exploited in the literature concerning the application of expert systems such as fuzzy systems and artificial neural net works on various control systems and industrial processes, [4, 14, 18, 19, 20, 22, 23]. In this thesis, we exploit fuzzy-based methods for modeling human mental behavior in complex systems. An application example is provided concerning the power de3

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mand prediction in the country of Jordan, [3, 4]. The recorded data in the Jordan Supervisory Control Center ( JSCC) were utilized to develop a fuzzy model capable of providing the system users with continuous predictions of system loads. Mean while, the main motivation to this research is to provide suitable measures of the sophisticated parameters influencing power demand, generation and distribution. Such measures are, indeed, the reflection of the skilled dispatcher's intuition to schedule the power system (expertise). This expertise is, actually, available in a form of data-base concerning power demand and generation at each load point and at every hour during an elapsed period of time which, in turn, would be of great use for the reflection of past knowledge on future processing. 1.2 Literature Survey Since the inception of fuzzy set theory, there have been numerous fuzzy-based indus trial applications all over the world, including building computer chips and systems that intelligently control subways, automobile systems and numerous consumer elec tronic devices, [32]. Today in Japan, for instance, high-speed fuzzy systems control subways, stabilize helicopters, and regulate air conditioners. In addition, the fuzzy logic control method introduced by Mamdani, [20, 21), has been successfully applied to various complex and nonlinear control problems. Such control methods are used in cam corders, washing machines, vacuum sweepers, and televisions, [32). A fuzzy washing machine, for example, uses load sensors to measure the size and texture of the wash load and uses a pulsing light sensor to measure the dirt in the wash water. Fuzzy rules turn these measurements into patterns of water agitation for different lengths of time. Furthermore, fuzzy systems have been used for system control and prediction based on the expert knowledge involved. They estimate functions and control sys-4

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terns with partial descriptions of a system behavior. However, in the area of power system applications, some research efforts have been devoted to load forecasting. G.Lambert, for example, [16) has proposed a load shape model which obtains the load forecasting through operation of base-load and deviation load. Then, fuzzy conditional statements (rules) were used for the decision making concerning the power prediction. The system variables taken into consideration were the dry bulb temperature, wet bulb temperature, relative humidity, wind direction and wind speed. These variables were related to the country Canada taken as a case study. The average error found in this approach was 2.01 %. Second, Young I1 Park, [23) has used two different methods for short'"term load forecasting: a simple moving average method is used for ordinary days and in the case of holidays and special days, a fuzzy estimation is made. In ordinary days, he assumed that the change of load pattern is not so abrupt and a simple moving average method which requires small calculations can be used. In special days, he assumed that there are impre cise fuzzy factors affecting the shape of load curves and these factors have to be combined with rules for load estimations. The fuzzy factors considered were hot weather, rainy weather, next to a national day and next to New Year's Day. A utility matrix was then constructed using fuzzy decisions. The degree of influence on load data was calculated for the computation of the load prediction. Percentage error was found to be about 1.4 for ordinary days and about 2 for special days. 1.3 Problem Formulation In this work, fuzzy logic, sets, and theory are exploited in the qualitative approach of system management and control design procedures. The research implies the following: First, definition of fuzziness in the terminology of system theory. Sec ond, presenting fuzzy set notations, terminologies, and theoretic operations. Third, 5

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design procedures of rule-based fuzzy systems for modeling human-like decision making within the conceptual framework of fuzzy logic and approximate reasoning. Fourth, various methods of fuzzy reasoning and different types of implication func tions, sentence connectives, and compositional operators are discussed. Finally, fuzzification and defuzzification criteria necessary for the system design interface are also discussed. An application example is also presented concerning the power demand predic tions in the country of Jordan. Fuzzy logic concepts are exploited, in this example, to predict the power demand in an efficient way to help in optimizing the power generation based on expert knowledge. The designed fuzzy system is used to mimic the action of a skilled dispatcher in predicting the power demand, [16, 17], [25]. Figure 3.1 shows the power demand predictor model. The input variables to the fuzzy model are the geographical location, population, climate, temperature, time and the industrial factor. The output of the fuzzy model is the current predicted area power demands (PAPD's) at each of then load points. Computation of the predicted power demand is done using an algorithm based on fuzzy logic. The algo rithm computes the power demand at each load point on an hourly basis concerning a target period of interest. Knowledge-base involves the use of system data-base that stores the data concerning the process for the years back. Jordan power system is believed to be a good application example for the control and prediction of system complexity and uncertainty. This is due to the following reasons: First, the population of Jordan being unevenly distributed. Second, the country's steady growth and modernization process. Third, the region political situation having resulted in mass people movement which, in turn, greatly disturbed the daily life resources and requirements. 6

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1.4 Thesis Outline In this thesis, fuzzy system design procedures are explained through providing an application example regarding the power demand prediction in the country ofJordan. Chapter two discusses fuzzy logic, sets, and theory an a substantial basis. It discusses the various types of fuzzification and defuzzi:fication interface criteria, knowledge base, engine inference, fuzzy reasoning, and implication functions. Chap ter three discusses the implementation of fuzzy logic approach for the prediction of power demand. The chapter also presents the detailed analytical method used in the design procedure. Chapter four gives conclusions and direction for future research. 7

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2 Fuzzy Logic, Sets, and Systems 2.1 Introduction Fuzzy set theory has been an active area of research in recent years. It has been suc cessfully used for various applications, especially in the realm of industrial processes, [14]. Fuzzy set theory describes the mathematical representation of a multi-level logic known as fuzzy logic. It is a logic which is closer in spirit to human think ing and natural language than a traditional logic. Fuzzy logic was first introduced by Zadeh, [29], in 1965 as a mathematical way to describe vagueness and uncer tainty in everyday life. Since then, fuzzy set theory has played an important role in control systems, system structure identification, state prediction, and in processes that lack quantitative data regarding the behavioral input-output relations. For instance, fuzzy logic-based controllers provide means of converting a linguistic con trol strategy into an automatic control strategy based on expert knowledge. On the other hand, rule-based fuzzy predictions reflect human expert's thought process in determining system future behavior. Meanwhile, the literature in fuzzy set theory and its application has grown beyond scientists' expectations to intervene almost in every aspect of human life. This chapter is organized as follows: Section two gives some important notations and terminologies on fuzzy sets. Theoretic operations are also provided in section three. Section four presents the definition of linguistic and fuzzy variables. Section five gives an overview on fuzzy systems. Finally, fuzzy system design procedure is given in section six. 8

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Domain=X Figure 2.1: Fuzzy sets are membership functions. 2. 2 Notation and Terminology A fuzzy set was defined by Zadeh in his pioneering research on fuzzy set theory,[29], as a linguistic approach for representing, manipulating, and utilizing data and in formation that posses non-statistical uncertainty. Definition 2.1 Fuzzy Set, [29]: A class or space of objects X with the generic elements x E X characterized by a membership function JLA(x) is called a fuzzy set A which associates with each point in X a real number in the interval [0, 1]. Unlike conventional sets in set theory, fuzzy sets are always functions from a universe of objects X into [0, 1]. This is depicted in Figure 2.1 which illustrates that the fuzzy set is the function JLF that carries X into [0, 1]. The support of A is the set of points in X at which JLA ( x) is positive. The crossover point in A is an element of X whose grade of membership in A is 0.5. A fuzzy singleton is a fuzzy set A whose support is point x defined as: X (2.1) where JL is the grade of membership of x in A. 9

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Corollary: A fuzzy set A in X is represented as a set of ordered pairs of a generic element x and its grade of membership: A {(x,JlA(x)); x EX} When X is continuous, a fuzzy set A can be written concisely a.S: A= f JlA(x) lx x (2.2) (2.3). Where the integral sign stands for the union of the fuzzy singletons When X is discrete, however, the fuzzy set A has a finite support { Xt, x2 Xn} which can then be represented by the summation. (2.4) As an illustration, suppose that the domain of objects X = {xi}; i = 1 ... oo is the interval [0, 120], with x t::. temperature, then the fuzzy sets of X labeled cold and hot may be represented as cold = r; + (1+ (y 40)'r fy hot = J.:"' ( 1+ (Y 60) 'r' fy Figure 2.2 illustrates the diagrammatic representation of cold and hot. 2.3 Set Theoretic Operations (2.5) (2.6) Definition 2.2 Emptiness: A fuzzy set is empty if and only if its membership function is identically zero on X, i,e {A={; 'v'xEX}} (2.7) 10

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Membership function cold hot 11-----.. OS Figure 2.2: Diagrammatic representation of cold and hot. Definition 2.3 Equivalence: The fuzzy sets A and B are said to be equal written as A = B if and only if /-l A (X) = /-l B (X) for all X in X (2.8) Definition 2.4 Complement: The complement of a fuzzy set A denoted by A defines the membership function 1-l ..4 ( x) for all x E X by 1-l..i(x) = 1JLA(x) (2.9) Definition 2.5 Containment:. A is said to be contained in B or equivalently A is a subset of B, if and only if 1-'A(x) 1-'B(x) for all x EX. In symbols A C B 1-'A(x) :::; 1-'B(x) (2.10) Definition 2.6 Union: The union of two fuzzy sets A and B ( AU B) is also a fuzzy set whose membership function is defined as /-lAuB(x) = max{/-lA(x),/-lB(x)} (2.11) 11

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Definition 2. 7 Intersection: The intersection of two fuzzy sets A and B ( An B) is also a fuzzy set whose membership function is defined as Proposition Given a fuzzy set A and its complement A then AnA -1-0 AUA -I X (2.12) (2.13) (2.14) Definition 2.8 Cartesian Product: The Cartesian product of the fuzzy sets Al ... An in xl, ... Xn is also a fuzzy set in the product space xl X ... X Xn with the membership function (2.15) Definition 2.9 Fuzzy Relation: An n-ary fuzzy relation in X is a fuzzy set R in the product space X x X x ... X, or symbolically Xi EX, i = 1 ... n (2.16) where J.LR is a membership function defined on the universe of discourse. Definition 2.10 Composition: Composition of a two binary fuzzy relations R1 and R2 denoted by R1 o R 2 is also a fuzzy relation in X whose membership function is defined by J.LR1oR2(x, y) =Sup-star {J.LA(x, v ), J.LB( v, y)} (2.17) 11 2.4 Linguistic and Fuzzy Variables A physical variable in fuzzy systems is identified as a linguistic or fuzzy variable whose values are labels or fuzzy subsets of the universe of objects. A fuzzy variable 12

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is usually characterized by a quintuple (y, T( x ), X) in which y is the system variable; T( x) is the term set of labels of y defined on the universe of discourse X. As an illustration consider the temperature variable in a temperature-controlled system as a fuzzy variable with a term set T(temp) where T(temp) ={cold, warm, hot} or T(temp) ={very cold, cold, warm, hot, very hot} where each term in T(temp) is characterized by a fuzzy set in X = (0, 120)F to a degree of membership in (0, 1]. In this contest, the first case, "cold" may be interpreted as "a temperature below about 40", "warm" may be interpreted as "a temperature over 30 and below 70", "hot" may be interpreted as "a temperature over 60 and below 120". In the second case, "very cold" may be interpreted as "a temperature below about 30", "cold" may be interpreted as "a temperature over 20 and below 50", "warm" may be interpreted as "a temperature over 40 and below 70", "hot" may be interpreted as "a temperature over 60 and below 90", and finally, "very hot" may be interpreted as "a temperature over 80 and below 120". 2.5 Fuzzy Systems Fuzzy systems are defined as systems of modeling human decision-making within the conceptual framework of fuzzy logic and approximate reasoning. They are rule based dynamic systems that can recognize complex input-output relations without definition. In conventional physical systems, the output value is often used to adjust the settings of the system states through some kind of process compensators to provide best performance. In contrast, fuzzy systems work on the basis of associa tive memories. They consist of rules that combine the system input and output. A typical fuzzy system consists of four principal components, (18, 19], a fuzzification 13

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interface, a knowledge base, decision-making logic, and a defuzzification interface. Figure 2.3 shows the basic configuration of a fuzzy system. FUzzification Interface: At this stage, values of the input variables are mea sured and transferred into fuzzy representations or linguistic variables viewed as labels of fuzzy sets. Symbolically: x-fuzzifier (x0 ) where x0 is a crisp input value from a process; x is a fuzzy set; and fuzzifier is a fuzzification process. Knowledge Base: The knowledge base, mainly, represents a classification skill of an intelligent system. It consists of expert domain knowledge and attendant control goals. In the knowledge base, the system variables are associated with a linguistic certainty value forming a set of linguistic control rules, and they are characterized by the control policy of the expert domain of the process. Symbolically: K = (U,C) where K is referred to as a knowledge base; U is a finite set representing the universe of objects. C = { c2 em} represents a family of classification patterns in which Ci is a disjoint subset in U. Decision-making: Decision-making is the area of command and control to select the most suitable guidance algorithm based on the system rules of inference and fuzzy control implications. 14

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Knowledge base .. -------. I I D
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Defuzzification Interface: This stage implies converting the inferred fuzzy system output to a non-fuzzy output. Symbolically: Yo= defuzzifier (y) where y is an output fuzzy set value; y0 is a crisp output value; and defuzzifier is a defuzzification process. 2.6 Fuzzy System Design Procedure An essential part in the design of fuzzy system is claimed to be understanding the system's behavior mechanism and identifying the system dynamics in terms of a conventional input-output model. Second, a conceptual design to formalize and structure a procedure which would then be modeled and simulated iteratively on a computer to obtain a desired system behavior. In fact, there are six steps considered in the design as illustrated in figure 2.4. First, define the model functional and operational characteristics. Second, define the system control variables. Third, select a fuzzification strategy. Fourth, define the behavior of the control surfaces. Fifth, synthesize the structure of the system model (decision-making logic). Finally, select a method of defuzzification. Each of the above steps is investigated separately in the following sections. 2.6.1 Model Functional and Operational Characteristics At this stage, the architectural characteristics of the system are identified in terms of an input-output process model. This basically, deals with three main points in the fuzzy system design: 1. What information and data points :flow into the system. 16

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"Define the functional and operational II characteristics of the model I 11Define the system control variables .. I 11Select a fuzziflcation strategy I_ 'Define the behavior of the control surfaces11 I 11Synlhesize the structure of the system .. I select a difuzzlflcatlon strategy Figure 2.4: Fuzzy system design procedure. 17

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2. What basic transformations are performed on the data. 3. What data elements are output from the system. 2.6.2 System Control Variables This step implies identifying the space of process state variables {X} and the space of output variables {Y}. However, the system outputs Y = {y1 Ym} are related to the system states X = { x1 . xm} by some type of transformation function which takes into account the system unobserved disturbances and the control ac tions applied. Definition 2.11 : Let X designate the space of observed process states, U the space of control actions with which a set of objectives { Z*} is satisfied, E the space of unobserved disturbances, and Y the space of process outputs, then Y = F0(X, U, E) where F0 is the operator transforming the input X into the output Y. In fuzzy sys tems the transformation operator F0 is the rule-base which presents the realization of the system strategies, symbolically where >is an operator > = >(., .) to process the information on the states of the input and output I(X, Y) and the system's desired objectives {Z*} 2.6.3 Fuzzification Strategies In a fuzzy model, a fuzzification strategy implies decomposing each process state (input) variable and output variable into a set of fuzzy representations known as fuzzy sets. A fuzzy set linguistically approximates uncertain or imprecise sampled 18

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values to a degree of membership in the interval [0, 1]. However, the fuzzy sets representing the system state variables form a fuzzy input space, while those repre senting the system output variables form a fuzzy output space. Noticeably, in the fuzzification process, the uncertain information of the system universe of discourse is either continuous or discrete. In the discrete universe, the fuzzification process is carried out based on labeling segments of intervals with fuzzy sets. Each generic element in the segment is identified with a grade of membership value in [0, 1]. On the other hand, if the universe is continuous, a discrete universe is formed by a discretization of the continuous universe, [18]. This is important in the system data quantification and representation with fuzzy sets. Quantification discretizes a universe into a certain number of segments known as quantization levels. Each segment forms a discrete universe labeled with a fuzzy set that identifies a grade of membership value in [0, 1]. Finding the number of fuzzy sets or linguistic terms associated with each variable defined on the same universe of discourse is believed to be a complex issue. Some rules of thumb have been suggested by [6], [8]. First, the number of fuzzy sets associated with a variable should generally be an odd number. Second, each set should overlap between 10 and 50 percent of the neighboring space. Finally, the density of the fuzzy sets should be highest around the optimal control point of the system. The grade of membership function defining a fuzzy set may be represented by a vector of numbers (denoted by [15] as a fit vector) or by a functional form (e.g. bell-shaped function, triangular-shaped function ... etc.). A fit vector represents a discrete universe of discourse X whose entries are the membership values at each sample datum in X. In this case, the fuzzy set Ai is written as in Eq. 2.4. On the other hand, a functional form represents a continuous universe of discourse X, and 19

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the fuzzy set Ai is of the form in Eq. 2.3. As an illustration, the fuzzy sets, in fuzzy logic controllers, usually have a mean ing such as NB: negative big; NM: negative medium; NS: negative small; ZE: zero; PS: positive small; PM: positive medium; PB: positive big; or any other common sense linguistic terms that best describe a physical variable in a control system. It is also important to mention that in some applications, the observed data are disturbed by a random noise which is measurable only in a statistical sense. The fuzzification strategies, in this case, should involve both uncertainty and ran domness. In this sense, Bharathi [7] and Dubois [10] have a broad explanation on the fuzzification of statistical data. Two methods were suggested. First, bijective transformation which transforms a probability measure into a possibility measure by using the concept of the degree of necessity. Second, hybrid number arithmetics regarding both probabilistic and possibilistic modes of characterization. 2.6.4 Behavior of the Control Surfaces A control surface is a hypothetical surface created by the interaction of rules with fuzzy sets. Defining the behavior of the control surfaces implies writing fuzzy con ditional statements of the form, "If-Then" rules which reflects or expresses system expert's domain knowledge. Fuzzy rules are usually formulated in linguistic terms simulating human decision-making behavior. Such a group of rules is deduced from the observation of human expert's actions in terms of the input-output data. A fuzzy rule is mainly defined by the relation between observation (antecedent) and action (conclusion or consequent). The maximum number of fuzzy rules is deter mined by the cardinality D, or the number of representations, of a term set in the fuzzy input space. For example, if the cardinality of two input fuzzy variables x1 and x2 in a fuzzy system are D(xi) = 5 and D(x2 ) = 7 respectively, then the 20

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maximum number of rules will be 5 x 7 which represent the total possible input combinations. In some cases, fewer rules may be used but in a compromise of knowledge. Meanwhile, based on expert knowledge and the fuzzy model of the process, the linguistic description of the process dynamic characteristics formulates a fuzzy set of control rules for optimizing the overall system performance. Therefore, expert experience and system engineering knowledge have a crucial role in the characteri zation of the operational rules in a fuzzy system. In this reasoning model, for instance, a multi-input single-output fuzzy model with an observation space U, and action space V, would be characterized by the fuzzy rules of the type Rk If Ail d Ai2 d Ain2 th Bi : X1 IS l an X2 IS 2 . an Xn IS n2 en y IS where Rk is the kth rule (1 :5 k :5 n1); n1:the cardinality of the observation space; x1 (1 :5 l :5 n2) are fuzzy input variables; y is the fuzzy output variable; Af and Bi are the fuzzy sets of xz andy in the universe of discourses U and V respectively. Rk can be rewritten as Rk : If X is Ai then Y is Bi where Ai = (Ai1 ... (1 :5 i :5 ml ; ml : number of fuzzy sets in x1) are the fuzzy sets of the linguistic variables X = (x11 x2 Xn2 ) in the universe of discourse U = (U11 U2 Un2). In fuzzy systems, a fuzzy rule is implemented by a fuzzy implication function in which Ri is written symbolically (2.19) (2.20) where (Ai1 and ... Bi) is the fuzzy association in ul X u2 ... Un X v, and 21

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denotes a fuzzy implication function. Example : A fuzzy rule in an anti-lock braking system might be: R1 = "If brake temperature is Warm and speed is Slow then brake pressure is Slightly Decreased". Here, the observation space U is characterized by the set F(X) = { degree of brake temperature} x { term for speed } and the action space V is characterized by the set F(Y) = {terms of brake pressure}. For x1 may have a term set as {Hot, Warm, Cold} of which all the elements represent a common sense linguistic representation over the universe of discourse of U1 x2 may have a term set as { Slow, Medium, Fast} of which all the elements represent a common sense linguistic representation over the universe of discourse of U2 Finally, y may have a term set as { Slightly Decreased, Decreased, Constant, Slightly Increased, Increased} of which all the elements represent a common sense linguistic representation over the universe of discourse of V. Hence, the cardinality representing the number of all possible system rules is equal to 3 x 3 = 9. Meanwhile, the fuzzy rule in the above example can be written symbolically as: ( ( U2)] JlBl ( V) 2.6.5 Decision-making Logic (2.21) (2.22) (2.23) The decision-making logic is an interface between the human model ari.d the formal model used to represent a particular judgment situation. This interface relates the human information processing activities with the imperfect knowledge of system information and organization. Decision-making logic involves concepts of system 22

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fuzzy relation, sentence connectives, Compositional operators, and interface mechamsms. Fuzzy Relation The concept of a fuzzy relation is a generalization of the concept of a function. In fuzzy logic, fuzzy relations are fuzzy system rules expressed with some type of fuzzy implication function. Moreover, there are many fuzzy implication functions proposed in literature with the following basic properties, [18, 19] : fundamen tal property, smoothness property, unrestricted inference, symmetry of generalized modus ponens and generalized modus tollens. Lee, [19] classified them into three categories: the fuzzy conjunction, the fuzzy disjunction, and the fuzzy implication. The former two are closely related to the fuzzy Cartesian product, and the latter is a generalization of implication in propositional calculus, modus ponens, and modus toll ens. Considering the fuzzy rule "If x is A then y is B" which can be represented by a fuzzy implication function A --+ B, where A is the antecedent and B is the consequent in the universes U and V with the membership functions /-LA and /-LB respectively. Then Definition 2.12 Fuzzy conjunction, [19]: The fuzzy conjunction for all u E U and v E V is defined by A--+B -AxB (2.24) f JLA(u) JLB(v) luxv (u,v) (2.25) Here is an operator representing intersection, algebraic product, bounded .product, or drastic product. Definition 2.13 Fuzzy disjunction, [19, 29]: The fuzzy disjunction for all u E U 23

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and v E V is defined by -AxB (2.26) f JLA(u) JLs(v) luxv (u,v) (2.27) Here is an operator representing union, algebraic sum, bounded sum, or drastic sum. For more details on the operators, refer to [19]. Definition 2.14 Fuzzy implication, [30, 18]: The fuzzy implication for all u E U and v E V is defined by 1. Material implication: AB (2.28) 2. Propositional calculus: (2.29) 3. Extended propositional calculus: (2.30) 4. Generalization of modus ponens: = sup{cE (2.31) 5. Generalization of modus tollens: B = inf{t E [0,1],Bt (2.32) where the operators and are defined as in definitions 2.12 and 2.13. 24

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Furthermore, here are some examples of fuzzy implications adopted in many practical applications as mentioned in, [19, 21]: Min-operation rule of fuzzy implication Rc -AxB (2.33) f JLA(u)nJLs(v) luxv u,v (2.34) Product operation rule of fuzzy implication RpAxB (2.35) f JLA(u)JLs(v) luxv u,v (2.36) Arithmetic rule of fuzzy implication Ra (A X V)(U X B) (2.37) { 1 n (1JLA(u) + JLs(v)) luxv u,v (2.38) Maximum rule of fuzzy implication Rm -(A X B) u (A X V) (2.39) f (JLA(u) n JLB(V)) U (1 JLA(u)) (2.4Q) luxv u,v Sentence Connectives The sentence connectives "and" and "else" are commonly used in system fuzzy rules. "and" connective is interpreted as an intersection (min operator) in the Cartesian product, and "else" is interpreted as a union (max operator). 25

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Definition 2.15 "and", [19, 29]: Let U and V be the universes of A and B respectively, then the antecedent of "if A and B then C" is also a fuzzy set in the product space U x V with J.LAxB(u,v) = min{J.LA(u),J.LB(v)} (2.41) The fuzzy relation in this context, is defined as R= (A X c (2.42) Definition 2.16 "else", [19, 29]: Let U and V be the universes of A and B respectively, then the connective "else" in the fuzzy rule "if A and B then C else D" is interpreted using the following fuzzy relations: Compositional Operator R = (A X c (A X D (A X D (A X D (2.43) (2.44) (2.45) (2.46) A Compositional operator is referred to as a Sup-Star composition, where star denotes an operator ( min, max, product, ... etc.). Definition 2.17 Compositional rule of inference, [19, 29]: Let R be a fuzzy relation from U to V and A is a fuzzy subset of U, then the Compositional rule of inference asserts that the solution of the fuzzy subset B of Vis B A oR (2.47) J.LAoR( V) Supstar {J.LA( u), J.LR(u, v)} ueU,vEV (2.48) 26

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where o is a Compositional operator. The main Compositional operators mentioned in the literature are the Sup-min, Sup-product, Sup-bounded and Sup-drastic product, [19, 29] Selection of a Compositional operator depends on the nature of some defined problems. Meanwhile, Sup-min and Sup-product operators are the most frequently used in fuzzy system applications. Inference Engine The inference engine is a mechanism for manipulating rules from the knowledge base, forming inferences and drawing conclusions. The conclusions can be deduced in a number of ways which depend on the structure of the engine. An uncertainty of information in the knowledge base would, however, certainly induce uncertainty in conclusions. Therefore, the inference has to provide a suitable measure of un certainty in the conclusion which can be comprehensive and interpretable to the system user. Lee, [19], stated four types of fuzzy reasoning. First, a reasoning based on Mamdani 's minimum operator rule. Second, a reasoning based on Larsen's product operation rule. Third, a reasoning based on Tsukamotos' method with a monotonic membership function. Finally, a reasoning which is based on taking the consequences of a rule as a function the input fuzzy variables. Definition 2.18, [19], Given an observation A' and B' in a 2-input, 1-output fuzzy system with A17 AN1 as fuzzy subsets in U and B17 BN2 as fuzzy subsets of v, and ... 'CNa as fuzzy subsets of w. Then the action inferred from the kth rule Rk of a set of N1 x N2 rules is defined as follows = (A', B') o Rk (2.49) where A' E U, B' E V, C' E W, and Rk E U X V X W JLc;. (w) -Sup-star (JLA'( u), JlB'( v ), JLR,.( u, v, w)) u.EU,vEV (2.50) 27

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-Sup-star (JLA'(u) JLB'(v),JLc,.(w)) ueU,veV here, ak : JLA'(u) JLB'(v), : implication operator, *" : fuzzy reasoning resulting from the implication function used. When the input data driven from the observation space is read, one or multiple rules will be activated in parallel but to different degrees, as shown in figure 2.5. The kth activated rule will produce a non-null output Ck with a non-negative weight ak depending on the fuzzy reasoning and the implication operator used. The inferred consequence Ck due to the kth rule is given by equations (2.49, 2.50). Finally, the resultant inferred consequence C' equals the union of the individual weighted outputs Ck. Hence, m c' = U (2.51) k=l and, the membership function of the inferred consequence C' can be written as m JLC' U (2.52) k=l m U ak *. JLc,.(w) (2.53) k=l m:number of activated rules due to the observations A' and B' In real time processes, the inputs are usually measured by sensors and are crisp values which will then be converted into fuzzy sets in the fuzzification process. In turn, these will activate a number of rules parallely in the knowledge base processing which will be interpreted to give a control action, prediction result or parameter estimation value. As an illustration, consider the input data u0 v0 which activate the following two control rules 28

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Input Data Rule 1 C2 1 1--0utput : C -i Defuzzlner Data ./ cotm Figure 2.5: Fuzzy system archeticture. Rl : if X is AI and y is Bl then z is cl R2 : if X is A2 and y is B2 then z is c2 Then the weighing factors a1 and a2 of the first and .second rule may be ex pressed as Here* could be any implication operator in the class of (minimum, algebraic prod uct, hounded product, or drastic product) or the class of (union, algebraic sum, bounded sum, drastic sum, or disjoint sum). Then the fuzzy reasoning due to the first and second rule will lead to the control decisions: 29

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with membership functions Then the inferred consequence C' is c' u /-lC{ ( W) U ( W) Selection of *' depends on the type of fuzzy reasoning .1. Type 1 "Mamdani's minimum operation rule": The individual inferred con sequence from the kth rule is computed by using the minimum operation rule (defined by an operator (A)) as an implication function in the fuzzy reasoning encountered. Hence, 1-lc;. (w) = O:k A J.lc,. (w) k = 1,2 (2.54) Then /-lC' U /-lC' 1 2 (2.55) /-lC' : The membership function of the resultant inferred consequence C'. 2. Type 2 Larsen's product operation rule": The individual inferred conse quence from the kth rule is computed by using the product operation rule (defined by an operator (.)) as an implication function in the fuzzy reasoning encountered. Hence, 30

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k = 1,2 (2.56) Then /-LC1 U /-LC1 1 2 (2.57) To obtain a deterministic control action for the first two types of reasoning a defuzzification is required, as will be discussed later in this chapter. 3. Type 3 "Tsukamoto's Method": The results inferred from the first and second rule are o:1 and o:2 such that (2.58) (2.59) And the crisp control action is the weighted combination (2.60) 4. Type 4 "The consequence of a rule is a function of input variables": In this mode of reasoning the fuzzy rules are of the form Rt : if x is At andy is B1 then z = ft(x, y) R2 : if xis A2 andy is B2 then z = h(x,y) fi : is a function of the process state variables x, y defined in the input subspaces. The control action inferred from the first rule is o:1f1 ( uo, vo) and that is inferred from the second rule is o:2 !2 ( u0 v0). The crisp value is then defined as (2.61) 31

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2.6.6 Defuzzification Strategies The defuzzification process is a mapping from a fuzzy output space defined over the universe of discourse U into a space of non fuzzy (crisp) control action. The mapping function is denoted by a defuzzification strategy. The most commonly used strategies are: the max criterion, the mean of the max, and the center of gravity, [12, 15]. 1. Max criterion: This is the simplest defuzzification strategy. The output crisp value w is computed by choosing the element that has a maximal membership value in the output fuzzy distribution C'. In particular, to find Wmax of the C' distribution where (2.62) If the output space W equals a finite set of values {WI, ... wp}, then the supremum in Eq. 2.62 is replaced with a maximum /-LC' ( Wmax) =m{IX {J.Lc' ( Wi)} (2.63) 2. Mean of the max: The crisp output value w generated from this strategy is based on computing the mean value of all fuzzy outputs whose membership value reaches the maximum. i.e. k ... "Wi w=LJi=l k (2.64) where Wi is the support value at which the membership function is maximum, and k is the number of these values. 3. Center of gravity: In this defuzzification strategy, all the information in the output fuzzy distribution C' is used to compute the output value as the cen-32

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troid w (2.65) In the case of discrete fuzzy systems, the fuzzy centroid w of the fit-vector C' IS (2.66) where Wk is the centroid of Ck. 33

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3 Application Example 3.1 Introduction This chapter discusses the implementation of a fuzzy-logic approach on the predic tion of power commitments. In fact, prediction of power demand plays an important role in various kinds of power system applications such as unit commitment and economic dispatch. It can be used for the system operation as an aid for the oper ator to know the power demand in the lead-time period. However, there have been a number of studies in this field during the past few decades. Traditional methods used are known to be inaccurate for days with special events. This is due to the fact that power prediction for those days requires the knowledge and experience of an expert in a control center. Due to this fact, several artificial intelligence (AI) methodologies have been proposed to replace the expert in the control cen ter. Some of these methods have shown encouraging results concerning short-term, medium-term, and long-term power demand predictions. In the literature, fuzzy systems and neural networks are important AI method ologies used for power demand prediction, (16, 23]. Fuzzy systems simulate human like performance. They are composed of rules that relate system inputs to system outputs using fuzzy logic. Meanwhile, fuzzy logic simulates human thinking that measures the degree of uncertainty and vagueness in a physical variable. Fuzzy sys tems are dynamic systems that recognize an input-output relationship using fuzzy associations (rules). They deal with issues at a linguistic level by providing a struc tural framework that defines each imprecise factor by a fuzzy set and when several fuzzy factors exist simultaneously, an action is taken accordingly with some rules. 34

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The proposed study was aimed at developing a fuzzy system for power demand predictions on both short-term and medium-term lead-time periods. The imprecise environmental, industrial, population, and weather related factors were considered. This chapter is organized as follows: section two gives a general description of the system development. The power demand prediction model is presented in the third section. System application is discussed in the fourth section. Finally, discussion and system evaluation are presented in section five. 3.2 System Development The computational unit (fuzzy model) depicted in Fig.3.1, is used to predict the total power demand based on expert knowledge. The input variables to the fuzzy model are the geographical location, population, climate, temperature (dry bulb temperature), time and the industrial factor. The outputs of the fuzzy model are the current predicted area power demands (PAPD 's) at each of the n load points. The presented model is used to predict the total power demand in the days ahead. The architectural characteristics of the model are designed in such a way to evolve the information :flowing at each load point then executing all the necessary rules combining the input values to the output model, [3, 4]. Computation of the predicted power demand has been implemented using an algorithm based on fuzzy logic. The algorithm computes the power demand indi vidually at each load point and on hourly basis. The input data fed into the model concerning each state variable are fuzzified, which, in turn, will activate at least one of the system rules. Each activated rule will emit a weighted fuzzy output variable. Summation of all weighted fuzzy outputs is then defuzzified to produce the power demanded by each load point and at every hour concerning the target period of interest. The system data taken from the Jordan Supervisory and Control Center 35

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Time ClimateTemperature -Geo.location Industrialization_ Population Power Demand Predictor Figure 3.1: System fuzzy model. PAPDl PAPDn (JSCC) were utilized in the derivation of fuzzy system rules based on the human operator experience. 3.3 Power Demand Prediction Model 3.3.1 System Analysis In this section, the fuzzy model used to predict the total power demand is presented. As shown in Fig.3.1, the input variables to the fuzzy model are: time of the day (T), geographical location (L), temperature (K), climate (C), population (P) and industrialization (I) These variables are treated as state fuzzy variables in the observation space. The outputs of the fuzzy model are the predicted area power demands (PAPD's). The amount of power consumption varies according to the variation of system's state fuzzy variables concerning a target period. Time of the day factor inherently imbeds human requirements for electricity as related to the day and night activities, weekdays, weekends, holidays and finally seasonal requirements. The temperature factor supports energy requirements re-36

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lated to weather conditions. The location factor involves special requirements, and people's habits that may influence energy needs. The climate also reflects the reac tion of humans to energy needs as a response to rainy, snowy and cloudy weather conditions. Finally, the population and industrialization factors have a significant effect on power consumption. For example, it could be expected that a certain load point which is lightly populated, highly industrial, and under cold and cloudy weather conditions, will have a medium power demand. Meanwhile, another load point under the same other conditions would require higher power demand, if it were highly populated. It should be noted that the model's main objective is to predict the total power demand within considerable bounds of uncertainty. The prediction of power demand is dependent on the system variables that may vary from time to time. Fuzzification of variables involves a trade off of precision in prediction and computation time. Increasing the number of fuzzy regions would improve the preci sion of power prediction but, on the other hand, would increase the computational time and hence the time between two adjustments becomes larger. For example, the fuzzy variable, temperature, may have three labels: Cold(C), Warm(W) and Hot(H), and in a more precise identification it may have five labels: Very Cold(VC), Cold( C), Warm(W), Hot(H), and Very Hot(VH). Then a 0C may be cold (the first case) or very cold (the second case). As a matter of fact, consumption of power is considerably dependent on temperature. That is, a decision is taken in favor of low power demand when the combination includes cold in the first case and in favor of very low power demand when the combination includes very cold in the second case. In summary, decomposing a certain fuzzy variable into large number of fuzzy regions leads to a more precise prediction of the power demand. However, the cost of better precision is a large processing time, large memory needed to absorb all data and temporal results, and large step size between two consecutive adjustments. 37

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3.3.2 Methodology The fuzzy system constructed here is a fuzzy associative memory (FAM) system, [15]. The fuzzy associative memories (FAMs) are transformations that map fuzzy sets to fuzzy sets. It encodes a bank of compound FAM rules that associate multiple outputs with multiple inputs. Fuzzification of System State Variables In a fuzzy system, the physical variables are identified as fuzzy variables. A fuzzy variable describes the discrete values of the physical variable with linguistic common sense terms. These linguistic terms are called fuzzy-set values or fuzzy regions. For example, the temperature variable taken in our study is a fuzzy variable that takes H (hot), W (warm), and C (cold) as fuzzy-set values. The set W covers temperatures between 10 and 30 degree celsius to a degree of membership between zero and one. Different fuzzy quantifications may be done depending on the number of fuzzy set values (fuzzy regions). In our application, prediction of power demand on a country-wide basis, the physical variables involved are chosen in a way that they are highly effective on the amount of power consumption. In the model, each of the system state fuzzy variables is decomposed into a reasonable number of fuzzy regions following the rules of thumb,[8], that is by choosing an odd number of labels associated with a variable and each label should overlap between 10 and 50 percent of the neighboring space, finally the density of fuzzy sets should be highest around the optimal control point of the system and should thin out as the distance from that point increases. The membership function associated with each of the state variables is designed to have the form of a triangular-shaped function. Figure 3.2 shows the fuzzification of the location variable where northern (N), 38

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center (C), and southern (S) parts of Jordan are the chosen fuzzy regions, and this is due to the fact that those regions are the most dynamic regions in Jordan. Fuzzification of the population and industrialization variables are shown in Fig. 3.3 and Fig. 3.4 respectively. The words slightly, normally and heavily are used to indicate how populated or how industrialized is the considered area. Figure 3.5 shows the fuzzification of the temperature variable. Since power consumption is not very responsive to a small variation in temperature in almost all regions, we chose the usual terms cold (C), warm (W), and hot (H) in representing the region's temperature (dry bulb temperature). The climate variable is fuzzified as clear (CR), partially cloudy (PD), and cloudy (CD) as shown in Fig. 3.6. Measures of cloudiness are indicated with numbers between one and eight (Meteorological Department). Finally, the fuzzification of time of the day variable being decomposed into the fuzzy regions: late at night (LN), early morning (EM), morning (M), midday (MD), afternoon (AF), evening (E), and finally night (N) as shown in Fig. 3.7. Fuzzification of System Output Variable The output of the designed fuzzy model is the amount of power demand at the associated area. In some areas (load points), the range of variation in power con sumption is so wide. Therefore, the fuzzy variable PAPD is decomposed into large number of fuzzy regions to improve the precision of the area power demand pre diction. There are seven fuzzy regions denoted as very very low (VVL), very low (VL), low (L), medium (M), high (H), very high (VH) and finally very very high (VVH) as shown in Fig. 3.8. Also the figure shows that the whole amount of power consumption to be covered by the fuzzy representations has been taken as a variable K that would be given a certain value depending on the area under consideration. The membership function associated with the output fuzzy variable is designed to 39

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have the form of a trapezoid-shaped function as in the state variables. Knowledge Base 1-Data-Base In this case, the FAM bank is a 6 -dimensional space with a 7 x (3)5 = 1701 possible fuzzy set entries. Let the domain of time (T), geographical location (L), temperature (K), climate (CL), population (P) and industrialization (I) be quanti fied to n points such that: U2=h,l2, ... ,ln u3 = kb k2, ... kn and the domain of PAP D to p points such that Let the subsets Ail Bi2 Ci3 Di4 Ei5 and pis represent the i1 th i2th i3th i4th ' ' ' i5th, and i6th fuzzy region in the state variables T,L,K,CL,P and I respectively. Finally, let Gi7 represent the i7th fuzzy region in the output variable PAP D, i1 and i7=1. .. 7, i2, i3, i4, i5, and i6=1...3. The subsets Ail, Bi2 Ci3 Di\ Ei5 pia and Gi7 define the membership functions J.LAil, J.LBi2, J.LOi3, J.LDi4, J.LEis, J.LFia and J.LGiT that map the elements of tj of ul, lj of u2, kj of u3, Tj of u4, Pi of Us, ij of Us 40

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'I . . . GeogrophlcGI locG tron Me,bersh;p N 1.00 c s 0.75 0.50 0.25 -, Crty c c Ill 5" 11'10: c: D CA ID c: !..: -c ::r::: z % c:c: ;; % w c: u ::r:::a -w i: a.: "' :a: "' c ... !..: c ::a: w oc: > "' lo.c: ::r::: c: c: c: ..:; ::r::: X IIIQ "' c 0 "' "' ::I c: IE Ill: Ill Figure 3.2: Fuzzification of the location variable. 41

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!=) 0 0 b N lJ1 Ul 0 Ul 0 :3: ,. ::J IT I ,. BROADCAST Ill ;r-RESHA '6 l-%j SHEDIAH ..... RU\JEISHED J l!tl '"I ,...,r ('!) c..:> RASHADIAH c..:> QATRANA G. SAFI N F'UHEIS N s; HASA 0 GAIA ..... AZRAQ 0 '1:1 I=' MRA r 0 'C *"" a. c p -I
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Tempera.ture MeMbership 1.00---1----------0.75 0.50 0.25 u 10 20 30 40 Figure 3.5: Fuzzification of the temperature variable. 44

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MeMbership CliMate CR PC CD 1.00 I I i0.75 1-0.50 0.25 ,I J I I I I I I I I I I Degree 0 .s I.S 2 2.S l J.s 4 s s.s 6.S 7 7.'5 8 Figure 3.6: Fuzzification of the climate variable. 45

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TlMe> Of' The Day "'bership LN EM H MD AF E N Hours 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 0 Figure 3. 7: Fuzzification of the time variable. 46

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Power DeMond VVL VL L M H VH VVH 1.00 0.75 0.50 0.25 l M\J \/ 0 k Figure 3.8: Fuzzification of system output variable: Area Power Demand. 47

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and pd; of V to degrees of membership in the interval [0,1]. The membership value indicates how much tj belongs to the subset Ail and how much lj belongs to the subset Bi2 etc. Each of the subsets described above can be represented by a fit vector whose entries are the membership values associated with the support values addressed to that fuzzy region, such that:[15] Ail(a a ) Bi2 = (bi1 bin) Ci3 c ) ""1 Di4(d d ) tl tn (3.1) Ei5 = ( ei1 ein) Fi6 -( .f. .f. ) Jm Gi7(g g ) tl . tp where ( i1 and i7=1. .. 7 i2, i3, i4, i5, and i6=1. .. 3 ) indicate the number of fuzzy regions in a particular fuzzy variable, As an illustrative example, let the fuzzy sets A3 B1 C1 D3 E2 F3 and G5 en code the area power demand-prediction association (Morning, North, Cold, Cloudy, Normally populated, Heavily industrial; High) that can be interpreted linguistically as R=" IF Tis A3 AND Lis B1 AND K is C1 AND CL is D3 AND Pis E2 AND I is F3 ; THEN PAPD is G5 2-Rule-Base The association combining the state variables with the output variable are the FAM system rules that define the behavior of the system. The number of these rules or associations is dependent on the system state variables, each of which is divided into fuzzy regions. 48

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Fuzzy system rules have been set in the algorithm and justified by analyzing and contrasting the behavior of the designed system. As a matter of fact, the rule-base of the system under study is said to have the form of a multi-input single-output (MISO) system, and is defined as follows: where R'Mrso ( k = 1 : n) represents the kth rule such that: R'Mrso=" IF Tis Ai1 AND Lis Bi2 AND K is Ci3 AND CL is Di4 AND P is Ei5 AND I is Fi6 ; THEN P APD is Gi1 The antecedent of R'M 150 forms a fuzzy set Ai1 x Bi2 x Ci3 x Di4 x Ei5 x Fi6 in the product space ul X u2 X u3 X u4 X Us X Us. The inferred consequence has a membership function defined by the fuzzy implication representing the kth rule as Rk -(Ai1 and Bi2 and Ci3 and Di4 and Ei5 and Fi6 ) --+ Gi1 MISO where (Ail and Bi2 ... Fi6 ; Gi7 ) is the fuzzy association in ul X u2 . Un X v, and --+ denotes a fuzzy implication function. Such types of rules, in the literature are referred to as state evaluation fuzzy rules. Decision-Making Logic As has been mentioned, the developed fuzzy system is aimed to emulate a skilled human operator in the control center. In other words, it is aimed to model human decision-making in the framework of approximate reasoning. In this application, fuzzy reasoning of the first type is based on Mamdani's minimum operation rule as a fuzzy implication function. In this mode of reasoning, 49

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the kth rule leads to the output prediction such that (pd) = O:k A J-la,. (pd) (3.2) and the resultant inferred prediction G' is the union of the individual output pre dictions When a set of input values is read, one or multiple rules will be activated in parallel but to a different degree, as shown in Fig. 2.5. The kth activated rule will produce a non-null output Gk with a nonnegative weight O:k The inferred consequence is then defined such that, (pd) O:k *' J-lG,. (pd) (3.3) O:k A J-lG,. (pd) where Then the resultant inferred output consequence equals the individual weighted outputs Glc, such that: m G' u (3.5) k=l m J-ta(pd) U (3.6) k=l m U O:k A J-tc,. (pd) k=l m:number of activated rules due to the observations A', B', C', D', E', and F'. 50

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Defuzzification In the defuzzification process, to produce a single numerical output corresponding to the predicted area power demand, a center of gravity defuzzification process is encountered. The fuzzy centroid pd of the output G' is: (3.7) where pdk is the centroid of 3.4 System Application Jordan has been taken as a case study for the proposed model. There are 25 main load points, geographically distributed all over the country and connected together by 132KV and 400KV national grid lines. Each load point experiences both different weather conditions and distinct variations of power demand. The implemented algorithm, shown in Fig.3.9, scans over all the load points on an hourly basis. At each load point the related data describing the system fuzzy variables are manipulated by calling the associated fuzzy variable function in such a way as to compose all the possible rules. A weighted output fuzzy label is then searched to satisfy an activated rule. As was explained in the previous section, different rules may be activated at the same time, and the combination of their outputs is then defuzzified to compute the predicted area power demand at that hour. Finally, the numerical value of the computed area power demand is added successively to the total power demand. Implementation of the algorithm, as depicted in Fig.3.9, is based on the following procedure: 51

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1. Decide on whether the computation is to be carried out for an ordinary day or a special day. 2. Begin computation at 1 o'clock with one hour increment based on a 24 hour system. 3. Scan over the 25 load points for each associated hour. 4. Call the time function which classifies day time into an associated fuzzy rep resentation(s) and determines its degree of membership between (0,1) to that representation( s). 5. Call the temperature function which classifies the weather dry bulb tempera ture at that hour and in that region into an associated fuzzy representation( s) and determines its degree of membership between (0,1) to that representa tion(s). 6. Call the climate function which classifies the degree of cloudiness at that hour and in that region into an associated fuzzy representation( s) and determines its degree of membership between (0,1) to that representation(s). 7. Call the location function which classifies the region's location into an associ ated fuzzy representation( s) and determines its degree of membership between (0,1) to that representation(s). 8. Call the population function which classifies population of the region into an associated fuzzy representation(s) and determines its degree of membership between (0,1) to that representation(s). 9. Call the industrialization function which classifies the region's industrializa tion into an associated fuzzy representation(s) and determines its degree of 52

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membership between (0,1) to that representation. 10. In the rule function, the input fuzzy representations activate at least one rule which, in turn, produces a weighted output fuzzy representation. 11. Finally, the weighted outputs are then defuzzified to produce a numerical value for the predicted area power demand using the defuzzification function. The defuzzification process is done in the same way as it was explained in the previous section. 12. The procedure above is repeated for each load point and at each hour. Functions in the items 4 through 9 convert the input values into fuzzy repre sentations with a degree of membership in the same way as they are repres-ented in the Figures 2 through 7. An example of power prediction Consider the forecasting for the Bayader load point for an ordinary day (April 11, 1993) and under the following conditions: 1. Degree of cloudiness is five. 2. Dry bulb temperature of 7.8C0 3. Time of the day is 9 o'clock. Solution Following the procedure described in sections 3.3 and 3.4, The input data set (Bayader, 5, 7.80, 9) activates two rules: 53

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Type of Day? Loopi .. 1 to 24 Loop L 1 to 35 Call Time Function Call Function Call Climate Fundlon Call Location Function Call Population Function Call Industrialization Function Call Rule Function Call Defuzzificatlon Function End Figure 3.9: Flowchart of the algorithm. 54

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1. (M, C, C, PC, NP, NI; M) rule which is interpreted as: IF time Tis morning (M) AND location L is center (C) AND temperature K is cold (C) AND climate CL is partly cloudy (PC) AND population P is normally populated (NP) AND industrialization I is normally industrial (NI) THEN area power demand (PAPD) should be medium (M) The output PAPD is then set to high but to a degree of membership given by : min(JLM(9), J.tc( Bayader ), J.tc(7.8), JLPc(5), JlNP( Bayader ), JlNI( Bayader)) w1 = min(0.7, 1, 1, 1, 0.334, 1) = 0.334 2. (M, C, C, PC, HP, NI; H) rule which is interpreted as: IF time Tis morning (M) AND location L is center (C) AND temperature K is cold (C) AND climate CL is patly cloudy (PC) AND population P is highly populated (HP) AND industrialization I is normally industrial (NI) THEN area power demand (PAPD) should be high (H)" The output PAPD is then set to high but to a degree of membership given by : min(JLM(9), J.tc( Bayader ), J.tc(7.8), JLPc(5), JlHP( Bayader ), JlNI(Bayader)) w2 = min(0.7, 1, 1, 1, 0.667, 1) = 0.667 The above two rules describe the area power demand for the observed values of the system state variables. The resulting outputs are combined, and fuzzy centroid is computed. The training parameter K has been assigned a value of 80 MW for the Bayader load point. Fig.3.10 illustrates the computational procedure for the above example. In this case the fuzzy centriodal area power demand value equals 47.315 MW. 55

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--T . .. t.L...f .... llii111Hfili1J11f!'J'IH I I .. .. I . I iiiiiiii iii Hir .. ..J 1 ---.. l -_ ...... .. f. .. Lk: .. = .. =r ':"' ;:::- +---r--r-........--r-L-rl---+14 7Jt-r--5 Hll+--rl ---r--r---11 80 "" .... ".. '.I.O_Lj 0 ...... -.... .l.. ..... ..... : I .. -0 I U a AS U I U S I d Figure 3.10: An example of power prediction. 56

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3.5 Discussion and System Evaluation In the preceding sections we have suggested a fuzzy model for power demand predic tion based on the fuzzy variables time, temperature, climate, location, population, and industrialization. Some of these variables have more effect than others on the amount of power demand. For instance, the time of the day variable has a high effect, and this is due to the fact that there are fixed working hours in public offices and some other industrial areas. However, temperature and climate variables have also strong effects. This can be noticed from the example given in the previous section, where a change of 2 MW at the Bayader load point is due to a change of 14C in temperature .and 5 degrees in cloudiness. Location, population, and in dustrialization are also relevant variables, but they are fixed for each load point in the short-term load forecasting computations. Hence, these variables can be frozen for short-term power-demand predictions. This, in turn, will facilitate prediction processing concerning the time, temperature, and climate variables. This has been attempted in the model, where each load point was denoted with a fixed variable covering its location, population, and industrialization, and has shown encouraging results. The system data concerning actual temperatures, actual degree of cloudi ness, and actual power demand at each hour and at each load point have been used to set up the rules. Data cover April 1 to May 15, 1993. Data concerning power demand have been taken from the JSCC, and weather-related data have been taken from the Meteorological Department in Jordan. As an application example, the load curve has been estimated for Tuesday, April 6, 1993, during the 24 hours, and with the actual weather readings as inputs to the system. Figure 3.11 shows the actual load curve (solid line), and the predicted load curve (dotted line) given by the model. The two curves have been compared for the indices: average error (eav9), maximum error (emax), minimum error (emin), root 57

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mean square error ( erms), and standard deviation error ( estd) The above indices were evaluated for the data obtained, and their values are: eavg of 0.5219%, emax of 9.8799%, emin of 0.1115%, erms of 4.4338%, and finally an estd of 4.4977%. The above indices indicate that the system is qualified to replace the work of an operator. This an interesting result, as it opens the door to industrial automation in a fuzzy environment. It is worth mentioning that one should not view the actual values (sold line) in Fig. 3.11 as optimal values. On the contrary, the results (dotted line) are closer to optimal values. The fact that a computer has a greater ability to compare, remember, and correlate the conditions for different times, faster and more accurately than humans, reinforces this belief. 58

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650 r___ _,_ _____ r--------,------.-----.:-..;....__---, 600 550 500 / \. ,,/ \l 350 5 10 15 20 Time (l-Iours) Figure 3.11: The actual load diagram (solid line) and the predicted load diagram (dotted line) for Tuesday, April 6, 1993. O'l lt:l

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4 Conclusions and Future Directions This thesis presents a fuzzy approach in the qualitative modeling of system control and management. However, the recent developments in the area of fuzzy systems include the application of fuzzy logic controllers (FLC) in numerous industrial appli cations, [14]. They have been applied in warm water process, [12], traffic junction, [22], power system applications, [4, 16], [23, 33], fuzzy memory devices, [28], etc. In these applications, the designed FLC hardware system can operate at a speed of 10 mega fuzzy logic inferences per second (FLIPS). Fuzzy chips and fuzzy computers have been also designed[28]. Fuzzy chips, for instance, can perform approximately 250 000 FLIPS at 16-MHz clock, and fuzzy computers which were first built by OMRON Tateishi Electric Co.Ltd are capable of processing fuzzy information at a speed of approximately 10 Mega FLIPS. A fuzzy computer comprises of a fuzzy memory, a set of inference engines, a MAX block, a defuzzifier, and a control unit processing data in a fuzzy environment. Further improvement in this direction for faster processing and production of reliable components may keep this field in pace with other sciences. However, the rise in the complication of data and controls with a diverse phys ical nature has significantly hampered the formulation of both quantitative and qualitative models, and consequently, the development of design methodology. As a consequence, much attention in system design has to be devoted to analytical fuzzy methods to simplify models of deterministic and stochastic nature. On the other hand, the concept of fuzzy systems being applied in this work through an adaptive approach based on the use of fuzzy logic for predicting hourly 60

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power demand in the country of Jordan has shown promising results. An initial evaluation of the approach has reflected reasonable results that are compatible with the actual needs. Further evaluation may be gained through the implemen tation of sufficient teaching of the model. Indeed, the implemented algorithm has shown little divergence in realizing the system performance and computational time. Meanwhile, it can still provide the control center operator with indications of load fluctuations, due to any inevitable conditions. The total predicted power demand, to help decide on an optimum number of generators to be committed, has also been investigated in substantial research in Jordan through the application of an arti ficial neural network. Training a neural network would help in scheduling weekly, monthly and yearly maintenance for all the available generators. Finally, we conclude this thesis by providing the following suggestions for future studies: Further investigation to derive knowledge base and fuzzy rules from the skilled operators decision-making structure in an efficient and systematic methodology. Development and structuring of an adaptive fuzzy system capable of providing varying fuzzy regions. This, in turn, will improve system precision relying on narrowing or widening regions. Another important topic for forthcoming investigation is to provide a sound theoretical background means in the development of a theory of deterministic and stochastic fuzzy systems. In response, an attempt is needed to search the application and extension of fundamental notations of system dynamic theory such as state space representation, controllability, observability, stability, etc. to the conditions and problems on which fuzzy systems are applicable. 61

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