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Introduction to fuzzy arithmetic—theory and applications
Book Reviews INTRODUCTION TO Fuzzy ARITHMETICTHEORY AND APPLICATIONS, by Arnold Kaufmann and Madan M. Gupta. 351 pages, diagrams, figures. Van Nostrand Reinhold Company, New York, 1985. The theory of fuzzy sets was first introduced in 1965 by Professor Lotfi A. Zadeh in the United States and has been studied and applied by many people in all parts of the world. More recently, the theory of fuzzy numbers has been introduced by S. Nahmais in the United States, and by H. Dubois and D. Prade in France. The present volume is the first to present a comprehensive and selfcontained theory of fuzzy numbers and its applications. Much of what has been developed is used by the authors, but Professors Kaufmann and Gupta have also introduced some novel concepts. To quote from Professor Zadeh’s Foreword : One cannot but be greatly impressed by the wide variety of concepts that Arnold Kaufmann and Madan M. Gupta discuss so well and with so many carefully worked-out examples. Many of the concepts and techniques they describe are original and appear in the literature for the first time; in particular, the concept of a hybrid number, which is both possibilistic and probabilistic in nature; algebraic systems of fuzzy numbers discussed in chapter 3 ; fuzzy trigonometric functions and complex numbers ; and a fuzzy generalization of catastrophe theory. In the Preface, the authors
stated :
The concept of fuzziness can be approached from several different avenues. The one that we have used in this book is based on the concept of the interval of confidence. We start with this concept and extend it to a range of levels between 0 and 1. Fuzzy arithmetic, a domain of fuzzy set theory, reveals many interesting results. Some of the results described here can be extended further. There are altogether four chapters three appendices. A brief Bibliography Vol. 321,No. 3, pp. 189-191,March Printed in Great Britain The Franklin Institute
and and
an Index are given at the end of the book. Chapter 1 states definitions and main properties of fuzzy numbers. After discussing the interval of confidence and uncertain numbers or fuzzy numbers, addition, multiplication, minimum and maximum, convolution and deconvolution of fuzzy numbers are developed. The chapter concludes with L-R, the fuzzy numbers of Dubois and Prade, and triarigular fuzzy numbers. Chapter 2 deals with a large sample of some novel concepts and tools. Hybrid numbers are first introduced. Sheaf and expectation of fuzzy numbers, and distance between two fuzzy numbers are given. Comparisons between hybrid numbers and simulation of hybrid data are covered. Then the authors extend the concept of the interval of confidence to type 2 which assumes that the lower and upper bounds of the interval of confidence, instead of being ordinary numbers, are fuzzy numbers that themselves have intervals of confidence. This chapter concludes with random fuzzy numbers and fuzzy numbers of dimension 2 and more. It is noted that random data differ from fuzzy material, and random data and fuzzy material can be used together. In Chapter 3, arithmetic and combinatorics with uncertain numbers are studied and examined. Fuzzy relative integers modulo n and fuzzy real numbers modulo 1 are presented. Factorials, sequences, and series with fuzzy numbers are then developed. Functions of fuzzy numbers and derivatives of such functions are discussed. The chapter concludes with fuzzy trigonometric functions, fuzzy hyperbolic functions and fuzzy complex numbers. The last chapter-Chapter 4-is concerned with deviation or divergence for a fuzzy number and agreement index for a fuzzy number. The chapter concludes with theory of catastrophe and fuzzy numbers. The three appendices are concerned with (a) the multiplication of intervals of confidence in R; (B) multiplication of fuzzy numbers in R; and (C) change of coordinates for a domain of confidence of level a. This book is clearly written and has many
,986
189
Book Reviews
illustrative examples. Since the book is a first of its kind, the authors should be complimented on their effort and readers are encouraged to use it as a good reference text. N.
Moore
X.
CHEN,
F. Y.
ZHU
and
School of Electrical University Philadelphia,
Y. H. Ku
Engineering,
of Pennsylvania, PA 19104,
U.S.A.
ELECTRIC POWER SYSTEM DYNAMICS, by Yao-Nan Yu. 6 x 9 in, 255 pages, diagrams, figures, $39.00. Academic Press, New York,, 1983. The objective of this book by Professor Yao-Nan Yu is to bridge a major gap in power system engineering between recent literature dealing with the dynamic performance of large electric power systems and classic books on power system stability before the inception of digital computers and modern control theory. New power system dynamic problems have been included, among them lowfrequency intersystem oscillations, torsional oscillations due to subsynchronous resonance of capacitor-compensated transmission lines and the derivation of dynamic equivalents for large electric power system dynamic studies. There are seven chapters and one appendix on computer programs. Chapter 1 begins with : basic components of an electric power plant; modern large electric power systems; and problems of electric power system dynamics. Chapter 2 discusses basic models for power system dynamic studies. Fundamental equations of synchronous machines are based on classic papers by R. H. Park, Trans. AIEE, Vol. 48, Part 1, July 1929, pp. 716730 and Vol. 52, part 2, June 1933, pp. 352-355. (The reviewer published a “companion” paper in July 1929 Trans. pp. 707-715.) Using the d, q components, five equations are given in Eq. (2-l) which include two damper windings D, Q. Reactances, transient and subtransient reac-
190
tances, transient and subtransient time constants are defined. (For other references, see Y. H. Ku, “Electric Energy Conversion”, Ronald Press, 1959). After discussing the second-order and third-order synchronous generator model, D. W. Olive’s seventhorder synchronous machine model has been derived. (For comparative study, see Y. H. Ku: “On nonlinear oscillations in electromechanical systems”, J. Franklin Inst., Vol. 272, Oct. 1961, pp. 2533274.) The rest of this chapter deals with exciter and voltage regulator models, hydraulic power and governor models and steam turbine and governor models. Chapter 3 deals with low-frequency intersystem oscillations and supplementary controls. The first example of such lowfrequency oscillations was observed during a trial interconnection of the Northwest Power Pool (predominantly hydroelectric) and the Southwest Power Pool (predominantly steam-electric) before the existence of the Western Systems Coordinating Council (WSCC). In order to improve system damping, supplementary excitation control has been developed. Further system damping can be improved by governor control. Chapter 4 deals with linear optimal stabilization of electric power systems. Choosing a performance index of the quadratic form, the linear optimal control (LOC) is derived from the minimization of the performance index (also known as the cost function) in conjunction with the state equation. A combination of modern control theory and classical mechanics involves a Hamiltonian generalizedenergy function H, which is related to the Lagrangian L, and the solution of the Riccati matrix equation. After discussing linear optimal stabilization of a one-machine, infinite-bus system, LOC is applied to a multimachine system. An eigenvalue assignment technique for the LOC design is developed in Section 4-6. In Section 47, other developments of LOC of electric power systems are briefly presented. Chapter 5 deals with subsynchronous ressonance and torsional oscillations. The first two shaft failures due to subsynchronous Journal of the
Franklin Insmute Pergamon Press Ltd.
stated :
The concept of fuzziness can be approached from several different avenues. The one that we have used in this book is based on the concept of the interval of confidence. We start with this concept and extend it to a range of levels between 0 and 1. Fuzzy arithmetic, a domain of fuzzy set theory, reveals many interesting results. Some of the results described here can be extended further. There are altogether four chapters three appendices. A brief Bibliography Vol. 321,No. 3, pp. 189-191,March Printed in Great Britain The Franklin Institute
and and
an Index are given at the end of the book. Chapter 1 states definitions and main properties of fuzzy numbers. After discussing the interval of confidence and uncertain numbers or fuzzy numbers, addition, multiplication, minimum and maximum, convolution and deconvolution of fuzzy numbers are developed. The chapter concludes with L-R, the fuzzy numbers of Dubois and Prade, and triarigular fuzzy numbers. Chapter 2 deals with a large sample of some novel concepts and tools. Hybrid numbers are first introduced. Sheaf and expectation of fuzzy numbers, and distance between two fuzzy numbers are given. Comparisons between hybrid numbers and simulation of hybrid data are covered. Then the authors extend the concept of the interval of confidence to type 2 which assumes that the lower and upper bounds of the interval of confidence, instead of being ordinary numbers, are fuzzy numbers that themselves have intervals of confidence. This chapter concludes with random fuzzy numbers and fuzzy numbers of dimension 2 and more. It is noted that random data differ from fuzzy material, and random data and fuzzy material can be used together. In Chapter 3, arithmetic and combinatorics with uncertain numbers are studied and examined. Fuzzy relative integers modulo n and fuzzy real numbers modulo 1 are presented. Factorials, sequences, and series with fuzzy numbers are then developed. Functions of fuzzy numbers and derivatives of such functions are discussed. The chapter concludes with fuzzy trigonometric functions, fuzzy hyperbolic functions and fuzzy complex numbers. The last chapter-Chapter 4-is concerned with deviation or divergence for a fuzzy number and agreement index for a fuzzy number. The chapter concludes with theory of catastrophe and fuzzy numbers. The three appendices are concerned with (a) the multiplication of intervals of confidence in R; (B) multiplication of fuzzy numbers in R; and (C) change of coordinates for a domain of confidence of level a. This book is clearly written and has many
,986
189
Book Reviews
illustrative examples. Since the book is a first of its kind, the authors should be complimented on their effort and readers are encouraged to use it as a good reference text. N.
Moore
X.
CHEN,
F. Y.
ZHU
and
School of Electrical University Philadelphia,
Y. H. Ku
Engineering,
of Pennsylvania, PA 19104,
U.S.A.
ELECTRIC POWER SYSTEM DYNAMICS, by Yao-Nan Yu. 6 x 9 in, 255 pages, diagrams, figures, $39.00. Academic Press, New York,, 1983. The objective of this book by Professor Yao-Nan Yu is to bridge a major gap in power system engineering between recent literature dealing with the dynamic performance of large electric power systems and classic books on power system stability before the inception of digital computers and modern control theory. New power system dynamic problems have been included, among them lowfrequency intersystem oscillations, torsional oscillations due to subsynchronous resonance of capacitor-compensated transmission lines and the derivation of dynamic equivalents for large electric power system dynamic studies. There are seven chapters and one appendix on computer programs. Chapter 1 begins with : basic components of an electric power plant; modern large electric power systems; and problems of electric power system dynamics. Chapter 2 discusses basic models for power system dynamic studies. Fundamental equations of synchronous machines are based on classic papers by R. H. Park, Trans. AIEE, Vol. 48, Part 1, July 1929, pp. 716730 and Vol. 52, part 2, June 1933, pp. 352-355. (The reviewer published a “companion” paper in July 1929 Trans. pp. 707-715.) Using the d, q components, five equations are given in Eq. (2-l) which include two damper windings D, Q. Reactances, transient and subtransient reac-
190
tances, transient and subtransient time constants are defined. (For other references, see Y. H. Ku, “Electric Energy Conversion”, Ronald Press, 1959). After discussing the second-order and third-order synchronous generator model, D. W. Olive’s seventhorder synchronous machine model has been derived. (For comparative study, see Y. H. Ku: “On nonlinear oscillations in electromechanical systems”, J. Franklin Inst., Vol. 272, Oct. 1961, pp. 2533274.) The rest of this chapter deals with exciter and voltage regulator models, hydraulic power and governor models and steam turbine and governor models. Chapter 3 deals with low-frequency intersystem oscillations and supplementary controls. The first example of such lowfrequency oscillations was observed during a trial interconnection of the Northwest Power Pool (predominantly hydroelectric) and the Southwest Power Pool (predominantly steam-electric) before the existence of the Western Systems Coordinating Council (WSCC). In order to improve system damping, supplementary excitation control has been developed. Further system damping can be improved by governor control. Chapter 4 deals with linear optimal stabilization of electric power systems. Choosing a performance index of the quadratic form, the linear optimal control (LOC) is derived from the minimization of the performance index (also known as the cost function) in conjunction with the state equation. A combination of modern control theory and classical mechanics involves a Hamiltonian generalizedenergy function H, which is related to the Lagrangian L, and the solution of the Riccati matrix equation. After discussing linear optimal stabilization of a one-machine, infinite-bus system, LOC is applied to a multimachine system. An eigenvalue assignment technique for the LOC design is developed in Section 4-6. In Section 47, other developments of LOC of electric power systems are briefly presented. Chapter 5 deals with subsynchronous ressonance and torsional oscillations. The first two shaft failures due to subsynchronous Journal of the
Franklin Insmute Pergamon Press Ltd.