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

NỘI DUNG GỒM 9 CHƯƠNG:Chapter 1. Mathematical Background and State of the Art. Chapter 2. Static State Estimation. Chapter 3. Load Modeling for ShortTerm Forecasting. Chapter 4. Fuzzy Regression Systems and Fuzzy Linear Models. Chapter 5. Dynamic State Estimation. Chapter 6. Load Forecasting Results Using Static State Estimation. Chapter 7. Load Forecasting Results Using Fuzzy Systems. Chapter 8. Dynamic Electric Load Forecasting. Chapter 9. Electric Load Modeling for LongTerm 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

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A catalogue record for this book is available from the British Library.

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Printed in the United States of America

10 11 12 13 14 10 9 8 7 6 5 4 3 2 1

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)

In the market and the community of electric power system engineers, there is a age of books focusing on short-term electric load forecasting Many papers have beenpublished in the literature, but no book is available that contains all these publica-tions The idea of writing this book came to my mind two or three years ago, butthe time was too limited to write such a big book In the spring of 2009, I was diag-nosed with liver cancer, and I have had to treat it locally through chemical therapyuntil a suitable donor is available and I can have a liver transplant The president

short-of Misr University for Science and Technology, Prshort-ofessor Mohammad Rafat, and

my brothers, Professor Mostafa Kamel, vice president for academic affairs, and fessor Kamal Al-Bedawy, dean of engineering, asked me to stay at home to eliminatephysical stress As such, I had a lot of time to write such a book, especially becausethere are many publications in the area of short-term load forecasting Indeed, myappreciation goes to them and chancellor of Misr University, Mr Khalied Al-Tokhay

Pro-My appreciation goes to my wife, Laila, who did not sleep, sitting beside me dayand night while I underwent 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, ProfessorMahmoud 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, myappreciation goes also to my sons-in-law, Ahmad Abdel-Azim and MohammadAbdel-Azim

My deep appreciation goes to Dr Helal Al-Hamadi of Kuwait University, who wasthe coauthor with me for some materials we used in this book

Many thanks also go to my friends and colleagues among the faculty of theengineering department at Misr University for Science and Technology To them,

I say, “You did something unbelievable.” In addition, many thanks to my friendsand colleagues among the faculty of the engineering department at Ain ShamsUniversity for their moral support Special thanks go to my good friends ProfessorMahmoud 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 sickand took me to his friend, Professor Mohammad Alwahash, who is a liver transplantexpert Professor M E El-Hawary, of Dalhousie University, Nova Scotia Canada, myspecial friend, I miss you MO; I did everything that makes you happy in Egypt andCanada

My deep appreciation goes to the team of liver transplantation and intensive careunits 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 withhis colleagues, who answered my calls any time during the day or night, and whosupported me and my family morally; Professor Salah Aiaad, the man, in my firstmeeting 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 whocan easily read what’s in my eyes; and Dr Mohammad Hesaan, who reminds me ofwhen I was in my forties—everything should go ideally for him.

Last, but not least, my deep appreciation and respect go to General 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 MayGod 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 SolimanAbdel-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 dothank Professor M.E El-Hawary for teaching me and guiding me in the scope of thematerial of this book Also, my appreciation goes to Professor Yacoub Al-Refae,general director of The Public Authority for Applied Education and Training inKuwait, for his encouragements and notes

A.M Al-KandariThe authors of this book would like to acknowledge the effort done by Ms SarahBinns for reviewing this book many times and we appreciate her time To her wesay, you did a good job for us, you were sincere and honest in every stage of this book

Economic development, throughout the world, depends directly on the availability ofelectric energy, especially because most industries depend almost entirely on its use.The availability of a source of continuous, cheap, and reliable energy is of foremosteconomic importance

Electrical load forecasting is an important tool used to ensure that the energy plied by utilities meets the load plus the energy lost in the system To this end, a staff

sup-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 agiven system, for a specified period of time ahead These predictions may be justfor a fraction of an hour ahead for operation purposes, or as much as 20 years intothe 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 thatexpansion 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 ning and dispatching costs can be minimized

run-In the preceding three categories, an accurate load model is required to tically represent the relationship between the load and influential variables such astime, weather, economic factors, etc The precise relationship between the load andthese 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

mathema-of estimation techniques

Extrapolating the mathematical relationship to the required lead time ahead andgiving the corresponding values of influential variables to be available or predictable,forecasts can be made Because factors such as weather and economic indices areincreasingly difficult to predict accurately for longer lead times ahead, the greaterthe lead time, the less accurate the prediction is likely to be

The final accuracy of any forecast thus depends on the load model employed, theaccuracy 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 predictionaccuracy

Over the past 50 years, the parameter estimation algorithms used in load forecastinghave been limited to those based on the least error squares minimization criterion, eventhough estimation theory indicates that algorithms based on the least absolute value cri-teria are viable alternatives Furthermore, the artificial neural network (ANN) hadshowed 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 precisevalue from one hour to another In this book, fuzzy systems theory is implemented toestimate the load model parameters, which are assumed to be fuzzy parameters having

a certain middle and spread Different membership functions, for load parameters, areused—namely, triangular membership and trapezoidal membership functions Theproblem of load forecasting in this book is restricted to short-term load forecastingand is formulated as a linear estimation problem 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 isincluded in the estimated membership

Outline of the Book

In this book, different techniques used in the past two decades are implemented toestimate the load model parameters, including fuzzy parameters with certain middleand certain spread The book contains nine chapters:

Chapter 1,“Mathematical Background and State of the Art.” This chapterintroduces mathematical background to help the reader understand the problems for-mulated in this book In this chapter, the reader will study matrices and their applica-tions in estimation theory and see that the use of matrix notation simplifies complexmathematical expressions The simplifying matrix notation may not reduce theamount of work required to solve mathematical equations, but it usually makes theequations much easier to handle and manipulate This chapter explains the vectorsand 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 chapteralso explains some optimization techniques and introduces the concept of a statespace model, which is commonly used in dynamic state estimation The reader willalso review different techniques that, developed for the short term, give the state ofthe 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 Advantagesand 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 availablealgorithms treat the parameters of the proposed load model as crisp parameters, which

is not the case in reality

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 includethe 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 casesfor the fuzzy parameters are discussed in this part The first case is for the fuzzy linearregression of the linear models having fuzzy parameters with nonfuzzy outputs, thesecond case is for the linear regression of fuzzy parameters with fuzzy output, andthe third case is for fuzzy parameters formulated with fuzzy output of left and righttype (LR-type)

Chapter 3,“Load Modeling for Short-Term Forecasting.” This chapter poses different load models used in short-term load forecasting for 24 hours

pro-• Three models are proposed in this chapter—namely, models A, B, and C Model A is a tiple linear regression model of the temperature deviation, base load, and either wind-chillfactor for winter load or temperature humidity factor for summer load The parameters ofload A are assumed to be crisp parameters in this chapter The termcrisp parametersmean clearly defined parameter values without ambiguity

mul-• 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 dailyload 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-varyingbase load and a weather-dependent component This model is developed with the aim ofeliminating the disadvantages of the other two models by combining their modelingapproaches After finding the parameter values, one uses them to determine the electricload from which these parameter values are extracted, and this value is called the estimatedload Then the parameter values are used to predict the electric load for a randomly chosenday in the future, and it is called the predicted load for that chosen day

Chapter 4,“Fuzzy Regression Systems and Fuzzy Linear Models.” The tive of this chapter is to introduce principal concepts and mathematical notions offuzzy set theory, a theory of classes of objects with non sharp boundaries

objec-• 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 theinterval [0, 1]

• A number of notions of fuzzy sets, such as representation support,α-cuts, convexity, andfuzzy numbers, are then introduced The resolution principle, which can be used to expand

a fuzzy set in terms of itsα-cuts, is discussed

• This chapter introduces fuzzy mathematical programming and fuzzy multiple-objective sion making We first introduce the required knowledge of fuzzy set theory and fuzzy mathe-matics in this chapter

deci-• Fuzzy linear regression also is introduced in this chapter; the first part is to estimate thefuzzy regression coefficients when the set of measurements available is crisp, whereas inthe second part the fuzzy regression coefficients are estimated when the available set ofmeasurements is a fuzzy set with a certain middle and spread

• Some simple examples for fuzzy linear regression are introduced in this chapter

• The models proposed inChapter 3for crisp parameters are used in this chapter Fuzzymodel A employs a multiple fuzzy linear regression model The membership function forthe model parameters is developed, where triangular membership functions are assumedfor 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 This modelinvolves fuzzy parameters having a certain median and certain spread

• Finally, a hybrid fuzzy model C, which is the combination of the multiple linear regressionmodel A and harmonic model B, is presented in this chapter

Chapter 5,“Dynamic State Estimation.” The objective of this chapter is to studythe dynamic state estimation problem and its applications to electric power systemanalysis, especially short-term load forecasting Furthermore, the different approachesused to solve this dynamic estimation problem are also discussed in this chapter Afterreading 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 ofinterest

• 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 filteringalgorithms

Chapter 6,“Load Forecasting Results Using Static State Estimation.” Theobjective of this chapter is as follows:

InChapter 3, the models are derived on the basis that the load powers are crisp in nature; thedata available from a big company in Canada are used to forecast the load power in the crispcase

• In this chapter, the results obtained for the crisp load power data for the different load modelsdeveloped inChapter 3are 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 pare between them for summer and winter

com-Chapter 7,“Load Forecasting Results Using Fuzzy Systems.”Chapter 6cusses the short-term load-forecasting problem, and the LES and LAV parameter esti-mation algorithms are used to estimate the load model parameters The error in theestimates is calculated for both techniques The three models, proposed earlier in

dis-Chapter 3, are used in that chapter to present the load in different days for differentseasons In this chapter, the fuzzy load models developed inChapter 5are tested Thefuzzy parameters of these models are estimated using the past history data for summerweekdays and weekend days as well as for winter weekdays and weekend days Thenthese models are used to predict the fuzzy load power for 24 hours ahead, in both

summer and winter seasons The results are given in the form of tables and figures forthe estimated and predicted loads.

Chapter 8,“Dynamic Electric Load Forecasting.” The main objectives of thischapter are as follows:

• A one-year long-term electric power load-forecasting problem is introduced as a first stepfor short-term load forecasting

• A dynamic algorithm, the Kalman filtering algorithm, is suitable to forecast daily load files with a lead-time from several weeks to a few years

pro-• The algorithm is based mainly on multiple simple linear regression models used to capturethe 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 theKalman filtering algorithm to improve the load-forecast accuracy obtained so far fromthe regression models

Chapter 9,“Electric Load Modeling for Long-Term Forecasting.” The tives of this chapter are as follows:

objec-• This chapter provides a comparative study between two static estimation algorithms—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, loadfactor, etc

• The problem turns out to be a linear estimation problem in the load parameters Differentmodels are developed and discussed in the text

1 Mathematical Background and

State of the Art

The objectives of this chapter are

• Introducing a mathematical background to help the reader understand the problemsformulated in this book

• Studying matrices and their applications in estimation theory and showing that the use ofmatrix notation simplifies complex mathematical expressions The simplifying matrix nota-tion may not reduce the amount of work required to solve mathematical equations, but itusually makes the equations much easier to handle and manipulate

• Explaining the vectors and the formulation of quadratic forms and, as we shall see, that mostobjective 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 stateestimation

• Reviewing the literature to introduce different techniques developed for short-term loadforecasting

• Explaining the merit of each technique used in the estimation of load forecasting and table places for implementation

sui-• In this chapter, we also try to compare different techniques used in electric load forecasting

A matrix is an array of elements [1] The elements of a matrix may be real orcomplex or functions of time A matrix that has n rows and m columns is called

ann
m (n by m) matrix If n ¼ m, the matrix is referred to as a square matrix If

A is an n
m matrix, then it can be written as

Note that the determinant is also an array of elements withn rows and n columns(always square) and has a value The matrix does not have a value but has adeterminant.

Column Matrix: This type of matrix has only one column and more than one row;that is, anm
1 matrix, m > 1 Quite often, a column matrix is referred to as a col-umn vector or simply anm-vector For example, the column vector X is written as

Y ¼ ½ y1,y2, , yn ¼ row.ðy1,y2, , ynÞ ð1:4ÞDiagonal Matrix: This is a square matrix with all elements equal to zero exceptfor the diagonal element; that is,aij¼ 0 for all i 6¼ j For example,

In terms of a shortcut:

AT ¼ A whereT 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 inA If A is an n
m matrix,which is represented by

A ¼ a i, j n
m

2 Electrical Load Forecasting: Modeling and Model Construction

then the transpose ofA, denoted by AT, is given by

3
4then

4
3The following are some operations using the transpose of a matrix:

1.3.2 Matrix Subtraction (Difference)

The subtraction (difference) of matrices is similar to the addition of matrices if all thesigns of the second matrix are changed from positive to negative and from negative topositive; that is,

The following rules hold true for addition and subtraction:

3
2

4 Electrical Load Forecasting: Modeling and Model Construction

B ¼ 12 13

ð2
2Þthen

2

6

37

If the matrixA is given by

A ¼ 3 40 1

ð2
2Þand the vector matrixX(t) is given by

XðtÞ ¼ x1ðtÞ

x2ðtÞ

ð2
1Þthen

5 ¼ 13 2244 59butBA will be

35

Although the commutative law does not hold in general for matrix multiplication,the associative and distributive laws still apply For the distributive law, we state that

provided that the product is conformable

For the associative law,

AB

if the product is conformable

1.3.4 Inverse of a Matrix (Matrix Division)

IfA is a square matrix of which the determinant exists, and if B is another squarematrix such that

where adj(A) is the adjoint of A, and it is the transpose of the matrix of cofactors of Awith elements

The minorsMijare determinants of the (n  1)
(n  1) matrices obtained by ing the ith row and jth column from A The following example explains the stepsinvolved

delet-6 Electrical Load Forecasting: Modeling and Model Construction

Calculate the determinant ofA as

35Thus, transposing Cof(A), we obtain adj(A) as

35The inverse ofA is obtained as

35

Some properties of the matrix inverse are

1.4 Rank of a Matrix

The rank of a matrixA is the maximum number of linearly independent columns of A,

or it is the order of the largest nonsingular matrix contained inA For example, thematrix

A ¼ 00 10

has a rank¼ 1and

A ¼ 03 55 13 42

has a rank¼ 2while

A ¼ 24 12

is a singular matrix becausejAj ¼ 44 ¼ 0

If the matrix is a singular matrix, the inverse of this matrix does not exist Even thoughthe matrixA 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 matrixA with characteristic values or eigenvalues λ1, , λn, the eigenvectors

of the matrix satisfy the relations

TheUis are called eigenvectors

The matrixU of the eigenvectors is nonsingular if the eigenvectors are linearlyindependent:

Let

U1¼ V ¼

VT 1

VT 2

VT n

2666

377

377

In terms of eigenvectors, we have

Usingequation (1.28), we obtain

Substituting forV in partitioned form, we get

The eigenvalues are obtained as

U11¼ U21ðor assume that U11 ¼ 1, then U21¼ 1Þ

Therefore, the first eigenvector is given by

Therefore, the transformed matrixA is given by

For example, the following 3
4 matrix is partitioned into four blocks:

5¼ B CD E

whereB, C, D, and E are the arrays indicated by dashed lines The matrix entries ofsuch a partitioned matrix are called submatrices The main matrix is sometimesreferred to as the supermatrix

IfA is square, and its only nonzero elements can be partitioned as principal matrices, then it is called a block diagonal A convenient notation that generalizes is

The only restriction is that the blocks must be conformable for multiplication, so thatall the productsBX1,CX2, , etc., exist This requires that in a product AX the num-ber of columns in each block ofA must equal the number of rows in the correspond-ing block of X.

It is difficult to obtain the inverse of matrices of high dimension by using the classicalmethod In this case, the partitioned form is useful Suppose thatF is a matrix in par-titioned form as

provided that the matrix A is nonsingular

F can be partitioned as shown.

Thus, we can find

2 c

264

37

5xy

 

or

fð ÞX ¼ XTAX

The preceding equation, as mentioned previously, is in quadratic form The matrixA

in this form is a symmetrical matrix

A more general form for the quadratic function can be written in a matrix form as

we need to write this function in a quadratic form

We define the vector

xiaijxj

¼Xn

i¼1

Xn j¼1

xiajixj

16 Electrical Load Forecasting: Modeling and Model Construction

• Ann
n matrix A is positive definite if, and only if,

The state space representation is an alternative method used to describe a system’sdynamics, in which a compact standard notation is used for this description [3,6].The concept of a system statex was defined by Kalman in 1963 [5] as:

The state of a system is a mathematical structure consisting of a set ofn time-dependentvariablesx1(t), x2(t), , xi(t), , xn(t), referred to as the state variables The systeminputs uj(t) together with the initial conditions of the state variables x1(0),

x2(0), , xi(0), ., xn(0) are sufficient to uniquely define the system’s future response.Let us explain this statement through an example, from a simple ac seriesR, L, Ccircuit

The total voltage is given as

u tð Þ ¼ uRð Þ þ ut Lð Þ þ ut cð Þt ð1:44Þ

¼ RiðtÞ þ Ldi tdtð ÞþC1

Z

Taking the Laplace transform of both sides, assuming zero initial conditions

VðsÞ ¼ RIðsÞ þ sLIðsÞ þCs1 IðsÞ

V sð Þ

I s ¼ R þ sL þ

1Cs

ð1:46Þ

The input to the system is assumed to be the voltageV(s), and the output is I(s).Then

5 _x1ðtÞ_x2ðtÞ

þ

01L

24

The vector-matrix form can also be applied to nonlinear state equations Thegeneral expression is

dxðtÞ

18 Electrical Load Forecasting: Modeling and Model Construction

1.12 Difference Equations

In modern control systems, digital processors are used to perform the task of control,

so it is important to establish equations that relate digital and discrete time signals.Whereas differential equations are used to represent systems with analog signals, dif-ference equations are used for systems with discrete data It is easier to use differenceequations than differential equations on a digital computer, and these equations aregenerally easier to solve

As an example of discrete approximation, we can use a forward difference process

to approximate the derivative of a function at a given instant; that is,

then the equation is written as

in vector-matrix form, we have

ð1:67Þ

In this section we discuss the general optimization problem without going into matical analysis details [2,4] The first part of the section introduces unconstrainedoptimization that has many applications throughout this book, and the second part

mathe-20 Electrical Load Forecasting: Modeling and Model Construction

introduces the constrained optimization problem Generally speaking, the tion problem has the following form:

gen-1.13.1 Unconstrained Optimization

In the unconstrained optimization problem, we need to find the value of the vector

X ¼ [x1, , xn]Tthat minimizes the function

provided that the function f is continuous and has a first-order derivative

To obtain the minimum and/or maximum of the functionf, we set its first tive, with respect to thexis, to zero:

Equations (1.75) to (1.77)representn equations in n unknowns The solution ofthese equations produces candidate solution points If the functionf has second partialderivatives, then we calculate the Hessian matrixH ¼∂

2f ðx1, , xnÞ

∂x2 i If the matrixH

is positive definite, then the functionf is a minimum at the candidate points, but if thematrix H is negative definite, then f is a maximum at the candidate points Thefollowing examples illustrate these steps

22 Electrical Load Forecasting: Modeling and Model Construction

XTHX > 0 ð∀X 6¼ 0Þ

soH is positive definite, and the function f is a minimum at the candidate point.Note that the positive definiteness ofH can also be verified just by calculatingthe values of the different determinants, produced fromH as

Therefore,H is a positive definite matrix; or we calculate the quadratic form

264

375The Hessian matrix is calculated as

35

24 Electrical Load Forecasting: Modeling and Model Construction

1,x 2

ψjðx1, , xnÞ  0, ð j ¼ 1, , mÞ ð1:80ÞLet us consider, for instance, a case when an objective function is subject only toequality constraints We form the augmented objective function by adjoining the equal-ity constraints to the function via Lagrange’s multipliers to obtain the alternative form:Minimize

λj∂∂xj

Equation (1.83) is a set of n equations in (n þ ℓ) unknowns ðxi; i ¼ 1, , n:

λj; j ¼ 1, , ‘Þ To obtain the solution, we must satisfy the equality constraints;that is,

iðx1, , xnÞ ¼ 0 i ¼ 1, , ‘ ð1:84Þ

Solvingequations (1.83)and(1.84), we obtainx

i andλ

j This scenario is illustrated

in the following examples

Example 1.8

Minimize

f ðx1,x2Þ ¼ x2

1þ x2 2Subject to

∂ef

∂λ¼ 0 ¼ x1þ 2x

2þ 1 ðequality constraintÞSolving the preceding three equations gives

The augmented cost function is

efðx1,x2,λÞ ¼ ð30 þ 5x1þ 0:2x2

1þ 3x2þ 0:1x2

2Þ þ λð10  x1 x2ÞPutting the first derivatives to zero, we obtain

x

1¼ 0, x

2¼ 10, λ ¼ 5and the minimum of the function is

f ð0, 10Þ ¼ 30 þ 30 þ 10 ¼ 70

If there are inequality constraints, then the augmented function is obtained byadjoining these inequality constraints via Kuhn-Tucker multipliers to obtainefðX, λ, μÞ ¼ f ðXÞ þ λTðXÞ þ μTψðXÞ ð1:85ÞPutting the first derivative to zero, we obtain

Solving the preceding equations gives the candidate solutionðX,λ, μÞ

Indeed, as we see, the variables do not violate their limits, and the optimalsolution in this case is

0 x2 8

then we can see that for the solution here,x

2¼ 10 violates the upper limit In thiscase we put

x

2¼ 8; with μ3ð8  x

2Þ ¼ 0and recalculatex

1¼ 0 and μ

2 ¼ 0 To calculate λ andμ

3, weuse the first two equations as

μ

3¼ 1:6 þ 3  5:8

¼ 1:2

Short-term load forecasting (STLF) is an integral part of power system operation,which is essential for securing an inexpensive supply of reliable electric energy.This type of forecasting is used to predict load demands up to a week ahead so

that the day-to-day operation of a power system can be efficiently planned and so thatthe operating costs are minimized.

Short-term load forecasting can be performed in one of two modes—namely, onlineand offline forecasting This categorization, as the names suggest, stems from the areas ofapplication of the load predictors Offline load forecasting is primarily implemented inthe scheduling of the large generating units of which the startup times may vary from

a few hours ahead to a few days ahead The scheduling process is termedunit ment and ensures that there is sufficient operating generation capacity to meet the variableload demand with specified reliability [7] When load forecasting is poor, incorrect sche-duling may occur, resulting in higher daily operational cost caused by use of higher-costquick-start units in the event of underscheduling or, alternatively, resulting in the uneco-nomic operation of large generating units in the event of overscheduling [50]

commit-Online operation of a power system, the economic load dispatching to various ating units, makes the generating mix dependent on calculations to minimize the costfunction, which is based on the characteristics of the generating units These calculationsare based on values of load demand predicted a few hours in advance, and as such theoptimum generating mix is dependent on the accuracy of the online forecasts

gener-It has been recognized for a long time that accurate short-term load predictors aswell as a load model are basic necessities for the optimum economic operation ofpower systems A prerequisite to the development of an accurate load-forecastingmodel is an understanding of the characteristics of the load to be modeled Thisknowledge of load behavior is gained from experience with the load and thoroughstatistical analysis of past load data Utilities with similar climatic and economicenvironments usually experience similar load behavior, and load models developedfor one utility can usually be modified to suit another

For short-term load forecasting, many factors should be included in the loadprediction model Reference [7] reviews the short-term load-demand modeling andforecasting for offline and online implementation Included also in [7] is a review ofmost techniques used at that time; the merits and drawbacks of each approach arepresented Reference [8] presents an algorithm based on curve fitting of past loadgrowth for forecasting distribution system loads The proposed algorithm in this refer-ence uses clustering of historical load at the small area level as the forecast algorithm.References [9,10] compare 14 methods of forecasting future distribution system loads

in terms of forecasting accuracy, data needs, and resources The tests of different cast methods were carried out in as uniform a manner as possible These referencesclaim that the selection of a forecast method is based on a great deal more criteriathan those discussed in the references Data availability is usually an important factor;choice of a distribution load-forecasting method may also be constrained by many otherfactors, including available computer resources and the level of expertise of the users.Reference [11] reviews some of the existing studies on 1- to 24-hour load-forecastingalgorithms and presents an expert system-based algorithm as an alternative This algo-rithm is developed on the logical and syntactical relationships between weather andload, and prevailing daily load shapes It is found in this reference that the proposed algo-rithm is robust and accurate and has yielded results that are equally good, if not better,when compared to the regression-based forecasting techniques

fore-30 Electrical Load Forecasting: Modeling and Model Construction