POWER MAGNETICDEVICES

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POWER MAGNETICDEVICESA Multi-Objective Design Approach

S. D. SUDHOFF

Copyright © 2014 by The Institute of Electrical and Electronics Engineers, Inc.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Sudhoff, Scott D.Power magnetic devices : a multi-objective design approach / S.D. Sudhoff.

pages cmIncludes bibliographical references.ISBN 978-1-118-48999-4 (cloth)1. Electromagnetic devices. 2. Power electronics. 3. Electric machinery.

I. Title.TK7872.M25S83 2013621.31 0042—dc23

2013029968

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

http://www.wiley.com/go/permissionhttp://www.copyright.comhttp://www.wiley.com

To my wife Julie for over thirty wonderful years, my daughter Emilyfor her unshakable optimism, and my daughter Samantha for her

determination. They make the world a better place.

CONTENTS

PREFACE XI

CHAPTER 1 OPTIMIZATION-BASED DESIGN 1

1.1 Design Approach 11.2 Mathematical Properties of Objective Functions 31.3 Single-Objective Optimization Using Newton’s Method 51.4 Genetic Algorithms: Review of Biological Genetics 81.5 The Canonical Genetic Algorithm 111.6 Real-Coded Genetic Algorithms 161.7 Multi-Objective Optimization and the Pareto-Optimal Front 271.8 Multi-Objective Optimization Using Genetic Algorithms 301.9 Formulation of Fitness Functions for Design Problems 341.10 A Design Example 36References 43Problems 44

CHAPTER 2 MAGNETICS AND MAGNETIC EQUIVALENT CIRCUITS 45

2.1 Ampere’s Law, Magnetomotive Force, and Kirchhoff’s MMF Law for MagneticCircuits 46

2.2 Magnetic Flux, Gauss’s Law, and Kirchhoff’s Flux Law for Magnetic Circuits 492.3 Magnetically Conductive Materials and Ohm’s Law for Magnetic Circuits 512.4 Construction of the Magnetic Equivalent Circuit 612.5 Translation of Magnetic Circuits to Electric Circuits: Flux Linkage

and Inductance 642.6 Representing Fringing Flux in Magnetic Circuits 702.7 Representing Leakage Flux in Magnetic Circuits 752.8 Numerical Solution of Nonlinear Magnetic Circuits 882.9 Permanent Magnet Materials and Their Magnetic Circuit Representation 1042.10 Finite Element Analysis 108References 109Problems 110

CHAPTER 3 INTRODUCTION TO INDUCTOR DESIGN 113

3.1 Common Inductor Architectures 1133.2 DC Coil Resistance 1153.3 DC Inductor Design 1183.4 Case Study 125

vii

3.5 Closing Remarks 131References 131Problems 132

CHAPTER 4 FORCE AND TORQUE 133

4.1 Energy Storage in Electromechanical Devices 1334.2 Calculation of Field Energy 1354.3 Force from Field Energy 1384.4 Co-Energy 1394.5 Force from Co-Energy 1434.6 Conditions for Conservative Fields 1444.7 Magnetically Linear Systems 1454.8 Torque 1474.9 Calculating Force Using Magnetic Equivalent Circuits 148References 151Problems 152

CHAPTER 5 INTRODUCTION TO ELECTROMAGNET DESIGN 154

5.1 Common Electromagnet Architectures 1545.2 Magnetic, Electric, and Force Analysis of an EI-Core Electromagnet 1555.3 EI-Core Electromagnet Design 1655.4 Case Study 171References 177Problems 177

CHAPTER 6 MAGNETIC CORE LOSS 178

6.1 Eddy Current Losses 1786.2 Hysteresis Loss and the B–H Loop 1866.3 Empirical Modeling of Core Loss 1916.4 Time Domain Modeling of Core Loss 197References 203Problems 203

CHAPTER 7 TRANSFORMER DESIGN 205

7.1 Common Transformer Architectures 2057.2 T-Equivalent Circuit Model 2077.3 Steady-State Analysis 2117.4 Transformer Performance Considerations 2137.5 Core-Type Transformer Configuration 2237.6 Core-Type Transformer MEC 2307.7 Core Loss 2367.8 Core-Type Transformer Design 2387.9 Case Study 2457.10 Closing Remarks 252References 252Problems 253

viii CONTENTS

CHAPTER 8 DISTRIBUTED WINDINGS AND ROTATING ELECTRIC MACHINERY 254

8.1 Describing Distributed Windings 2548.2 Winding Functions 2658.3 Air-Gap Magnetomotive Force 2688.4 Rotating MMF 2728.5 Flux Linkage and Inductance 2738.6 Slot Effects and Carter’s Coefficient 2768.7 Leakage Inductance 2788.8 Resistance 2838.9 Introduction to Reference Frame Theory 2848.10 Expressions for Torque 290References 295Problems 296

CHAPTER 9 INTRODUCTION TO PERMANENT MAGNET AC MACHINE DESIGN 299

9.1 Permanent Magnet Synchronous Machines 2999.2 Operating Characteristics of PMAC Machines 3019.3 Machine Geometry 3099.4 Stator Winding 3159.5 Material Parameters 3189.6 Stator Currents and Control Philosophy 3199.7 Radial Field Analysis 3209.8 Lumped Parameters 3259.9 Ferromagnetic Field Analysis 3279.10 Formulation of Design Problem 3339.11 Case Study 3399.12 Extensions 346References 346Problems 347

CHAPTER 10 INTRODUCTION TO THERMAL EQUIVALENT CIRCUITS 349

10.1 Heat Energy, Heat Flow, and the Heat Equation 34910.2 Thermal Equivalent Circuit of One-Dimensional Heat Flow 35210.3 Thermal Equivalent Circuit of a Cuboidal Region 35910.4 Thermal Equivalent Circuit of a Cylindrical Region 36310.5 Inhomogeneous Regions 37010.6 Material Boundaries 37710.7 Thermal Equivalent Circuit Networks 37910.8 Case Study: Thermal Model of Electromagnet 385References 401Problems 401

CHAPTER 11 AC CONDUCTOR LOSSES 403

11.1 Skin Effect in Strip Conductors 40311.2 Skin Effect in Cylindrical Conductors 41011.3 Proximity Effect in a Single Conductor 41411.4 Independence of Skin and Proximity Effects 416

CONTENTS ix

11.5 Proximity Effect in a Group of Conductors 41811.6 Relating Mean-Squared Field and Leakage Permeance 42211.7 Mean-Squared Field for Select Geometries 42311.8 Conductor Losses in Rotating Machinery 42811.9 Conductor Losses in a UI-Core Inductor 43211.10 Closing Remarks 436References 437Problems 437

APPENDIX A CONDUCTOR DATA AND WIRE GAUGES 439

References 440

APPENDIX B SELECTED FERRIMAGNETIC CORE DATA 441

Reference 442

APPENDIX C SELECTED MAGNETIC STEEL DATA 443

Reference 444

APPENDIX D SELECTED PERMANENT MAGNET DATA 445

Reference 445

APPENDIX E PHASOR ANALYSIS 446

APPENDIX F TRIGONOMETRIC IDENTITIES 453

INDEX 455

x CONTENTS

PREFACE

This work is intended either as a textbook for a senior-level or beginning graduatecourse or as a resource for the practicing engineer. There are three objectives for thetext. The first is to set forth a systematic multi-objective optimization approach for thesemi-automated design of power magnet components. A second objective of the textis to discuss physical principles and analysis necessary for the design of powermagnetic devices including fields, magnet equivalent circuit analysis, core loss, eddycurrent losses, thermal analysis, skin and proximity effect, and the like. The thirdobjective of the text is to provide some fundamental background in a variety ofdevices including inductors, electromagnets, transformers, and rotating electricmachinery. It is not the intent of this book to provide a cookbook of designinformation for all devices. Rather, it is the intent that after taking this course thereader be well-poised to start adapting the approach to specific devices of interest totheir work—whether it be a transformer or a novel type of rotating machine.

From a pedagogical point of view, the organization of the text is designed toinvolve the reader in the design process as rapidly as possible. While it might be moreefficient to discuss all relevant physical effects and then to discuss the design, such anapproach is not always satisfying in that it leaves the reader hungry for a meal a longtime before dinner is served. For that reason, the text generally alternates betweendiscussing a physical effect and then considering a design problem in which that effectis considered.

Chapter 1 provides an introduction to multi-objective optimization usinggenetic algorithms. This chapter is not a comprehensive discussion of optimization,but does provide a sufficient background to conduct formal multi-objectiveoptimization-based design. Chapter 2 provides a background in magnetic analysisthat is used throughout the book. Formal design is introduced in Chapter 3 whereininductor design is considered, using the material from Chapters 1 and 2. In Chapter 4,force and torque production are considered, and in Chapter 5 this material is used inelectromagnet design. Chapter 6 concerns magnetic core loss; this is used in Chapter 7in transformer design. The alternating pattern of analysis and design continues, withChapters 8 and 9 focusing on rotating machinery. The book concludes with twoprimarily analysis chapters to supplement the earlier work, with Chapter 10 focusingon thermal analysis (and revisiting some of the earlier design efforts) and Chapter 11discussing skin and proximity effect losses.

The amount of material is in excess of what can be used in a one-semestercourse. The author recommends that Chapter 1 through Chapter 6 be covered as astarting point. The remaining chapters could be covered at the discretion of theinstructor or the reader, though Chapter 9 is heavily dependent upon Chapter 8.

xi

Chapters 8 and 9 are based on Chapters 2 and 15 of Analysis of ElectricMachinery and Drive Systems, 3rd edition, by Paul Krause, Oleg Wasynczuk, ScottSudhoff, and Steve Pekarek. This work is also published by IEEE/Wiley.

MATLAB source code to support this book is given in S. D. Sudhoff,MATLABCodes for Power Magnetic Devices: A Multi-Objective Design Approach [online],available at http ://books upport.wi ley.com. Thi s code inclu des the Genetic Optim iza-tion System Engineering Toolbox, a Magnetic Equivalent Circuit Toolbox, a ThermalEquivalent Circuit Toolbox, and all the design examples discussed in the book. Thisshould greatly reduce the amount of work needed to either teach from the book or touse the principles taught in this text for the reader’s own purposes. Partially annotatedslides are also available.

Throughout this work, scalar variables are normally in italic font (for example,x), while vector and matrices are bold nonitalic (for example, x). Functions of alldimensionalities are denoted by nonitalic nonbold font (for example, xq). Bracketsin equations are associated with iteration number in iterative methods.

The author of this book is deeply indebted to many individuals. First, to myparents, who gave me the time, support, and indulgence to pursue my interests.While in high-school, I was blessed with many excellent teachers. I am particularlyindebted to Sister Thomasita Hayes. As an undergraduate, I was also fortunate tohave some outstanding instructors—particularly Stanislaw Zak, in control andoptimization; Fred Mowle, who taught me how to code; and Paul Krause, whotaught me electric machinery. Paul would later become my major professor, and, asone of his other students put it, is somewhere between a somewhat mischievouslittle brother and a grandfather to many at Purdue. I spent the beginning part of mycareer at the University of Missouri—Rolla (now Missouri University of Scienceand Technology). Here, I was fortunate to have a number of excellent mentors,including Keith Stanek, Max Anderson, and especially Mariesa Crow and JimDrewniak.

I would like to thank many current and former students, post docs, and researchscientists who directly or indirectly contributed to this book. These include BenjaminLoop, Chunki Kwon, Jim Cale, Aaron Cramer, Brandon Cassimere, Brant Cassimere,Chuck Sullivan, Ricky Chan, Shengyu Wang, Yonggon Lee, Cahya Harianto,Jacob Krizan, Grant Shane, Omar Laldin, Ahmed Taher, Jamal Alsawhali, HarishSuryanarayana, and Jonathan Crider. Jamal Alsawalhi, Grant Shane, Jonathan Crider,Ahmed Taher, Ruiyang Lin, David Loder, and Andrew Kasha, who contributed inperforming many of the FEA and/or experimental results in the book. Finally, I wouldlike to thank Julie Sudhoff, Jamal Alsawalhi, Rachel Grossman, David Loder, andAndrew Kasha for their efforts in reading the first drafts. I would like to especiallythank Dionysios Aliprantis for sparking my interest in genetic algorithms.

A variety of U.S. government agencies have contributed to research efforts thatcontributed to this book, including the Army, Navy, and NASA. The Office of NavalResearch in particular has provided steady support for my entire career, which directlyand indirectly supported this work, and without which this work would not have beenpossible. The support of the Grainger Foundation has also been very important to theprogram at Purdue.

xii PREFACE

http://booksupport.wiley.com

Finally, I would like to thank my colleagues at Purdue. A key attribute of anyinstitution is the people you work with. With this regard, my colleagues at Purdue,namely, Paul Krause, Oleg Wasynczuk, Steve Pekarek, Chee-Mun Ong, DionysiosAliprantis, Maryam Saaedifard, and Ragu Balakrishnan, are a great group. I am alsofortunate to have great friends at other institutions, particularly Paula and Ed Zivi.

S. D. SUDHOFF

PREFACE xiii

CHA P T E R 1OPTIMIZATION-BASED DESIGN

WE WILL begin our study of power magnetic device design with a generalconsideration of the design process. A case will be made to approach the design

process rather formally by converting the design problem into an optimization prob-

lem. Next, single-objective optimization is discussed, with particular emphasis on

optimization using genetic algorithms. This is followed by a discussion of multi-

objective optimization. Practical aspects of formulating design problems as optimi-

zation problems are then discussed. The chapter concludes with a design example

that focuses on a UI-core inductor.

1.1 DESIGN APPROACH

It is appropriate to begin this work by considering the design process. Clearly, thereare a myriad of different approaches by which components may be designed. Forexample, a possible manual design process is illustrated in Figure 1.1. In order toconsider this process in a more concrete way, suppose that the component we aredesigning is an electromagnet and that we wish to design an electromagnet so that acertain set of specifications are met.

Using the design process in Figure 1.1, our first step would be to perform adetailed mathematical analysis of the device. Typically, when we analyze a device,our analysis predicts device performance (mass, loss, force) in terms of the deviceparameters (geometry, materials) rather than directly addressing the design problemby deriving expressions for what the device parameters should be in terms of thedevice specifications (allowed loss, required force). Therefore, we must manipulateour detailed analysis into a set of design equations that are used to calculate the designparameters as a function of device specifications. However, going from detailedanalysis to design equations invariably requires numerous assumptions and approxi-mations, even beyond the ones found in our original “detailed” analysis. As a result,we check our design, either against our original analysis or using some numerical toolsuch as a finite element analysis. Based on the results from the numerical analysis, wewill revise the design and repeat the numerical analysis until specifications are met, atwhich point we have arrived at a final design. Of course, we often use a more involveddesign process; for example, another iteration of the design may be made based onphysical prototypes.

1

Power Magnetic Devices: A Multi-Objective Design Approach, First Edition. S. D. Sudhoff.© 2014 by The Institute of Electrical and Electronics Engineers, Inc. Published 2014 by John Wiley & Sons, Inc.

The manual design process we have been considering involves an engineerin the iteration process. Variations of this process are successfully used ubiqui-tously throughout the engineering community. However, the process has somesignificant drawbacks. First, it requires a great deal of engineering time. Second, itrequires a great deal of engineering experience. This experience comes into play inthe development of the design equations, which often take the form of rules-of-thumb based at least partially on experience. Experience is also a factor in makingchanges to the design based on the numerical analysis. Finally, while the processhas been very successful in yielding working designs, it may not lead to the bestdesign.

An alternate design process is illustrated in Figure 1.2. Therein, anoptimization-based design process is shown. In this case, the process is not illustratedin a sequential manner as in Figure 1.1, but rather in an organizational manner. Theprocess again starts with a detailed analysis of the device or component. However,unlike the manual design process, in the optimization-based process the detailedanalysis is not used to formulate design equations. Instead, the detailed analysis isused to calculate design metrics such as mass, cost, and loss. The detailed analysis isalso used to check constraints such as achieving some minimum acceptable level ofperformance. The metrics and constraints are combined into an objective or fitnessfunction. This function is defined so that its optimization results in optimization of thedesign metrics subject to all design metrics being met.

At the outermost level of this design process, an optimization engine will selectthe parameters of the design (geometry, materials, etc.) so as to maximize theobjective function. In terms of computational algorithm, Figure 1.2 depicts anoptimization engine at the outer level. This engine operates on an objective functionthat is calculated based on the detailed analysis.

There are several advantages of this approach. First, it is unnecessary toformulate design equations. This is beneficial in that it reduces the number ofapproximations and assumptions made and reduces the amount of design experi-ence needed for a good design. A second advantage of the optimization-basedapproach is that the design is formally optimized with regard to the design metrics,potentially leading to better designs, at least in terms of the design metrics. A thirdadvantage of the approach is that, since the engineer is out of the optimization loop,

DetailedAnalysis

DesignRevisions

DesignEquations

NumericalAnalysis

FinalDesign

Figure 1.1 A manual design process.

DetailedAnalysis

Objective Function

Optimization Engine

Figure 1.2 Optimization-based design process.

2 CHAPTER 1 OPTIMIZATION-BASED DESIGN

less engineering time is generally required. There are some disadvantages of theprocedure. First, the process can be numerically intense and require significantcomputing time, sometimes on the order of hours and, in extreme cases, days.Fortunately, computer time is significantly less expensive than engineering time.Second, the quality of the result depends upon the quality of the detailed analysis. Inthis regard, design experience is still valuable, though not as critical as in themanual design approach.

In order to utilize the optimization based design process, it is clearly necessaryto be able to optimize mathematical functions. For design purposes, we will beoptimizing the objective function, which we will also refer to as a fitness function.Optimization is a broad subject which has been the subject of a strong and sustainedinterest of a host of researchers over the years. The purpose of this chapter is tointroduce the subject to an extent sufficient to enable the reader to utilize anoptimization based design process for power magnetic devices. More thorough studyof optimization methods will serve every engineer well; for a good textbook devotedto the subject the reader is referred to Chong and Zak [1].

1.2 MATHEMATICAL PROPERTIES OFOBJECTIVE FUNCTIONS

Before discussing optimization algorithms, it is appropriate to discuss some propertiesof objective functions that are relevant to their optimization, as these propertiesdetermine the effectiveness of one optimization approach relative to another.

As we proceed to do this, note that throughout this work, scalar variables arenormally in italic font (for example, x) while vector and matrices are bold nonitalic(for example, x). Functions of all dimensionalities are denoted by nonitalic nonboldfont (for example, xq). Brackets in equations are associated with iteration number initerative methods.

In considering the properties of the objective function, it is appropriate tobegin by defining our parameter vector, which will be denoted x. The domain of x isreferred to as the search space and will be denoted W, which is to say that we requirex∈W. The elements of parameter vector x will include those variables of a designwhich we are free to select. In general, some elements of x will be discrete in naturewhile others will be continuous. An example of a discrete element might be one thatdesignates a material type from a list of available materials. A geometricalparameter such as the length of a motor would be an example of an elementthat can be selected from a continuous range. If all members of the parameter vectorare discrete, the search space is described as being discrete. If all members of thesearch space are continuous (in the set of real numbers), the search space is said tobe continuous. If the elements of x include both discrete and continuous elements,the search space is said to be mixed. It is assumed that the function that we wish tooptimize is denoted fx. We will assume that fx returns a vector of dimension mof real numbers, that is, fx∈Rm, where m is the number of objectives we areconsidering. For most of this chapter, we will merely consider fx to be amathematical function for which we wish to identify the optimizer of; however,

1.2 MATHEMATICAL PROPERTIES OF OBJECTIVE FUNCTIONS 3

in Section 1.9, and in the rest of this book for that matter, we will focus on how toconstruct fx so as to serve as an instrument of engineering design.

For this section, let us focus on the case where all elements of x are real numbersso that x∈Rn, whereRn denotes the set of real numbers of dimension n and where thenumber of objectives is one (i.e., m= 1) so that fx is a scalar function of a vectorargument. Finally, let us suppose we wish to minimize fx. A point x* is said to be theglobal minimizer of f over W provided that

fx*£ fx ∀x∈Wfx*g (1.2-1)where∀ is read as “for all” andWfx*g denotes the setW less the point x*. If the £ isreplaced by

for any two points xa; xb ∈Q, then Q is convex. This is illustrated in Figure 1.4 forQ⊂R2.

In order to determine if a function is convex, it is necessary to consider itsepigraph (Figure 1.5). The epigraph of fx is simply the set of points greater than orequal to fx. A function is considered convex if its epigraph is a convex set. Note thatthis set will be in Rn+1, where n is the dimension of x.

If the function being optimized is convex, the optimization process becomesmuch easier. This is because it can be shown that any local minimizer of a convexfunction is also a global minimizer. Thus the situation shown in Figure 1.3(b) cannotoccur. As a result, the minimization of continuous convex functions is straightforwardand computationally tractable.

1.3 SINGLE-OBJECTIVE OPTIMIZATION USINGNEWTON’S METHOD

Let us consider a method to find the extrema of an objective function fx. Let usfocus our attention on the case where fx∈R and x∈Rn. Algorithms to solve thisproblem include gradient methods, Newton’s method, conjugate direction methods,quasi-Newton methods, and the Nelder–Mead simplex method, to name a few. Let usfocus on Newton’s method as being somewhat representative.

(b) nonconvex set(a) convex set

x1 x1

x2 x2

x (x ,x )a a,1 a,2 x (x ,x )a a,1 a,2

x (x ,x )b b,1 b,2x (x ,x )b b,1 b,2

Θ Θ

Figure 1.4 Definition of a convex set.

(a) convex function (b) nonconvex function

x x

f(x) f(x)epigraph of f(x) epigraph of f(x)

Figure 1.5 Definition of aconvex function.

1.3 SINGLE-OBJECTIVE OPTIMIZATION USING NEWTON’S METHOD 5

In order to set the stage for Newton’s method, let us first define some operators.The first derivative or gradient of our objective function is denoted rfx and isdefined as

rfx= ∂fx∂x1

∂fx∂x2

∙ ∙ ∙∂fx∂xn

� �T(1.3-1)

The second derivative or Hessian of fx is defined as

Fx=

∂2fx∂2x1

∂2fx∂x2∂x1

∙ ∙ ∙∂2fx∂xn∂x1

∂2fx∂x1∂x2

∂2fx∂2x2

∙ ∙ ∙∂2fx∂xn∂x2

..

. ...

O ...

∂2fx∂x1∂xn

∂2fx∂x2∂xn

∙ ∙ ∙∂2fx∂2xn

2666666666666664

3777777777777775

(1.3-2)

If x* is a local minimizer of f, and if x* is in the interior ofW, it can be shown that

rfx*= 0 (1.3-3)and that

Fx*³ 0 (1.3-4)Note that the statement Fx*³ 0 means that Fx* is positive semi-definite, which isto say that yTFx*y³ 0∀y∈Rn. This second condition verifies that a minimumrather than a maximum has been found. It is important to understand that theconditions (1.3-3) and (1.3-4) are necessary but not sufficient conditions for x* tobe a mimimizer, unless f is a convex function. If f is convex, (1.3-3) and (1.3-4) arenecessary and sufficient conditions for x* to be a global minimizer.

At this point, we are posed to set forth Newton’s method of finding functionminimizers. This method is iterative and is based on a kth estimate of the solutiondenoted xk. Then an update formula is applied to generate a (hopefully) improvedestimate x k+ 1 of the minimizer. The update formula is derived by firstapproximating f as a Taylor series about the current estimated solution x k . Inparticular,

fx= fx k +rfx k x − x k + ∙ ∙ ∙+

12x−x k TFx k x − x k +H (1.3-5)

where H denotes higher-order terms. Neglecting these higher-order terms and findingthe gradient of fx based on (1.3-5), we obtain

rfx ≈ rfx k + Fx k x − x k (1.3-6)

6 CHAPTER 1 OPTIMIZATION-BASED DESIGN

From the necessary condition (1.3-3) we next take the right-hand side of (1.3-6) andequate it with zero; then we replace x, which will be our improved estimate, withx k+ 1 . Manipulating the resulting expression yields

x k+ 1 = x k − Fx k −1rfx k (1.3-7)which is Newton’s method.

Clearly, Newton’s method requires f ⊂ C2, which is to say that f is in the set oftwice differentiable functions. Note that the selection of the initial solution, x 1 , canhave a significant impact on which (if any) local solution is found. Also note thatmethod is equally likely to yield a local maximizer as a local minimizer.

Example 1.3A. Let us apply Newton’s method to find the minimizer of the function

fx= 2x1−24 + 3ex2−x12 + 8 (1.3A-1)This example is an arbitrary mathematical function; we will consider how to constructfx so as to serve a design purpose in Section 1.9. Inspection of (1.3A-1) reveals thatthe global minimum is at x1 = 2 and x2 = ln2. However, let us apply Newton’smethod to find the minimum. Our first step is to obtain the gradient and the Hessian.From (1.3A-1) we have

rfx= 8x1−23 − 6ex2 − x1

6ex2 − x1ex2� �

(1.3A-2)

and

Fx 24x1−22 6 − 6ex2

− 6ex2 12e2x2 − 6x1ex2

� �(1.3A-3)

Let us arbitrarily take our initial estimate of the solution to be x 1 = 0 0 T .Table 1.1 lists the numerical results from the repeated application of (1.3-7). Ascan be seen, during the first three iterations, the value of the function decreasesrapidly. However, then the rate of reduction of the function slows. Observe that onthe 10th iteration the value of the objective function is the minimum value to threesignificant digits, though there is still some discrepancy in the estimate of the

TABLE 1.1 Newton’s Method Results

k x k fx k rfx k Fx k 1 0

0

� �43 − 70

6

� �102 − 6− 6 12

� �2 0:677

− 0:162

� �14.2 − 19:6

0:888

� �48:0 − 5:10− 5:10 5:23

� �3 1:11

0:0938

� �9.25 − 5:53

− 0:0938

� �24:9 − 6:58− 6:58 7:13

� �10 1:95

0:666

� �8.00 − 2:23 ? 10− 3

− 2:00 ? 10− 3

� �6:07 − 11:7− 11:7 22:7

� �

1.3 SINGLE-OBJECTIVE OPTIMIZATION USING NEWTON’S METHOD 7

minimizer. In this problem, the minimum is quite shallow, which reduces the speed ofconvergence.

Newton’s method can be extremely effective on some problems, but proveproblematic on others. For example, if fx is not twice differentiable for some x,difficulties arise since Newton’s method requires the function, its gradient, and itsHessian. Many optimization methods require similar information and share similardrawbacks. There are optimization methods that do not require derivative informa-tion. One example is the Nelder–Mead simplex method. Even so, this algorithm canstill become trapped at local minimizers if the function is not convex.

One feature that makes these methods susceptible to becoming trapped at a localminimum is that they take the approach of starting with a single estimated solution andattempt to refine that estimate. If the single estimate is close to a local extrema, it willtend to converge to that extrema. There is another class of optimization methods thatare not based on a single estimate of the solution but on a large number (a population)of estimates. These population-based methods are not as susceptible to convergenceto a nonglobal local extrema because there are a multitude of candidate optimizers.

Genetic algorithms are a population-based optimization algorithm that hasproven very effective in solving design optimization problems. Other population-based optimization methods, such as particle swarm optimization, have also been usedsuccessfully. While one can engage in a lengthy debate over which algorithm issuperior, such a debate is unlikely to be fruitful. The focus of this text is on posing thedesign problem as a formal optimization problem; once the problem is so posed, anyoptimization algorithm can be used. A discussion of genetic algorithms is includedherein in order to provide the reader with a background in at least one method that canbe used for the optimization process.

1.4 GENETIC ALGORITHMS: REVIEW OFBIOLOGICAL GENETICS

In this section, we will set the stage for the use of genetic algorithms (GAs) asoptimization engines by reviewing some principles of biological genetics. All livingthings have a set of instructions on how they are constructed. These instructions arewritten in the deoxyribonucleic acid (DNA) contained within each cell of that livingbeing. The structure of this molecule was first determined by James Watson andFrancis Crick in the 1950s and is depicted in Figure 1.6. Therein, the horizontalstrands are made of phosphate and a sugar called deoxyribose. These strands are

A

T A

T

A

T

A

TG

C G

C{

gene

Figure 1.6 Deoxyribonuleic acid(DNA).

8 CHAPTER 1 OPTIMIZATION-BASED DESIGN

wound into a helix structure. The short vertical dashed lines in Figure 1.6 indicateweak hydrogen bonds that are instrumental in the duplication of DNA. The letters Aand G stand for adenine and guanine, respectfully, which are compounds known aspurines. The letters T and C designate thymine and cytosine, which are pyrimadines.The combinations AT, TA, GC, and CG form a four-letter alphabet. A sequence ofletters from this alphabet forms a gene of a living being. In terms of our discussion ondesign, we may view the gene as a design parameter of a living organism.

Each DNA molecule in a living organism is known as a chromosome. Livingorganisms generally have multiple chromosomes. For example, humans have 46chromosomes per cell. These chromosomes are arranged into 22 pairs (one of eachpair contributed by the father and one of each pair contributed by the mother). Inaddition, there are the two sex chromosomes denoted X and Y. In humans and manyother organisms, the existence of chromosomes in pairs leads to dominant andrecessive genes, as discovered by Gregory Mendel, a Roman Catholic monk andbotanist who studied the propagation of traits in pea plants. However, not all livingcreatures have chromosomes organized in pairs; ants, wasps, and bees are haploid andhave only one occurrence of each chromosome, while strawberries are octaploid witheight occurrences of each chromosome. This provides something to contemplatewhile eating strawberry pie.

In addition to some understanding of genes, chromosomes, and DNA, it isimportant to consider sexual reproduction and, in particular, the formation of gametes(sperm and egg cells). The formation of gametes is through a process known asmeiosis, which is illustrated in Figure 1.7. Let us consider a diploid organismwith twopairs of chromosomes, a long set and short set as shown in Figure 1.7. This is consistent

(a) lengthening (b) replication (c) pairing

(d) possible chromosome distributions in gametes without crossover

(e) additional possible chromosome distributions in gametes with crossover

Figure 1.7 Meiosis.

1.4 GENETIC ALGORITHMS: REVIEW OF BIOLOGICAL GENETICS 9

with chromosomes in cells that vary in length. The production of gametes begins withthe chromosomes lengthening out within the cell as shown in Figure 1.7(a). Note thereare two copies of this chromosome, one contributed by the father (darkly shaded) andone contributed by the mother (lightly shaded). Next replication of the chromosomesoccurs as seen in Figure 1.7(b). The two copies of a chromosome are referred to aschromatids, and they are connected at a point called the centromere. At this point,meiosis starts with the pairing of the chromosomes given by mother and father. In thispairing process, it is possible for the arms of the chromosomes to interchange, therebyleading to a new chromosome that consists of some genes from the father and somegenes from the mother. Note that the crossover point is normally between genes; this isbecause of the large amount of (apparently) nonfunctional DNA inmost chromosomes.While one crossover of one chromosome is shown, multiple crossovers in allchromosomes are possible.

After the chromosome pairing and crossover, the cell, which now contains fourversions of each chromosome, splits into four cells. The four chromosomes segregateinto these four cells. If crossover did not occur, this two-chromosome organism couldproduce four genotype (genetically unique) gametes; thus a single mother–father paircould produce sixteen genotypes. However, because of the crossover, many addi-tional genotypes of gametes can be produced. In fact, because of crossover the numberof genotypes that can be produced becomes related to the number of genes, not just thenumber of chromosomes. Thus, crossover is very important in achieving geneticdiversity.

Of course, in the case of humans, chromosome distribution alone yields 23 pairsof chromosomes, yielding 223 = 8; 388; 608 genotypes for the gametes. A set ofparents could thus produce 8,388,6082 genetically different children, which wouldseem to be an impressively diverse set, even without crossover. However, in artificialgenetic algorithms, the number of chromosomes is much smaller, often consisting of asingle chromosome.

Beyond increasing the sheer number of genotypes of the gametes, crossoverplays another critical role because it allows beneficial genes (traits) on a givenchromosome to be decoupled from detrimental genes (traits). Crossover will play avery important role in the operation of genetic algorithms.

In addition to gamete diversity due to genetic crossover and chromosomalsegregation into gametes, additional diversity is brought about because of mutation.Mutation arises from errors in copyingDNA. Inmitosis, or cell division, mutation oftenhas little effect, since the mutated cell will often die. However, in meiosis, mutation canhave a significant impact since the mutated genetic code will propagate to the geneticcode of every cell in the child. Even then, many mutations are not noticeable becausethey are a part of the genetic code that is unused.Whenmutation has a noticeable effect,it is generally for the worse. However, occasionally beneficial mutations occur whichimprove the ability of an individual (and eventually a species) to survive.

A final concept from biology that will serve our needs for an optimizationengine is the idea of natural selection and the survival of the fittest, an idea stemmingfrom Charles Darwin’s voyages of the H.M.S Beagle, during a period of time roughlycontemporary with the work of Mendel and the American Civil War. The idea that themost fit individuals of a population survive to reproduce is directly used in genetic

10 CHAPTER 1 OPTIMIZATION-BASED DESIGN

algorithms. These algorithms are based on an explicit fitness function, which will beused to determine which individuals “survive” and will be placed into a mating pool.

Clearly, the discussion in this section is at a high level and has been greatlysimplified. The interested reader is referred to Crow [2] for a more thoroughintroduction to the topic.

1.5 THE CANONICAL GENETIC ALGORITHM

A century after the work of Mendel and Darwin, but a mere decade after the work ofWatson and Crick, John Holland, a professor at the University of Michigan, proposedusing the principles of biological genetics as a computation algorithm for optimiza-tion, a concept termed a genetic algorithm or GA [3]. In this section we will begin ourconsideration of genetic algorithms with a canonical genetic algorithm similar toHolland’s original vision.

Genetic algorithms are quite different from traditional optimization algorithms.First of all, GAs operate not on the argument of the function being optimized, but ratheron an encoding of the argument. Second, rather than iterating to improve an estimate foran optimizer, GAs iterate to improve a large number of different estimates of theoptimizer. This collection of estimates will be referred to as a population. The use of apopulation of estimated solutions improves the chances of finding a global optimum.Third, GA operations are based only on the values of the objective function—gradientsand Hessians are not used, nor even estimated. This property is useful in function withdiscontinuities or with a discrete or mixed search space. Finally, GA operations arebased on probabilistic rather than deterministic computations.

The first concept that must be set forth in a genetic algorithm is that it, likeevolution, operates on a population, not on an individual. We will denote thepopulation within the genetic algorithm as P k , where k is the generation number.The kth generation consists of a number of individuals, that is,

P k = fq1; q2; ∙ ∙ ∙ qNpg (1.5-1)where qi is the genetic code for the ith individual in the kth generation of thepopulation and where Np denotes the number of individuals in the population, whichshould be an even number. The genetic code for the ith individual may be organized as

qi =

chromosome 1qi1qi2

(

chromosome 2

qi3qi4qi5

8>><>>:...

chromosome Nc qiNgn

266666666666666664

377777777777777775

(1.5-2)

1.5 THE CANONICAL GENETIC ALGORITHM 11

where Nc is the number of chromosomes, Ng is the number of genes, and qij is the jthgene of the ith individual (and it is understood that that we are referring to the kthgeneration). Eachgene is a string sequence.Recall thatDNAconsists of an alphabetAT,TA, CG, and GC. In the case of the canonical genetic algorithm, the string is mosttypically a binary sequence.Thus,qi takes the formof a binary number. The significanceof the chromosome organization will come into play when we consider reproduction.

The fact that the genes are encoded results in a limitation of the domain of theparameter vector. In other words, the domain of possible values of each element of theparameter vector is inherently limited. In some cases, this property is very convenient,but in other cases this limitation on the domain of the parameter vectors isdisadvantageous.

Associated with the genetic code for each population member, we will have adecoding function that translates the genetic code into a parameter vector. Inparticular,

xi = dqi (1.5-3)where xi is the parameter vector of the ith member of the population and is structured as

xi =

xi1

xi2

..

.

xiNg

26666664

37777775 (1.5-4)

As can be seen, xi has one element (denotedwith a subscript) for each gene.However, itis not partitioned into chromosomes.

Based on the parameter vector of ith population member, the objective functioncan be evaluated. In particular,

f i = fxi (1.5-5)In the case of a genetic algorithm, the objective function is referred to as a fitnessfunction. It will be used in a “survival of the fittest” sense to determine whichmembers of the population will mate to form the next generation. In the context of agenetic algorithm, fitness is viewed in a positive sense, thus it is assumed that wewish to maximize the fitness function. Fortunately, it is a straightforward matter toconvert between maximization of a function and the minimization of a function. Atthis point, we have enough background to discuss the computational aspects of aGA. However, before doing this, it is appropriate to briefly pause in our develop-ment and consider an example.

Example 1.5A. Suppose the 13th member of the population has the genetic code

q13 =0 1 0|fflfflfflfflffl{zfflfflfflfflffl}

gene 1

1 1 0 1|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}gene 2

0 1 1|fflfflfflfflffl{zfflfflfflfflffl}gene 3

� �T(1.5A-1)

12 CHAPTER 1 OPTIMIZATION-BASED DESIGN

The decoding algorithm is on a gene-by-gene basis and is of the form

xi = xmn;i +xmx;i − xmn;i

2l − 1

Xlm= 1

bm2m−1 (1.5A-2)

where l is the number of bits of the lth gene, m is an index ranging from leastsignificant bit to most significant bit for a given gene, and xmn;i and xmx;i are theminimum and maximum values of the ith element of the parameter vector. For theproblem at hand, we assume xmn;1 = 5, xmx;1 = 10, xmn;2 = − 2, xmx;2 = 0, xmn;3 = 0,and xmx;3 = 1. In Section 1.9 we will formally consider the construction of fitnessfunctions, and in Section 1.10 we will consider an engineering example. However, forthe moment we will assume a purely mathematical fitness function given by

fx= 1x1−12 + x2+22 + x23 + 1(1.5A-3)

Our goal is to compute the fitness of the 13th member of the population.The solution to this problem is straightforward. Using (1.5A-2), we obtain

x131 = 6:43, x132 = − 0:267, and x

133 = 0:429. Substitution of these values into (1.5A-3)

yields f 13 = 0:0297. Note that while the division of the bits of (1.5A-1) into genes isvery important to determine the fitness, the organization of genes into chromosomes isirrelevant for fitness evaluation and so has not been specified in this example.

At this point, we can now consider the primary aspects of a genetic algorithm.These are illustrated in Figure 1.8. Therein, the first step is initialization that yields aninitial population denoted P 1 . Next, the fitness of every member of the population isevaluated. Based on the fitness, a mating pool M k is determined by the selectionprocess. The individuals in this population will mate and genetic operators such ascrossover, segregation, and mutation will be used to produce children who will formthe next generation P k+ 1 . A stopping criterion is then checked; this can be as

M[k]

P[k+1]

P[k]No

k=k+1

InitializationP[1]

Fitness Evaluationi i i ix =d(θ ) f =f(x )

Selection

Mating:Chromosome Crossover, Segregation,

and Mutation

Yes

Stopping Criteria Met ?

Select Solutionii=argmax(f )

* ix =x

Figure 1.8 Canonicalgenetic algorithm.

1.5 THE CANONICAL GENETIC ALGORITHM 13

simple as checking a generation number. Once the stopping criterion is met, thealgorithm concludes by selection of the most fit individual of the final population to bethe optimizer. This process is implemented by the argmax() operator which returns theargument that maximizes its objective (which, in this case, is carried about byinspection of the finite population).

It is nowappropriate to consider each of these operations inmore detail, beginningwith the initialization step. The genetic code of every member of the population isinitialized at random. This yields an initial population of designs, denotedP 1 . The nextstep in the algorithm is to compute thefitness of everymember of the population. This isaccomplished by applying (1.5-3) and (1.5-5) to every member of the population.Example 1.5A illustrates this step for a single member of the population.

The next step in the process is selection. In this step, members of the populationP k are placed into the mating poolM k . Two algorithms to do this are roulette wheelselection and tournament selection. In both methods, the mating pool is initiallyempty and is filled one member at a time by repeatedly applying the selectionmechanism. In roulette wheel selection, members of the population are drawn into themating pool with a probability proportional to their fitness. In particular, theprobability of individual i being drawn into the mating pool on a given draw isgiven by

pi =f iPNp

i= 1f i

(1.5-6)

When applying this particular algorithm, it is important that the fitness function beconstructed so that f i ³ 0. If this is not the case, it is possible to scale/adjust the fitnessso that the condition is satisfied (for example, by adding a constant). Note that once apopulation member has been copied to the mating pool, it is not removed from thepopulation. Thus, it can be copied to the mating pool multiple times.

In n-way tournament selection, n members of the population are selected atrandom, and the member of this subset with the highest function is put into the matingpool. Here again, the member placed into the mating pool is not removed from thepopulation. In tournament selection, there is no restriction on the range of the fitnessfunction, which provides a slight simplification.

The next step in process is mating, which is comprised of chromosomecrossover, segregation, and mutation. In this step, pairs of parents are used to createpairs of children. Pseudo-code for this step appears in Table 1.2. Therein, underlinedtext denotes comments. Referring to the pseudo-code, q p1 and q p2 denote the geneticcode from parents taken from the mating pool. A crossover operator is applied to thecodes to form intermediate genetic codes qa1 and qa2. Crossover occurs at randompoints within a given chromosome. If it occurs, then all elements of the portion of thechromosome strings of the two parents past the crossover point are interchanged. Thisis similar to biological crossover but not identical: In the case of biological crossover,this process occurs in the formation of the gametes. The net result is the same. Next,the chromosomes of qa1 and qa2 are randomly segregated to form the next stage of

14 CHAPTER 1 OPTIMIZATION-BASED DESIGN