CÁC PHƯƠNG PHÁP DỰ BÁO PHỤ TẢI ĐIỆN TRONG HỆ THỐNG ĐIỆN (Electrical Load Forecasting)

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Library of Congress Cataloging-in-Publication Data
Soliman, S.A.
Electrical load forecasting : modeling and model construction / Soliman Abdel-hady Soliman (S.A. Soliman),
Ahmad M. Al-Kandari.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-12-381543-9 (alk. paper)
1. Electric power-plants–Load–Forecasting–Mathematics. 2. Electric power systems–Mathematical models.
3. Electric power consumption–Forecasting–Mathematics. I. Al-Kandari, Ahmad M. II. Title.
TK1005.S64 2010

333.793'213015195–dc22 2009048799
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
For information on all Butterworth–Heinemann publications
visit our Web site at www.elsevierdirect.com
Typeset by: diacriTech, India
Printed in the United States of America
1011121314 10987654321
To my parents, I was in need of them during my operation
To my wife, Laila, with love and respect
To my kids, Rasha, Shady, Samia, Hadeer,
and Ahmad, I love you all
To everyone who has the same liver problem,
please do not lose hope in God
(S. A. Soliman)
To my parents, who raised me
To my wife, Noureyah, with great love and respect
To my sons, Eng.Bader and Eng.Khalied,
for their encouragement
To my beloved family and friends
(A. M. Al-Kandari)
Acknowledgments
In the market and the community of electric power system engineers, there is a short-
age of books focusing on short-term electric load forecasting. Many papers have been
published in the literature, but no book is available that contains all these publica-
tions.Theideaofwritingthisbookcametomymindtwoorthreeyearsago,but
the time was too limited to write such a big book. In the spring of 2009, I was diag-
nosed with liver cance r, and I have had to treat it locally through chemical therapy
until a suitable donor is available and I can have a liver transplant. The president
of Misr University for Science and Technology, Professor Mohammad Rafat, and

my brothers, Professor Mostafa Kamel, vice president for academic affairs, and Pro-
fessor Kamal Al-Bedawy, dean of engineering, asked me to stay at home to eliminate
physical stress. As such, I had a lot of time to write such a book, especially because
there are many publications in the area of short-term l oa d fo rec astin g. I ndee d, my
appreciation goes to them and chancellor o f Misr University, Mr. Khalied Al-Tokhay.
My appreciation goes to my wife, Laila, who did not sleep, sitting beside me day
and night while I underw ent therapy. My appreciation also goes to my kids Rasha,
Shady, Samia, Hadeer, and Ahmad, who raced to be the first donor for their dad.
My appreciation goes to my brothers-in-law, Eng. Ahmad Nabil Mousa, Professor
Mahmoud Rashad, and Dr. Samy Mousa, who had a hard time because of my illness;
he never left me alone even though he was out of the city of Cairo. Furthermore, my
appreciation goes also to my sons-in-law, A hmad Abdel-Az im an d Mohammad
Abdel-Azim.
My deep appreciation goes to Dr. Helal Al-Hamadi of Kuwait University, who was
the coauthor with me for some materials we used in this book.
Many thanks also go to my friends and coll eagues among the fa culty of the
engineering department at Misr University for Science and Technology. To them,
Isay,“You did something unbelievable.” In addition, many thanks to m y friends
and colleagues among the faculty of the engineering department at Ain Shams
University for their moral support. Special thanks g o to my good friends Professor
Mahmoud Abdel-Hamid and Professor Ibrahim Helal, who forgot the misunderstand-
ing between us and came to visit me at home on the same day he heard that I was sick
and took me to his friend, Professor Mohammad Alwahash, who is a liver transplant
expert. Professor M. E. El-Hawary, of Dalhousie University, Nova Scotia Canada, my
special friend, I miss you MO; I did everything that makes you happy in Egypt and
Canada.
My deep appreciation goes to the team of liver transplantation and intensive care
units at the liver and kidney hospital of Al-Madi Military Medical Complex; Profes-
sor Kareem Bodjema, the French excellent expert in liver transplantation; Professor
Magdy Amin, the man with whom I felt secure when he visited me in my room with

his colleagues, who answered my calls any time during the day or night, and who
supported me and my fa mily morally; Professor Salah Aiaad, the man, in my first
meeting with him, whom I felt I knew for a long time; Professor Ali Albadry; Profes-
sor Mahmoud Negm, who has a beautiful smile; Professor Ehab Sabry, the man who
can easily read what’ s in my eyes; and Dr. Mohammad Hesaan, who reminds me of
when I was in my forties—everything should go ideally for him.
Last, but not least, my deep appreciation and respect go to Gene ral Samir Hamam ,
the manager of Al-Madi Military Medical Complex, for helping to make everything
go smoothly. To all, I say you did a good job in every position at the hospital. May
God keep you all healthy and wealthy and remember these good things you did for
me to the day after.
S.A. Soliman
It is a privilege to be a coauthor with as great a professor as Professor Soliman
Abdel-hady Soliman. I learned a lot from him. I thank him for giving me the oppor-
tunity to coauthor this book, which will cover a needed area in load forecasting. I do
thank Professor M.E. El-Hawary for teaching me and guiding me in the scope of the
material of this book. Also, my app reciat ion goes to Professor Yacoub Al-Re fae,
general director of The Public Authority for Applied Education and Training in
Kuwait, for his encouragements and notes.
A.M. Al-Kandari
The authors of this book would like to acknowledge the effort done by Ms. Sarah
Binns for reviewing this book many times and we appreciate her time. To her we
say, you did a good job for us, you were sincere and honest in every stage of this book.
xiv Acknowledgments
Introduction
Economic development, throughout the world, depends directly on the availability of
electric energy, especially because most industries depend almost entirely on its use.
The availability of a source of continuous, cheap, and reliable energy is of foremost
economic importance.
Electrical load forecasting is an importan t tool used to ensure that the energy sup-

plied by utilities meets the load plus the energy lost in the system. To this end, a staff
of trained personnel is needed to carry out this specialized function. Load forecasting
is always defined as basically the science or art of predicting the future load on a
givensystem,foraspecifiedperiodoftime ahead. These predicti ons may be just
for a fraction of an hour ahead for operation purposes, or as much as 20 years in to
the future for planning purposes.
Load forecasting can be categorized into three subject areas—namely,
1. Long-range forecasting, which is used to predict loads as distant as 50 years ahead so that
expansion planning can be facilitated.
2. Medium-range forecasting, which is used to predict weekly, monthly, and yearly peak loads
up to 10 years ahead so that efficient operational planning can be carried out.
3. Short-range forecasting, which is used to predict loads up to a week ahead so that daily run-
ning and dispatching costs can be minimized.
In the preceding three categories, an accurate load model is required to mathema-
tically represent the relationship between the load a nd influential variables such as
time, weather, economic factors, etc. The precise relationship between the load and
these variables is usually determined by their role in the load model. After the math-
ematical model is constructed, the model parameters are determined through the use
of estimation techniques.
Extrapolating the mathematical relationship to the required lead time ahead and
giving the corresponding values of influential variables to be available or predictable,
forecasts can be made. Because factors such as weather and economic indices are
increasingly difficult to predict accurately for longer lead times ahead, the greater
the lead time, the less accurate the prediction is likely to be.
The final accuracy of any forecast thus depends on the load model employed, the
accuracy of predicted variables, and the parameters assigned by the relevant estima-
tion technique. Because different methods of estimation will result in different values
of estimated parameters, it follows that the resulting forecasts will differ in prediction
accuracy.
Over the past 50 years, the parameter estimation algorithms used in load forecasting

have been limited to those based on the least error squares minimization criterion, even
though estimation theory indicates that algorithms based on the least absolute value cri-
teria are viable alternatives. Furthermore, the artificial neural network (ANN) had
showed success in estimating the load for the next hour. However, the ANN used
by a utility is not necessarily suitable for another utility and should be retrained to
be suitable for that utility.
It is well known that the electric load is a dynamic one and does not have a precise
value from one hour to another. In this book, fuzzy systems theory is implemented to
estimate the load model parameters, which are assumed to be fuzzy parameters having
a certain middle and spread. Different membership functions, for load parameters, are
used—namely, triangular membership and trapezoidal membership functions. The
problem of load forecasting in this book is restricted to short-term load forecasting
and is formulated as a lin ear estimation p roblem in the parameters to be estimated.
In this book , the parameters in the first part are assumed to be crisp parameters,
whereas in the rest of the book these parameters are assumed to be fuzzy parameters.
The objective is to minimize the spread of the available data points, taking into con-
sideration the type of membership of the fuzzy parameters, subject to satisfying con-
straints on each measurement point, to ensure that the original membership is
included in the estimated membership.
Outline of the Book
In this book, different techniques used in the past two decades are implemented to
estimate the load model parameters, including fuzzy parameters with certain middle
and certain spread. The book contains nine ch apters:
Chapter 1, “Mathematical Background and State of the Art.” This chapter
introduces mathematical background to help the reader understand the problems for-
mulated in this book. In this chapter, the reader will study matrices and thei r applica-
tions in estimation theory and see that the use of matrix notation simplifies complex
mathematical expressions. The simplifying matrix notation may not reduce the
amount of work required to solv e mathematical equations, but it usually makes the
equations much easier to handle and manipulate. This chapter explains the vectors

and the formulation of quadratic forms, and, as we shall see, that most objective func-
tions to be minimized (least errors square criteria) are quadratic in nature. This chapter
also explains some optimization techniques and introduces the concept of a state
space model, which is commonly used in dynamic state estimation. The reader will
also review different techniques that, developed for the short term, give the state of
the art of the various algorithms used during the past decades for short-term load fore-
casting. A brief discussion for each algorithm is presented in this chapter. Advantages
and disadvantages of each algorithm are discussed. Reviewing the most recent pub-
lications in the area of short-term load forecasting indicates that most of the available
algorithms treat the parameters of the proposed load model as crisp parameters, which
is not the case in reality.
xvi Introduction
Chapter 2, “Static State Estimation.” This chapter presents the theory involved
in different approaches that use parameter estimation algorithms. In the first part
of the chapter, the crisp parameter estimation algorithms are presented; they include
the least error squares (LES) algorithm and the least absolute value (LAV) algorithm.
The second part of the chapter presents an introduction to fuzzy set theory and sys-
tems, followed by a discussion of fuzzy linear regression algorithms. Different cases
for the fuzzy parameters are discussed in this part. The first case is for the fuzzy linear
regression of the linear models having fuzzy parameters with nonfuzzy outputs, the
second case is for the linear regression of fuzzy parameters with fuzzy output, and
the third case is for fuzzy parameters formulated with fuzzy output of left and right
type (LR-type).
Chapter 3, “Load Modeling for Short-Term Forecasting.” This chapter pro-
poses different load models used in short-term load forecasting for 24 hours.
•
Three models are proposed in this chapter—namely, models A, B, and C. Model A is a mul-
tiple linear regression model of the temperature deviation, base load, and either wind-chill
factor for winter load or temperature humidity factor for summer load. The parameters of
load A are assumed t o be crisp parameters in this chapter. The term crisp parameters

mean clearly defined parameter values without ambiguity.
•
Load model B is a harmonic decomposition model that expresses the load at any instant, t,
as a harmonic series. In this model, the weekly cycle is accounted for through use of a daily
load model, the parameters of which are estimated seven times weekly. Again, the param-
eters of this model are assumed to be crisp.
•
Load model C is a hybrid load model that expresses the load as the sum of a time-varying
base load and a weather-dependent component. This model is developed with the aim of
eliminating the disadvantages of the other two models by combining their modeling
approaches. After finding the parameter values, one uses them to determine the electric
load from which these parameter values are extracted, and this value is called the estimated
load. Then the parameter values are used to predict the electric load for a randomly chosen
day in the future, and it is called the predicted load for that chosen day.
Chapter 4, “Fuzzy Regression Systems and Fuzzy Linear Models.” The objec-
tive of this chapter is to introduce principal concepts and mathematical notions of
fuzzy set theory, a theory of classes of objects with non sharp boundaries.
•
We first review fuzzy sets as a generalization of classical crisp sets by extending the range
of the membership function (or characteristic function) from [0, 1] to all real numbers in the
interval [0, 1].
•
A number of notions of fuzzy sets, such as representation support, α-cuts, conve xity, and
fuzzy numbers, are then introduced. The resolution principle, which can be used to expand
a fuzzy set in terms of its α-cuts, is discussed.
•
This chapter introduces fuzzy mathematical programming and fuzzy multiple-objective deci-
sion making. We first introduce the required knowledge of fuzzy set theory and fuzzy mathe-
matics in this chapter.
•

Fuzzy line ar reg ress ion also is introduced in this chapter; the first part is to estimate the
fuzzy regression coefficients when the set of measurements av ailable is crisp, whereas in
the second part the fuzzy regression coeffi cie nts are estimated when the available set of
measurements is a fuzzy set with a certain middle and spread.
•
Some simple examples for fuzzy linear regression are introduced in this chapter.
Introduction xvii
•
The models proposed in Chapter 3 for crisp par ameters are us ed in this chapter. Fuzzy
model A employs a multiple fuzzy linear regression model. The membership function for
the model parameters is developed, whe re triangular membership functions are assumed
for each parameter of the load model. Two constraints are imposed on each load measure-
ment to ensure that the original membership is included in the estimated membership.
•
Fuzzy model B, which is a harmonic model, also is proposed in this chapter. Th is model
involves fuzzy parameters having a certain median and certain spread.
•
Finally, a hybrid fuzzy model C, which is the combination of the multiple linear regression
model A and harmonic model B, is presented in this chapter.
Chapter 5, “Dynamic State Estimation.” The objective of this chapter is to study
the dynamic state estimation problem and its applications to electric power system
analysis, especially short-term load forecasting. Furthermore, the different approaches
used to solve this dynam ic estimation problem are also discussed in this chapter. After
reading this chapter, the reader will be familiar with
The five fundamental components of an estimation problem:
•
The variables to be estimated.
•
The measurements or observations available.
•

The mathematical model describing how the measurements are related to the variable of
interest.
•
The mathematical model of the uncertainties present.
•
The performance evaluation criterion to judge which estimation algorithms are “best.”
Formulation of the dynamic state estimation problem:
•
Kalman filtering algorithm as a recursive filter used to solve a problem.
•
Weighted least absolute value filter.
•
Different problems that face Kalman filtering and weighted least absolute value filtering
algorithms.
Chapter 6, “Load Forecasting Results Using Static State Estimation.” The
objective of this chapter is as follows:
In Chapter 3, the models are derived on the basis that the load powers are crisp in nature; the
data available from a big company in Canada are used to forecast the load power in the crisp
case.
•
In this chapter, the results obtained for the crisp load power data for the different load models
developed in Chapter 3 are shown.
•
A comparison is performed between the two static LES and LAV estimation techniques.
•
The parameters estimated are used to predict a load using both techniques, where we com-
pare between them for summer and winter.
Chapter 7, “Loa d Forecasting Results Using Fuzzy Systems.” Chapter 6 dis-
cusses the short-term load-forecasting problem, and the LES and LAV parameter esti-
mation algorith ms are used to estimate the load model parameters. The error in the

estimates is ca lculated for both tech niques. The three mode ls, prop osed earlie r in
Chapter 3, are used in that chapter to present the load in different days for different
seasons. In this chapter, the fuzzy load models developed in Chapter 5 are tested. The
fuzzy parameters of these models are estimated using the past history data for summer
weekdays and weekend days as well as for winter weekdays and weekend days. The n
these models are used to predict the fuzzy load power for 24 hours ahead, in both
xviii Introduction
summer and winter seasons. The results are given in the form of tables and figures for
the estimated and predicted loads.
Chapter 8, “Dynamic Electric Load Forecasting.” The main objectives of this
chapter are as follows:
•
A one-year long-term electric power load-forecasting problem is introduced as a first step
for short-term load forecasting.
•
A dynamic algorithm, the Kalman filtering algorithm, is suitable to forecast daily load pro-
files with a lead-time from several weeks to a few years.
•
The algorithm is based mainly on multiple simple linear regression models used to capture
the shape of the load over a certain period of time (one year) in a two-dimensional layout
(24 hours  52 weeks).
•
The regression models are recursively used to project the 2D load shape for the next period
of time (next year). Load-demand annual growth is estimated and incorporated into the
Kalman filtering algorithm to improve the load-forecast accuracy obtained so far from
the regression models.
Chapter 9, “ Electric Load Modeling for Long-Term Forecasting.” The objec-
tives of this chapter are as follows:
•
This chapter provides a comparative study between two static estimation a lgorithms—

namely, the least error squares (LES) and least absolute value (LAV) algorithms—for esti-
mating the parameters of different load models for peak-load forecasting necessary for long-
term power system planning.
•
The proposed algorithms use the past history data for the load and the influence factors ,
such as gross domestic product (GDP), population, GDP per capita, system losses, load
factor, etc.
•
The problem turns out to be a linear estimation problem in the load parameters. Different
models are developed and discussed in the text.
Introduction xix
1 Mathematical Background and
State of the Art
1.1 Objectives
The objectives of this chapter are
•
Introducing a mathematical background to help the reader understand the problems
formulated in this book.
•
Studying matrices and their applications in estimation theory and showing that the use of
matrix notation simplifies complex mathematical expressions. The simplifying matrix nota-
tion may not reduce the amount of work required to solve mathematical equations, but it
usually makes the equations much easier to handle and manipulate.
•
Explaining the vectors and the formulation of quadratic forms and, as we shall see, that most
objective functions to be minimized (least error squares criteria) are quadratic in nature.
•
Explaining some optimization techniques.
•
Introducing the concept of a state space model, which is commonly used in dynamic state

estimation.
•
Reviewing the literature to introduce different techniques developed for short-term load
forecasting.
•
Explaining the merit of each technique used in the estimation of load forecasting and sui-
table places for implementation.
•
In this chapter, we also try to compare different techniques used in electric load forecasting.
1.2 Matrices and Vectors
A matrix is an array of elements [1]. The elements of a matrix may be real or
complex or functions of time. A matrix that has n rows and m columns is called
an n  m (n by m) matrix. If n ¼m, the matrix is referred to as a square matrix. If
A is an n  m matrix, then it can be written as
A ¼
a
11
a
12
a
1m
a
21
a
22
a
2m
.
.
.

.
.
.

.
.
.
a
n1
a
n2
a
nm
2
6
6
6
4
3
7
7
7
5
ð1:1Þ
In shorthand,
A ¼ a
ij
ÂÃ
nÂm
i ¼ 1, , n

j ¼ 1, , m
ð1:2Þ
Copyright © 2010 by Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-381543-9.00001-4
Note that the determinant is also an array of elements with n rows and n columns
(always squar e) and has a v alue. The matrix does not have a value but ha s a
determinant.
Column Matrix: This type of matrix has only one column and more than one row;
that is, an m  1 matrix, m > 1. Quite often, a column matrix is referred to as a col-
umn vector or simply an m-vector. For example, the column vector X is written as
X ¼
x
1
x
2
.
.
.
x
m
2
6
6
6
4
3
7
7
7
5

¼ col.ðx
1
, x
2
, , x
n
Þð1:3Þ
Row Matrix: This type of matrix has only one row and more than one column;
that is, an 1 Â n matrix, n > 1. Quite often, we call it a row vector. For example,
the row vector Y is given by
Y ¼½y
1
, y
2
, , y
n
¼row.ðy
1
, y
2
, , y
n
Þð1:4Þ
Diagonal Matrix: This is a square matrix with all elements equal to zero except
for the diagonal element; that is, a
ij
¼0 for all i 6¼ j. For example,
A ¼
a
11

00
0 a
22
0
00a
33
2
4
3
5
ð1:5aÞ
or, in terms of a shortcut,
A ¼ diag.ða
11
, a
22
, a
33
Þð1:5bÞ
Symmetric Matrix: This type of matrix is a square matrix that satisfies the
relation
a
ij
¼ a
ji
for all i, j
The following example indicates this matrix:
A ¼
135
304

542
2
4
3
5
In terms of a shortcut:
A
T
¼ A where T ¼ transpose of
Transpose of a Matrix: The transpose of a matrix is defined as a matrix obtained
by interchanging the corresponding rows and columns in A.IfA is an n  m matrix,
which is represented by
A ¼ a
i, j
ÂÃ
nÂm
2 Electrical Load Forecasting: Modeling and Model Construction
then the transpose of A, denoted by A
T
, is given by
A
T
¼ a
j, i
ÂÃ
mÂn
ð1:6Þ
Note that the order of A is n  m, while the order of A
T
is m  n. For example, if

A ¼
432 1
560À2
614 3
2
4
3
5
3Â4
then
A ¼
456
361
204
1 À23
2
6
6
4
3
7
7
5
4Â3
The following are some operations using the transpose of a matrix:
1: A
T
ÂÃ
T
¼ A ð1:7Þ

2: kA½
T
¼ kA
T
, k ¼ scalar number ð1:8Þ
3: A þ BðÞ
T
¼ A
T
þ B
T
ð1:9Þ
4: ABðÞ
T
¼ B
T
ÂA
T
ð1:10Þ
1.3 Matrix Algebra
1.3.1 Addition of Matrices
If A is an n  m matrix, and B is also an n  m matrix, then the sum of the two
matrices is given by
C ¼ A þB ð1:11aÞ
where the elements of the matrix C are given by
c
ij
¼ a
ij
þ b

ij
for all i, j ð1:11bÞ
For example, if
A ¼
230
451
!
and
B ¼
À124
0 À23
!
Mathematical Background and State of the Art 3
then
C ¼
2 À1ðÞ3 þ2ðÞ0 þ4ðÞ
4 þ0ðÞ5 À2ðÞ1 þ 3ðÞ
"#
C ¼
154
434
!
1.3.2 Matrix Subtraction (Difference)
The subtraction (difference) of matrices is similar to the addition of matrices if all the
signs of the second matrix are changed from positive to negative and from negative to
positive; that is,
C
nÂmðÞ
¼ A
nÂmðÞ

ÀB
nÂmðÞ
ð1:12Þ
where
½c
ij

nÂm
¼½a
ij

nÂm
þ½Àb
ij

nÂm
ð1:13aÞ
½c
ij

nÂm
¼½a
ij

nÂm
À½b
ij

nÂm
ð1:13bÞ

or
c
ij
¼ a
ij
À b
ij
for all i and j ð1:13cÞ
The following rules hold true for addition and subtraction:
1: A þ BðÞþC ¼ A þ B þ CðÞ associate lawðÞ ð1:14Þ
2: A þB þC ¼ B þC þ A ¼ C þ A þ B commutative lawðÞð1:15Þ
1.3.3 Matrix Multiplication
Let A be an n  m matrix and B be an m  p matrix. Then the product of A and B is
defined as
C
nÂpðÞ
¼ A
nÂmðÞ
ÂB
mÂpðÞ
ð1:16Þ
Note that the number of columns in the first matrix, m, must be equal to the number of
rows in the second matrix to carry out the multiplication.
The elements of the matrix C are given by
c
ij
¼
X
r
k¼1

a
ik
b
kj
i ¼ 1, , n
j ¼ 1, , p
ð1:17Þ
If, for example, the matrix A is given by
A ¼
23
14
À13
2
4
3
5
3Â2
4 Electrical Load Forecasting: Modeling and Model Construction
and
B ¼
2 À1
À13
!
ð2Â2Þ
then
C
ð3Â2Þ
¼ A
ð3Â2Þ
ÂB

ð2Â2Þ
C
ð3Â2Þ
¼
23
14
À13
2
4
3
5
ð3Â2Þ
2 À1
À13
"#
ð2Â2Þ
¼
ð2 Â2Þþð3ÞðÀ1Þð2 ÂÀ1 þ 3 Â3Þ
ð1 Â2 À1 Â4Þð1 ÂÀ1 þ4 Â3Þ
ðÀ1 Â2 À1 Â3ÞðÀ1 ÂÀ1 þ 3 Â3Þ
2
6
4
3
7
5
C
ð3Â2Þ
¼
17

À211
À510
2
4
3
5
If the matrix A is given by
A ¼
01
À3 À4
!
ð2Â2Þ
and the vector matrix X(t) is given by
XðtÞ¼
x
1
ðtÞ
x
2
ðtÞ
!
ð2Â1Þ
then
C ¼ AXðtÞ¼
01
À3 À4
!
ð2Â2Þ
x
1

ðtÞ
x
2
ðtÞ
!
ð2Â1Þ
¼
x
2
ðtÞ
À3x
1
ðtÞÀ4x
2
ðtÞ
!
ð2Â1Þ
It is possible in some cases to obtain the two products AB and BA. This could hap-
pen if A is an r  n matrix, and B is an n  r matrix. In this case, AB is an r  r
matrix, whereas BA is an n  n matrix. Obviously, AB 6¼ BA, and we say that A
and B do not commute, but if AB ¼BA, we say that A and B commute. If A and B
are of the order n  n, then AB, BA will be of the order n  n. For example,
A ¼
132
567
!
ð2Â3Þ
B ¼
23
15

42
2
4
3
5
ð3Â2Þ
Mathematical Background and State of the Art 5
Then AB will be
AB
2Â2
¼
132
567
!
23
15
42
2
4
3
5
¼
13 22
44 59
!
but BA will be
BA
3Â3
¼
23

15
42
2
4
3
5
132
567
!
¼
17 24 25
26 33 37
14 24 22
2
4
3
5
Although the comm utative law does not hold in general for mat rix multiplication,
the associative and distributive laws still apply. For the distributive law, we state that
ABþ CðÞ¼AB þAC ð1:18Þ
provided that the product is conformable.
For the associative law,
ABðÞC ¼ ABCðÞ ð1:19Þ
if the product is conformable.
1.3.4 Inverse of a Matrix (Matrix Division)
If A is a square matrix of which the determinant exists, and if B is another square
matrix such that
AB ¼ BA ¼ I
then B is called the inverse of A, denoted by A
À1

. Thus,
A ÂA
À1
¼ A
À1
A ¼ I
For matrices with low dimension, a straightforward procedure for matrix inversion
is given by
A
À1
¼
adjðAÞ
jAj
ð1:20Þ
where adj(A) is the adjoint of A, and it is the transpose of the matrix of cofactors of A
with elements
A
ij
¼À1ðÞ
iþj
M
ij
ð1:21Þ
The minors M
ij
are determinants of the (n À 1) Â ( n À 1) matrices obtained by delet-
ing the ith row and jth column from A. The following example explains the steps
involved.
6 Electrical Load Forecasting: Modeling and Model Construction
Example 1.1

Find the inverse of A, wher e
A ¼
1 À23
456
789
2
4
3
5
Calculate the determinant of A as
jAj¼1ð45 À48Þþ2ð36 À42Þþ3ð32 À35Þ
¼À3À12À9 ¼À24
The matrix of cofactors is obtained as
CofðAÞ¼
À36À3
42 À12 À22
À27 6 13
2
4
3
5
Thus, transposing Cof(A), we obtain adj(A)as
adjðAÞ¼
À342À27
6 À12 6
À3 À22 13
2
4
3
5

The inverse of A is obtained as
A
À1
¼
adj AðÞ
jAj
¼
À1
24
À342À27
6 À12 À6
À3 À22 13
2
4
3
5
¼
0:125 À1:75 1:125
À0:25 0:50 À0:25
0:125 0:91667 À0:54166
2
4
3
5
Some properties of the matrix inverse are
AA
À1
¼ A
À1
A ¼ I ð1:22Þ

A
À1
ÀÁ
À1
¼ A ð1:23Þ
If A and B are square matrices and are nonsingular, then
ABðÞ
À1
¼ B
À1
A
À1
ð1:24Þ
Mathematical Background and State of the Art 7
1.4 Rank of a Matrix
The rank of a matrix A is the maximum number of linearly independent columns of A,
or it is the order of the largest nonsingular matrix contained in A. For example, the
matrix
A ¼
01
00
!
has a rank ¼ 1
and
A ¼
0514
3532
!
has a rank ¼ 2
while

A ¼
392
130
261
2
4
3
5
has a rank ¼ 2
but
B ¼
300
120
001
2
4
3
5
has a rank ¼ 3
The following properties of the rank are useful in the determination of the rank of a
matrix. Given an nÂm matrix A,
1: Rank of A ¼ Rank of A
T
ð1:25Þ
2: Rank of A ¼ Rank of AA
T
ð1:26Þ
3: Rank of A ¼ Rank of A
T
A ð1:27Þ

The rank of a matrix is of great importance in electric load forecasting using the static
estimation algorithms.
1.5 Singular Matrix
If A is a square matrix, and if the determinant of A equals zero (i.e., jAj¼0), then the
matrix A is called a singular matrix. On the other hand, if jAj exists, the matrix A is
called a nonsingular matrix. For example, the matrix A is
A ¼
21
42
!
is a singular matrix because jAj¼4À4 ¼ 0.
If the matrix is a singular matrix, the inverse of this matrix does not exist. Even though
the matrix A is singular, it has a rank. The preceding matrix A has a rank of one.
8 Electrical Load Forecasting: Modeling and Model Construction
1.6 Characteristic Vectors of a Matrix
Given a matrix A with characteristic values or eigenvalues λ
1
, , λ
n
, the eigenvectors
of the matrix satisfy the relations
AU
i
¼ λ
i
U
i
ð1:28Þ
The U
i

s are called eigenvectors.
The matrix U of the eigenve ctors is nonsingular if the eigenve ctors ar e linearly
independent:
U ¼ U
1
U
2
U
n
½ ð1:29Þ
Let
U
À1
¼ V ¼
V
T
1
V
T
2
.
.
.
V
T
n
2
6
6
6

6
6
4
3
7
7
7
7
7
5
ð1:30Þ
because
U
À1
U ¼ I
Thus, in expanded form, we have
V
T
1
V
T
2
.
.
.
V
T
n
2
6

6
6
6
6
4
3
7
7
7
7
7
5
U
1
U
2
U
n
½¼
V
T
1
U
1
V
T
1
U
2
V

T
1
U
n
V
T
2
U
1
V
T
2
U
2
V
T
2
U
n
.
.
.
V
T
n
U
1
V
T
n

U
2
V
T
n
U
n
2
6
6
6
6
6
4
3
7
7
7
7
7
5
ð1:31Þ
Therefore, we can conclude that component-wise
V
T
i
U
i
¼ 1; j ¼ 1, , n ð1:32Þ
V

T
i
U
j
¼ 0; j ¼ 1, , n ð1:33Þ
1.7 Diagonalization
Consider now the matrix product
e
A ¼ U
À1
AU ð1:34Þ
Using equation (1.30), we have
e
A ¼ VAU ð1:35Þ
Mathematical Background and State of the Art 9
In terms of eigenvectors, we have
e
A ¼ VAU
1
, AU
2
, , AU
n
½ ð1:36Þ
Using equation (1.28), we obtain
e
A ¼ V λU
1
, λU
2

, , λU
n
½ ð1:37Þ
Substituting for V in partitioned form, we get
e
A ¼
V
T
1
V
T
2
.
.
.
V
T
n
2
6
6
6
6
6
4
3
7
7
7
7

7
5
½λ
1
U
1
, λ
2
U
2
, , λ
n
U
n
ð1:38Þ
Performing the multiplication, we obtain
e
A ¼
λ
1
V
T
1
U
1
λ
2
V
T
1

U
2
λ
n
V
T
1
U
n
λ
1
V
T
2
U
1
λ
2
V
T
2
U
2
λ
n
V
T
2
U
n

.
.
.
λ
1
V
T
n
U
1
λ
2
V
T
n
U
2
λ
n
V
T
n
U
n
2
6
6
6
6
6

6
4
3
7
7
7
7
7
7
5
ð1:39Þ
Substituting equations (1.32) and (1.33) into equation (1.39), we obtain
e
A ¼
λ
1
000
0 λ
2
00
.
.
.
.
.
.
.
.
.
.

.
.
000λ
n
2
6
6
6
4
3
7
7
7
5
ð1:40Þ
Thus, the matrix
e
A is a diagonal matrix of which the elements are the eigenvalues
of A:
e
A ¼ U
À1
AU ð1:41Þ
The expression in terms of the transformation T is
e
A ¼ TAT
À1
ð1:42Þ
where
T

À1
¼ U ð1:43Þ
The following example illustrates the preceding steps. Consider the matrix
A ¼
À21
1 À2
!
10 Electrical Load Forecasting: Modeling and Model Construction
The eigenvalues are obtained as
λI ÀA ¼
λ þ2 À1
À1 λ þ2
!
Thus, the characteristic polynomial is given by
PðλÞ¼jλI ÀAj¼ðλ þ 2Þ
2
À1
¼ λ
2
þ 4λ þ3
¼ðλ þ 1Þðλ þ3Þ
Thus, the eigenvalues are given by
λ
1
¼À1
λ
2
¼À3
Next, we compute the eigenvectors as follows:
AU

1
¼ λ
1
U
1
for λ
1
¼À1
Thus,
À21
1 À2
!
U
11
U
21
!
¼
ÀU
11
ÀU
21
!
which gives
U
11
¼ U
21
or assume that U
11

¼ 1, then U
21
¼ 1ðÞ
Therefore, the first eigenvector is given by
U
1
¼
1
1
!
Following the same steps, we obtain the second eigenvector as
U
2
¼
À1
1
!
and hence the matrix U is given by
T
À1
¼ U
1
U
2
½¼
1 À1
11
!
Thus,
T ¼

0:5 þ0:5
À0:50:5
!
Mathematical Background and State of the Art 11
Therefore, the transformed matrix A is given by
e
A ¼ TAT
À1
¼
0:50:5
À0:50:5
!
À21
1 À2
!
1 À1
11
!
¼
À10
À2 À1
!
1.8 Partitioned Matrices
Partitioning is useful when applied to large matrices because manipulations can be
carried out on the smaller blocks. More importan tly, when one is multiplying parti-
tioned matrices, the basic rule can be applied to the blocks as though they were single
elements.
For example, the following 3Â4 matrix is partitioned into four blocks:
A ¼
23

.
.
.
15
0 À2 .
.
.
20
ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ
10
.
.
.
À410
2
6
6
6
6
4
3
7
7
7
7
5
¼
BC
DE
!

where B, C, D, and E are the arrays indicated by dashed lines. The matrix entries of
such a partitioned matrix are called submatrices. The main matrix is sometimes
referred to as the supermatrix.
If A is square, and its only nonzero elements can be partitioned as principal sub-
matrices, then it is called a block diagonal. A convenient notation that generalizes is
to write A as
A ¼ diag. A
1
, A
2
, , A
n
ðÞ
where th e s u bm at ri ces A
1
, A
2
, , A
n
are square matrices, not necessarily of equal
dimension, which appear on the major diagonal. The inverse A
À1
of A ¼ diag.
A
1
, , A
n
ðÞis
A
À1

¼ diag. A
À1
1
, A
À1
2
, , A
À1
n
ÀÁ
Another advantage of partitioned matrices is that if the partitioned matrix A is mul-
tiplied by the matrix X, which is given by
X ¼
X
1
X
2
!
then
AX ¼
BC
DE
!
X
1
X
2
!
¼
BX

1
þ CX
2
DX
1
þ EX
2
!
12 Electrical Load Forecasting: Modeling and Model Construction
The only restriction is that the blocks must be conformable for multiplication, so that
all the products BX
1
, CX
2
, , etc., exist. This requires that in a product AX the num-
ber of columns in each block of A must equal the number of rows in the correspond-
ing block of X.
1.9 Partitioned Matrix Inversion
It is difficult to obtain the inverse of matrices of high dimension by using the classical
method. In this case, the partitioned form is useful. Suppose that F is a matrix in par-
titioned form as
F ¼
A
nÂn
B
nÂn
C
mÂm
D
mÂm

!
and
F
À1
¼
W
nÂn
X
nÂn
Y
mÂm
Z
mÂm
!
By definition of the matrix inverse,
FF
À1
¼ I
so
AB
CD
!
WX
YZ
!
¼
I
n
0
0 I

m
!
and applying the rules of partitioned multiplication produces
AW þ BY ¼ I
n
AX þBZ ¼ 0
CW þ DY ¼ 0
CX þDZ ¼ I
m
By solving the preceding equations, we can obtain
W ¼ A
À1
ÀA
À1
BY
Y ¼ÀD ÀCA
À1
BðÞ
À1
CA
À1
Z ¼ D ÀCA
À1
BðÞ
À1
X ¼ÀA
À1
BDÀCA
À1
BðÞ

À1
provided that the matrix A is nonsingular.
Mathematical Background and State of the Art 13
Example 1.2
Consider the matrix F given by
F ¼
23 4À1
12 5À2
3240
01À23
2
6
6
4
3
7
7
5
F can be partitioned as shown.
Thus, we can find
A ¼
23
12
!
, A
À1
¼
2 À3
À12
!

B ¼
4 À1
5 À2
!
, C ¼
32
01
!
and D ¼
40
À23
!
Then the matrices Y, W, Z, and X are given by
Y ¼À
40
À23
!
À
32
01
!
2 À3
À12
!
4 À1
5 À2
! !
À1
Â
32

01
!
2 À3
À12
!
¼
À1
30
18 jÀ18
19 jÀ14
!
W ¼
2 À3
À12
!
þ
1
30
2 À3
À12
!
4 À1
5 À2
!
18 À18
19 À14
!
¼
À1
30

10 À20
21 À6
!
Z ¼
40
À23
!
À
32
01
!
2 À3
À12
!
4 À1
5 À2
! !
À1
¼
13 À6
À86
!
À1
¼
1
30
66
813
!
X ¼

À1
30
2 À3
À12
!
4 À1
5 À2
!
66
813
!
¼
1
30
10 À10
À12 3
!
14 Electrical Load Forecasting: Modeling and Model Construction
Therefore,
F
À1
¼
0:333 À0:6667 0:333 À0:333
0:7 À0:2 À0:40:1
À0:60:60:20:2
À0:633 0:466 0:266 0:433
2
6
6
4

3
7
7
5
1.10 Quadratic Forms
An algebraic expression of the form
f
x,yðÞ
¼ ax
2
þ bxy þ cy
2
is said to be a quadratic form. If we let
X ¼
x
y
!
then we obtain
f
x, yðÞ
¼½xy
a
b
2
b
2
c
2
6
6

4
3
7
7
5
x
y
!
or
f
XðÞ
¼ X
T
AX
The preceding equation, as mentioned previously, is in quadratic form. The matrix A
in this form is a symmetrical matrix.
A more general form for the quadratic function can be written in a matrix form as
F
XðÞ
¼ X
T
AX þB
T
X þ C
where X is an n Â1vector,A is an n Ân matrix, B is an n Â1 vector, and C is 1 Â1vector.
Example 1.3
Given the function
f
x,yðÞ
¼ 2x

2
þ 4xy Ày
2
¼ 0
we need to write this function in a quadratic form.
Mathematical Background and State of the Art 15