ECONOMICS 
 
 OF 
 
 ELECTRICAL DISTRIBUTION 
 
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ECONOMICS 
 
 OF - : ' 
 
 ELECTRICAL DISTRIBUTION 
 
 BY 
 
 P. O. REYNEAU, M.E. 
 
 MEMBER A.I.E.E. 
 ASSISTANT ELECTRICAL ENGINEER, THE DETROIT EDISON COMPANY 
 
 AND 
 
 H. P. SEELYE, B.C.E. 
 
 A8SO. MEMBER A.I.E.E. 
 DISTRIBUTION ENGINEERING DEPARTMENT, THE DETROIT EDISON COMPANY 
 
 FIRST EDITION 
 
 McGRAW-HILL BOOK COMPANY, INC. 
 NEW YORK: 370 SEVENTH AVENUE 
 
 LONDON: 6 & 8 BOUVERIE ST., E. C. 4 
 1922 
 
- - - . 
 
 COPYRIGHT, 1922, BY THE 
 MCGRAW-HILL BOOK COMPANY, INC. 
 
 THE MAPLE PRESS YORK FA. 
 
PREFACE 
 
 Electrical Engineering is more than the application of the 
 principles of electricity to the design, construction and operation 
 of a machine, a power plant or a distribution system to the end 
 that it fulfills satisfactorily its intended purpose. It is highly 
 desirable that efficiency be also accomplished. Further than 
 this, it should be the aim of all engineers to develop efficiency in 
 its broader sense, that is, by the realization of maximum economy 
 in both design, construction and operation. 
 
 In designing, constructing, or operating an electrical distribu- 
 tion system, the object to be striven for is to provide all customers 
 with a good quality of service at the least possible cost over the 
 system as a whole. This result can be attained only through 
 a careful and conscientious application of the principles of 
 economics to all parts of the system. It is to define those 
 principles and to indicate methods for their application that 
 this book is presented. 
 
 This work has been prompted by the realization that the trans- 
 mission and distribution system, representing a large part of the 
 total investment in any central station and offering a wide field 
 for economic study has not often been given sufficient attention. 
 Much valuable information may be found scattered through the 
 engineering literature, but it is thought that there is need of 
 bringing together the factors involved in such a study and of 
 discussing the subject as a whole. 
 
 This book is by no means an attempt to cover the whole field 
 of economics as applied to the central station system. It is 
 well recognized that the range of problems encountered is very 
 broad and varied and that, as yet, comparatively little progress 
 has been made in such work. It would be impossible in a work of 
 this kind to cover in detail even the field of distribution and trans- 
 mission economics. The purpose of the book is to present the 
 need for the application of economic principles to the design of 
 distribution systems, to explain the fundamental principles in- 
 volved, to indicate the types of problems most often encountered 
 and to offer methods of studying such problems and reaching 
 their solution. 
 
 <: 
 
vi PREFACE 
 
 A considerable part of the material presented in this book 
 has appeared previously in the Electrical World. The chapter 
 on single-phase secondaries is a rearrangement of material 
 published in the Proceedings of the American Institute of Elec- 
 trical Engineers. 
 
 Merrill W. De Merit has furnished valuable assistance par- 
 ticularly in the chapters on Energy Cost and Underground 
 Lines. 
 
 DETROIT, MICH., THE AUTHORS. 
 
 Dec. 1921. 
 
CONTENTS 
 
 PAGE 
 
 PREFACE v 
 
 CHAPTER 
 
 I. INTRODUCTORY 1 
 
 II. APPLICATION OF ENGINEERING ECONOMICS TO TRANSMISSION AND 
 
 DISTRIBUTION PROBLEMS 5 
 
 III. COSTS 11 
 
 Principles underlying the determination of true costs 
 Formulas for unit labor costs Cost records Annual costs. 
 
 IV. ENERGY COST 22 
 
 Principles and methods involved in the determination of 
 the cost of energy and of energy losses. 
 
 V. LOAD CHARACTERISTICS 37 
 
 Power factor Balance factor Demand factor Diversity 
 factor Load factor Equivalent hours. 
 
 VI. GENERAL EQUATIONS 48 
 
 Kelvin's law General method of solving problems Presen- 
 tation of results. 
 
 VII. POWER Loss AND VOLTAGE DROP 55 
 
 Charts for simplified solution for power loss and voltage 
 drop. 
 VIII. TRANSMISSION LINE PROBLEMS "BACKBONE" TRANSMISSION 
 
 LINES 71 
 
 Method of determining most economical design for main or 
 "backbone" transmission lines. 
 
 IX. TRANSMISSION LINE PROBLEMS SECONDARY TRANSMISSION 
 
 LINES . . . . 81 
 
 Determination of most economical standards of construction, 
 conductor size, loading, route, etc. on lesser or secondary 
 transmission lines. 
 
 X. RECONSTRUCTION PROBLEMS 96 
 
 Principles involved in the solution of problems dealing with 
 the alteration or reconstruction of lines already built 
 Method of including value of salvaged material in cost study. 
 
 XI. POWER CIRCUITS 109 
 
 Problems relating to lines carrying power load chiefly 
 Voltage Economical conductor size Use of two lines in 
 place of one Distribution of load over several lines. 
 
 XII. LIGHTING CIRCUITS : . . 128 
 
 Economical studies on circuits carrying lighting only Pre- 
 diction of load Conductor size Increasing capacity of 
 overloaded systems. 
 
 vii 
 
viii CONTENTS 
 
 XIII. SECONDARY DISTRIBUTION SINGLE PHASE 139 
 
 Study of most economical design for secondaries Voltage 
 drop Conductor size Transformer size Length of 
 secondary. 
 
 XIV. POWER SECONDARIES 174 
 
 Power secondary vs. separate transformers Economical size 
 of 3</> secondaries. 
 
 XV. UNDERGROUND LINES 183 
 
 Voltage Cable size Route Number of ducts in a duct 
 line Arrangement of ducts and cables. 
 
 XVI. THE SYSTEM AS A WHOLE 194 
 
 XVII. INDUSTRIAL PLANT PROBLEMS 197 
 
 Application of industrial plant problems of the principles 
 of economics outlined for electrical distribution. 
 
 APPENDIX 201 
 
 INDEX. . 207 
 
ECONOMICS OF 
 ELECTRICAL DISTRIBUTION 
 
 PART I 
 
 CHAPTER I 
 INTRODUCTORY 
 
 Electrical engineering design has three necessary components, 
 i.e. electrical design, structural design and economical design. 
 They are equally important and no electrical engineering problem 
 can be solved properly without consideration of all three compo- 
 nents in a thorough manner. The solution of any problem will 
 come nearest to perfection as each of these forms of design is 
 more soundly analysed and the combination of the three more 
 intelligently applied. 
 
 Structural design calls for the study of materials used and their 
 combination into the desired structures in such a way that proper 
 factors of safety may be obtained. Electrical design calls for a 
 knowledge of electrical phenomena and the application of this 
 knowledge to achieving results satisfactory from an operating 
 standpoint. Economical design calls for the knowledge of costs 
 and their application to the determination of the most economical 
 design possible. It is evident that there must be overlapping 
 between these three fields. Structural design depends in most 
 cases on the size of electrical apparatus, conductors, etc., which 
 in turn is governed by the electrical design. Both structural 
 and electrical features, on the other hand, should be planned with 
 a view toward economy. Where there is a choice of more than 
 one possible design which is satisfactory from a structural and 
 an electrical standpoint, the decision should be based on study of 
 the relative economy of all the alternatives considered. It is pro- 
 posed in this book to consider electrical distribution from the 
 standpoint of economy principally. There is no intention 
 
 1 
 
2 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 however of minimizing the importance of structural and elec- 
 trical design. Economical design has probably received less 
 attention than the two other forms and the emphasis brought on 
 it here may help to give it its proper place in the work of obtaining 
 correct solutions to electrical distribution problems. 
 
 Scientific books in general may be divided into two classes. 
 The first class gives facts and data obtained from practical experi- 
 ence. The second class gives methods and suggestions of means 
 of attacking the study of certain problems. This book belongs 
 to the second class. In work with distribution systems the 
 conditions met with in different localities are diverse. Costs of 
 material and labor and methods of construction and operation 
 are varied and changeable. It follows that if definite figures 
 and results are given which might apply to one system at one 
 particular time, these same figures might be very misleading if 
 applied indiscriminately to any other locality, system or period of 
 time. It is necessary, therefore, that any figures given here, 
 particularly those referring to costs, should be considered only as 
 examples of the methods presented and not as material on which 
 to base any calculations. 
 
 Furthermore, there is no intention to cover fully either in 
 detail, or in a general descriptive manner all the problems, which 
 might be encountered in electrical distribution, viewed from the 
 economic standpoint. It is desired to present some general 
 methods that can be applied to the solution of most of these 
 problems, to give examples of the application of these principles 
 to some ordinary cases, and to indicate the types of questions 
 most often arising in this work. 
 
 The book is made up of two parts. The first part including 
 Chapters II to VII is intended to give working methods, or so to 
 speak, to furnish the tools to be used in solving electrical dis- 
 tribution problems economically. The authors have tried to 
 present the underlying principles which are the basis of economic 
 study of this kind. The fundamentals are not new but their 
 application to the design of a distribution system is not so 
 generally understood. The use of accurate annual costs instead 
 of first costs (or loose estimates) as a basis for determining the 
 most economical installation goes back to Lord Kelvin and 
 further. But, we see many engineers today ignoring the 
 economical factors of the problem entirely and others basing all 
 comparisons on first cost only, or attempting to apply the 
 
INTRODUCTORY 3 
 
 so-called Kelvin's Law indiscriminately. A knowledge of the 
 fundamentals is absolutely essential if any such study is to be 
 worth the time spent upon it. Otherwise the engineer may be 
 guilty of deluding himself and others with figures which are 
 entirely inapplicable to the case in hand. 
 
 Chapter II attempts to create a point of view regarding the 
 subject at hand. Then in succession are taken up means of 
 analyzing and tabulating costs of material and labor, of deter- 
 mining the cost of energy to be used under various conditions, 
 of studying the characteristics of loads to be handled, and of 
 forming general equations for solving problems in economical 
 design. Finally a chapter on power loss and voltage drop is 
 included to provide convenient means of handling these most 
 important electrical phenomena which must always be considered 
 in connection with economical design. 
 
 The second part of the book consists in presenting the applica- 
 tion of the methods, or the use of the tools described in the first 
 part. For convenience a division of the subject has been made 
 into transmission lines, power circuits, lighting circuits, second- 
 aries and underground lines. Under each heading some of the 
 general problems encountered are indicated and some particular 
 problems are solved in detail as definite examples. Special or 
 unusual problems have not been considered, the work for the 
 most part holding to the everyday questions met with in practice. 
 Chapter XVI in conclusion, touches on a few of the general prob- 
 lems which apply to the system as a whole, such as the location 
 of generating station and substations, the relation of one part of 
 the system to the remainder, etc. Chapter XVII takes up 
 briefly a few of the problems in distribution pertaining to 
 industrial plants. 
 
 It will be seen that the possibilities for study in the field of 
 distribution economics are boundless. The deeper one goes into 
 the subject the more problems present themselves and the more 
 evident becomes the need for careful and accurate investigation. 
 The objection may be raised that the changeable character of 
 the loads carried, and the fluctuations of prices make a too 
 detailed study of economies impracticable. That is no doubt 
 true sometimes, as regards individual problems. In the long 
 run, however, the knowledge gained is always valuable and of 
 great assistance in deciding general policies. As our information 
 on the behavior of materials and on the characteristics of the 
 various types of loads becomes more accurate, the practicability 
 
4 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 of the application of detailed economies will increase. With the 
 increasing demand for efficiency and economy in all lines of 
 endeavor, the necessity for the economical operation of distri- 
 bution systems becomes more imperative. In any case, an exact 
 knowledge of the true cost will always lead to efforts to reduce 
 that cost if possible. 
 
CHAPTER II 
 
 APPLICATION OF ENGINEERING ECONOMICS TO 
 TRANSMISSION AND DISTRIBUTION PROBLEMS 
 
 As stated in the first chapter, the purpose of this book is to 
 present some of the problems encountered in the design of lines 
 for transmitting electrical energy and to approach their solution 
 from an economic point of view. More specifically it is purposed 
 to apply engineering economics to distribution- and transmission- 
 line layouts indicating how the most economical installation 
 consistent with good service can be determined. 
 
 No argument should be required as to the advisability, or 
 rather necessity, for the application of economic principles to 
 engineering design. Engineering should make for efficiency but 
 there can be no real efficiency unless there also is accomplished 
 economy. Economy cannot often be recognized at first glance. 
 A working knowledge of its fundamentals at least is required. 
 Engineering and engineering economics are therefore synonymous 
 in all problems directly involved in the production or distribution 
 of a commercial commodity. This includes the great majority 
 of our present everyday engineering problems and naturally 
 those of transmission and distribution. 
 
 It is not within the scope of this book to consider unusual 
 problems such as very high-voltage transmissions for example, or 
 installations of special apparatus whose use has not become a 
 matter of accepted practice. Rather, the general purpose will 
 be to make a few analyses that will indicate methods of finding 
 the most economical design under usual conditions with material 
 of known characteristics and at known prices. It is especially 
 for the improvement of the design of the large class of everyday 
 jobs that this work is written. Such analyses will, naturally, 
 not only indicate the most economical installation with materials 
 and construction methods already in use but will make it possible 
 to determine where any economy can be effected by changes in 
 these methods or materials. It must be recognized that the 
 
 5 
 
6 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 majority of the transmission and distribution lines have been 
 laid out by rule-of -thumb method, with no definite conception 
 of actual costs or economy. The methods here presented have 
 been developed by the study of actual problems encountered in 
 practice. It is hoped that these methods will on the one hand 
 show the magnitude of the savings possible by the use of engineer- 
 ing economics and on the other hand be simple enough in their 
 application to be considered worth using in many places where 
 more empirical methods are used today. 
 
 METHOD OF TREATMENT 
 
 Practically all such problems of engineering design require 
 treatment in three distinct stages for a complete solution. At 
 first all the data must be obtained that can be determined with 
 exactness. Such elements as material costs, strength of parts of 
 construction, load tests, etc., are here included. Second, those 
 elements which are not subject to exact measurement or compu- 
 tation must be decided upon. These are usually a development 
 from practical experience over a number of years and are often 
 matters of accepted standard practice. They are such items as 
 probable increase in load, allowable voltage regulation, safety 
 factors, standards of construction, etc. Empirical methods must 
 here be employed. The third element is the most intangible 
 the good judgment of the engineer based on his knowledge and 
 experience and applied to the particular problem in hand to so 
 utilize the first two elements as to create an efficient and econom- 
 ical design best adapted to the situation. All these elements in 
 their proper proportions are equally important in attaining the 
 most satisfactory solution for any p'roblem.. Empirical methods 
 if applied blindly may lead far away from the purpose for which 
 they were originally intended. Judgment based on experience 
 alone is liable to become a mere guess unless supported by exact 
 knowledge and good practice. It would appear therefore, that 
 the further the exact data can be carried in a problem the less 
 dependence need be placed on the more intangible elements and 
 hence the greater certainty of the best solution. The extension 
 of this exact knowledge is the chief purpose of the study of the 
 economic features of a problem. In many cases economy may 
 be made the deciding factor between two or more designs appar- 
 ently equally good from other points of view. 
 
APPLICATION OF ENGINEERING ECONOMICS 7 
 
 Exact Data. For any given system there are a number of 
 limiting conditions which reduce the unknown factors in the 
 design. There are certain standards such as transformer sizes, 
 wire and cable sizes, etc., which are established by the manufac- 
 turer. Other limitations are established by the accepted good 
 practice of the profession in general. Also each company has 
 certain standards of materials and construction to which it is 
 wise to adhere unless there is a clear advantage in making a 
 change. The most complete information obtainable on all such 
 data relating to the particular system in hand should be kept 
 available for ready reference. Also other items which can be 
 obtained exactly for only the one locality or company such as 
 costs of material, labor and energy should be determined as 
 accurately as possible and be revised from time to time to con- 
 form to changing prices, wage scales and costs of production. 
 This information can be readily kept up to date and tabulated or 
 drawn up in curves. Then only such factors as are inherent to 
 the particular line itself need be determined in making the 
 design. 
 
 Empirical Data. Some of the empirical elements of the 
 problem of transmission and distribution of energy are worthy 
 of considerable attention. One of these is the element of "good 
 service." It is understood that in any design or layout the 
 greatest economy " consistent with good service" is the ob- 
 ject. Just what is good service may be a matter of considerable 
 question. 
 
 Good Service. Good service is not dependent on the ideas of 
 the engineer or the "server" as to what it should be. Service is 
 good when it satisfies the one that is served. That man who at 
 the receipt of his bill feels that he has gotten all that was due 
 him is receiving good service. He is willing to exchange his 
 money for what he has received from the server. 
 
 The human element is therefore the most important one in 
 determining good service. Customers will expect as good or 
 better operating conditions than they have been in the habit of 
 getting. Companies have, so to speak, educated their public 
 to certain expectations and they must live up to such expecta- 
 tions. It is evident however that it is impossible to eliminate all 
 interruptions or conditions of poor regulation. In thickly 
 populated districts with a high-load density, practically continu- 
 ous service at good voltage is easily given. In outlying, scattered 
 
8 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 districts the cost of the same quality of service would be pro- 
 hibitive. From the company's point of view the service must be 
 of good enough quality to insure earnings and yet not so good as 
 to necessitate too large expenditures. It is therefore necessary, 
 before attempting to solve a transmission or distribution problem 
 economically, to ascertain what quality of service is required 
 both from the customer's and the company's point of view. 
 
 Good service can therefore only be determined in a general 
 way and will vary with locality, company, kind of load, etc. 
 It might be said, however, that with the modern refinement of 
 methods and equipment and the nearly universal high require- 
 ment of the customer we are approaching more and more a 
 condition when wide variation of voltage and long or frequent 
 interruptions will hardly be permissible. 
 
 Increase in Load. A second quantity which must be empiri- 
 cally determined is that of load increase. The variation of the 
 loads to be carried and the growth of energy demand that is 
 prevalent in nearly all localities are always matters for consider- 
 able study in connection with a layout. There are some cases 
 where it is possible to design for a certain demand that can be 
 assumed to remain constant for a period of years, possibly for the life 
 of some part of the equipment. Such would be the case in planning 
 lines to a large power installation where growth could be taken 
 care of by new circuits. On the other hand such load as house 
 lighting increases continuously and its increase can only be esti- 
 mated from the figures for the past few years and general business 
 conditions. In any case we must design for a system that will 
 be most economical over its useful life, in other words when the 
 sum of the annual costs for all years under consideration will be a 
 minimum. It is further necessary to study as exactly as possible 
 the number of years to be cared for by the present design and 
 minor changes that can be made at various times during that 
 period for taking care of the changes in load. Generally the 
 design should cover the expected life of that part of the equipment 
 whose life is shortest and which would require a large expenditure 
 for its replacement. Any considerable change necessary to 
 care for larger load could then be made at the same time with 
 relatively smaller cost. In some cases, however, where the 
 probable increase in load is more definitely known, the economical 
 life of the present design can be accurately determined even 
 well within the expected life of all the important parts. 
 
APPLICATION OF ENGINEERING ECONOMICS 9 
 
 Financial Conditions. The condition of the money market 
 may be another determining factor in an economical design. 
 Often the difficulty in securing necessary funds for a job may 
 require the engineer to spend less at the time of installation than 
 is consistent with economy over the period the design covers. 
 However the condition should be carefully considered in order 
 to still obtain the best design as limited by the money available. 
 
 Relation of Parts to Whole System. The fact that the trans- 
 mission or distribution line is only one part of the system trans- 
 mitting energy from the turbine to the customer must be kept in 
 mind. Its design will affect and be affected by existing or 
 planned conditions in the other parts. It is therefore always 
 necessary to treat a line not only as an independent unit for 
 some specific purpose but also as a working part of the whole 
 system. 
 
 Bearing in mind the general considerations brought out above, 
 the table given below will show the most important points to 
 consider in studying the layout of an economical line. The 
 following chapters will discuss in more detail the methods of 
 obtaining this data and its application to particular classes of 
 problems such as transmission lines, power lines, secondaries, 
 etc. 
 
 1. Load 
 
 2. Route of line 
 
 Location. 
 
 Present size. 
 
 Probable increase and rate of increase. 
 
 Characteristics: Phase requirements. 
 
 Power factor. 
 
 Variation, daily, weekly, seasonal, etc., 
 and relation to station variations and 
 peak. 
 
 Unbalance factor. 
 
 Maximum. 
 
 Load factor. 
 
 Investigation of advisability of possible routes, affected 
 by : Available pole or duct space. 
 Purchase of right-of-way. 
 Difficulties in construction. 
 Interference with or from other lines, physically 
 
 or electrically. 
 Convenience of operation. 
 Possible future extensions. 
 Effect on system as a whole. 
 Cost as compared with other possible routes. 
 
10 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 3. Physical 
 constants 
 
 4. Costs 
 
 5. Relation to 
 System 
 
 Length of line 
 Spacing 
 Volfcage: desired, 
 
 available. 
 Resistance of line 
 Reactance 
 Voltage drop 
 Regulation: desired, 
 available 
 
 Under various line conditions as 
 limited 
 
 1. By good operation. 
 
 2. By standard practice. 
 
 3. By routing. 
 
 4. By economical considerations. 
 
 5. By good service. 
 
 1. Determination of 
 labor costs, 
 
 material costs, and | For all conditions considered 
 energy loss costs ( 
 
 2. Application of those costs to determine the best line for 
 the purpose. 
 
 General considerations that may affect most economical 
 line found above under (4) due to its being a part of a 
 large system. 
 Financial considerations. 
 
 From inspection of this table it is seen that the five divisions 
 in it are in one way or another interdependent so that each one 
 must be considered in relation to all the others. It is only by a 
 careful study of the problem from all angles and a coordination 
 of results that a satisfactory solution can be obtained. In this 
 book, however, the economic features of the design have been 
 emphasized rather than the mechanical or electrical. These 
 latter have been covered quite thoroughly in other works. 
 
CHAPTER III 
 COSTS 
 
 PRINCIPLES UNDERLYING THE DETERMINATION OF TRUE COSTS 
 FORMULAS FOR UNIT LABOR COSTS COST RECORDS 
 
 ANNUAL COSTS 
 
 The fundamental basis underlying the whole question of econo- 
 mic design is the accurate determination of costs, i.e., costs of 
 construction and cost of energy. If these are not correctly 
 determined any conclusions drawn from their use will have little 
 value. Further it is generally inadvisable to accept for this 
 purpose any cost data which has not been locally derived, as 
 every power company has its own methods and standards of 
 construction, its own labor costs, its own efficiencies of operation. 
 These may differ widely. Hence, it is absolutely necessary 
 to make as complete a determination of local costs as possible as 
 a basis for any economic study. 
 
 Evidently when it is decided to establish a cost record on con- 
 struction, material and labor, difficulties will be encountered in 
 obtaining correct information regarding all details. Much 
 valuable information will be found in appraisals and much can 
 be gotten from the accounting records by a skillful economist. 
 However, in most cases it will take several years to establish a 
 complete record and special studies will be necessary. This 
 record, when obtained, is an easy thing to keep up to date and 
 in convenient form for use. In the meantime certain make- 
 shifts will be necessary. If a complete record of unit construc- 
 tion and operating costs is not obtainable or when lack of time 
 or of available records requires any item of cost to be estimated, 
 the estimate should be based on known conditions as far as 
 possible. Further than this there should be determined by what 
 percentage the final result will be affected by any reasonable 
 variation in the assumed cost. For example, the cost of energy 
 at any point may be assumed to be 1 ct. per kilowatt-hour. If 
 this is not an accurately determined figure however, it would be 
 well to determine how the result would be affected if the cost of 
 energy were say % ct. or 1J4 cts. per kilowatt-hour. Such a 
 comparison will at least place the solution of the problem within 
 definite limits. 
 
 11 
 
12 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 It is a generally recognized fact that the computation of first 
 cost is only a step in the investigation of the real cost of a proposi- 
 tion. The economy in comparison with any other alternative 
 proposition can only be determined when the annual cost is 
 investigated. The annual cost must include all annual charges 
 against the investment itself and all operating costs, maintenance, 
 repair and energy losses. Of course annual costs cannot always 
 be considered the determining factor, since on items involving 
 large expenditure the possible difficulty of obtaining capital 
 may have considerable weight. Usually, however, annual cost 
 may be accepted as the criterion. 
 
 Before annual costs on a piece of property such as a trans- 
 mission line can be obtained it is necessary to determine its total 
 value or first cost. 
 
 First Cost. The determination of cost figures for general use 
 is greatly facilitated if line construction, materials and methods 
 are standardized. This allows a fairly accurate figure to be 
 obtained for standard units as per pole, or per 1,000 ft. or per 
 mile for any type of construction. Otherwise smaller units must 
 be depended upon such as per crossarm, per insulator, per 100 
 ft. of wire, etc. In any case it will be found exceedingly valuable 
 to have as complete a record as possible of itemized costs from 
 the smallest part, such as a bolt, up to an average cost for a large 
 assembled unit such as per mile of line. This should include 
 both material and labor costs with overhead expense all shown 
 separately. Provision should also be made for easy revision of 
 costs as prices change. This revision should be frequently made 
 when prices of labor and material are fluctuating to any extent 
 since a large part of economic study deals with possible new 
 construction which will be at present or future prices or with old 
 construction which represents a value equal to new construction 
 less a certain percentage for its age. Certain types of problems 
 require the consideration of the actual cost of the old construction 
 at the time it was installed but these are not so general. 
 
 There are a number of items which must be included in addi- 
 tion to current quotations on material or actual labor time in 
 erecting. To material price may be added such items as freight, 
 treating material for poles, tie wires on line, etc. To actual 
 unit labor costs should be added a proportional amount for 
 unoccupied time, rainy days, vacations, transportation 
 inspection, etc. For example: 
 
COSTS 
 
 13 
 
 
 Weight 
 
 Cost at 
 
 Freight 
 
 Injury 
 
 Ties 
 
 Total 
 
 Per cent 
 
 
 per 
 
 21 cts. 
 
 on re- 
 
 to reels 
 
 per mile 
 
 per mile 
 
 of wire 
 
 
 mile, 
 
 per 
 
 turned 
 
 
 
 
 cost 
 
 No. 2 Solid wire 
 
 one 
 
 pound 
 
 reels 
 
 
 
 
 
 
 wire, 
 
 
 
 
 
 
 
 
 pounds 
 
 
 
 
 
 
 
 
 1,066 Ib. 
 
 $223.86 
 
 1.86 
 
 2.32 
 
 3.63 
 
 $231.67 
 
 103.4 
 
 Naturally these additional items vary for each unit and each 
 locality. It is usually possible to obtain a percentage which 
 may be added to actual price to include such incidental material. 
 Loading. The item of overhead expense or loading is one 
 which must be included in nearly all problems involving costs. 
 There are various methods of applying this loading but for the 
 purpose under consideration the method of unit loading is 
 probably the most satisfactory, i.e., apportioning to each indi- 
 vidual item its pro rata cost for different items of overhead 
 expense, both on material and labor. It is evident, for example, 
 that the item of breakage will be greater for insulators than for 
 wire. The items which may legitimately be included in loading 
 are as follows: 
 
 Waste end trimmings of wire, cable cut back for splicing, incompetent 
 labor, etc. 
 
 Loss and breakage theft, broken insulators, etc. 
 
 Tool expense tools used up, broken, stolen and repairs and depreciation 
 on them. 
 
 Direct supervision engineering, heads of departments, general foremen, 
 office expense, clerks, stenographers. 
 
 Injuries and damages doctor and hospital expense, liability, insurance, 
 tc. 
 
 Purchasing expense all expenses of purchasing department. 
 
 Stores and supplies expense all expenses of stores department and 
 handling. 
 
 The percentage to be applied to any unit to cover any one of 
 the above items will depend entirely on local conditions and no 
 figures could be given here which would be of any value. The 
 total loading percentage is usually between 10 and 25 per cent 
 depending upon the class of property and upon local conditions. 
 
 As will be seen later all problems do not require a detailed 
 application of loading. Many problems involving the compari- 
 son of relatively small amounts of construction may be safely 
 considered from the point of actual material and labor costs 
 
14 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 alone, considering overhead to be equal in both cases. The 
 question of whether or not to apply loading must be determined 
 by the conditions involved in the particular problem under 
 consideration. In general where good cost records are kept 
 it will be just as easy and more accurate to include it in all 
 problems. 
 
 Labor costs on any unit may usually be reduced to an equiva- 
 lent formula of man hours for various classes of labor as foreman, 
 lineman, groundman, etc., for each unit plus a percentage for 
 incidental expenses as above. Such formulas facilitate revision 
 of prices when necessary as the current wages may be substituted 
 and the labor cost on the unit easily obtained. 
 
 For example, if a gang consisting of 1 foreman, 1 truck, 1 
 chauffeur, 4 linemen and 3 groundmen can string 1.6 miles of 
 No. bare stranded wire per day, on an average, the cost per 
 mile of wire will be .728 times the cost of the gang per day 
 
 + IT + ICh + 4L + 3Gr) 
 
 where the symbols represent the daily wage of the various 
 classes of labor included, foreman, truck, chauffeur, linemen and 
 groundmen. 
 
 Some of the labor formulas are more complicated but all are 
 computed on the same basis. For poles, for example, several 
 different gangs are included in the labor charge, i.e., unloading, 
 distribution, framing and roofing, digging hole, and setting and 
 the gangs for each operation may be different for different sizes 
 of poles. A formula similar to the above may be determined for 
 each operation and the total labor formula for the pole is a 
 composite of these. For example: 
 
 if Gi represents the daily cost of the unloading gang =(17V -f ICh + 3GV), 
 
 Ga represents the daily cost of the distributing gang =(17V + ICh + 3(?r), 
 
 Gi repres nts the daily cost of framing and roofing =(HL + 2Gr), 
 
 Gi represents the daily cost of the digging gang = (H F + MTr + MCh + IGr), 
 
 G& represents the daily cost of the setting gang =( 6 <,F-\-%Tr+tfCh+5Gr) 
 
 (The H f, % Tr, etc., are occasioned by the fact that one foreman and 
 one truck serve several gangs at one time.) 
 
 Then the total labor cost on one 35-ft. pole equals 
 
 .OlGOGj + .1403G 2 + .063G 3 + .2243 4 + .0468G 5 
 
 Cost Records. For a complete record of line costs there are 
 necessary the following items: 
 
COSTS 
 
 15 
 
 (a) Current prices of material and labor of all kinds. 
 (6) Labor formulas and constant multipliers for various classes of 
 materials with loading percentages for both. 
 
 (c) Current material and labor costs on units of construction, as per cross- 
 arm, per pole, per insulator, etc. 
 
 (d) Current costs on assemblies. Assemblies may range from small items 
 such as a crossarm erected with braces, bolts, etc., or a ground connection 
 with wire, ground rod, wood moulding, etc., up to large items such as cost 
 per mile of transmission line on cost of a railroad crossing, etc. 
 
 Some examples from such a record are given in the following: 
 
 (a) Prices (at warehouse) 1 
 
 No. 3 Porcelain insulators $.16 each 
 
 No. 2 T. B. W. P. wire 2348 per pound 
 
 Primary fuse boxes 3 . 94 each 
 
 30-ft. 6-in. pole, rough 7 . 28 each 
 
 etc 
 
 (b) Labor formulas. 
 
 Wire stringing (single wire) Gang = (IF + 17V + ICh + 4L + 3Gr) = G 
 
 
 Miles per 
 day 
 
 Labor cost 
 per mile 
 without 
 loading 
 
 Plus 16.55 
 per cent 
 for loading 
 
 No. 6 Solid 
 
 2 8 
 
 3575G 
 
 4165G 
 
 No. Stranded 
 
 1 6 
 
 625(7 
 
 728G 
 
 
 
 
 
 Ground connections Gang = (17V + 
 
 23,000- volt ground connections 
 
 + 3L + IGr) = G 
 
 Labor cost 
 
 each without Plus 18.75 per 
 loading cent 
 
 .02075G .0248G 
 
 Anchors and guys Gang = (2/7F + 2/77V + 2/7Ch + 2L + IGr) = G 
 
 Pole to pole one % in ... 
 Pole to anchor two % in . 
 Stub to anchor one % in 
 
 Labor cost Plus 17.65 per 
 
 each without 
 loading 
 .0833G 
 . 2833G 
 .25G 
 
 cent for 
 loading 
 .098 
 .3333G 
 .2941 
 
 1 The figures given below must not be taken as representing current 
 prices. They are given for example only. 
 
16 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Crossarms Gang = 1/3F + 1/37Y + l/3Ch + L + Gr) =G 
 
 Labor cost 
 
 each without 
 
 loading 
 
 23,000-volt, single-arm, 36 in 
 
 23,000-volt, double-arm, 64 in 
 
 23,000-volt, double-arm, 64 in. (strain 
 insulators, both ways) etc 
 
 (c) Unit costs. 
 
 .131G 
 
 .138G 
 
 Plus 
 
 17.65 per cent 
 for loading 
 .047G 
 . 154G 
 
 .162G 
 
 
 
 Plus 
 
 
 Plus 
 
 
 
 Mate- 
 rial 
 
 per 
 cent. 
 
 Labor 
 
 per 
 cent 
 
 Total 
 
 
 
 loading 
 
 
 loading 
 
 
 3 No. 6 Secondary, 1,000 ft 
 
 $99.00 
 
 (21.15) 
 
 $120.00 
 
 $34.35 
 
 (16.55) 
 
 $40.00 
 
 $160.00 
 
 2 No. 4 Primary, 1,000 ft 
 
 91.40 
 
 110.80 
 
 25.46 
 
 29.70 
 
 140.50 
 
 
 
 (22.64) 
 
 
 (17.65) 
 
 
 3M X 4M X 92 in., six-pin 
 
 .74) 
 
 
 
 
 
 crossarm and hardware 
 
 M) 
 
 1.64 
 
 1.04 
 
 1.22 
 
 2.86 
 
 
 
 (21.80) 
 
 
 
 
 No. 3 porcelain insulator 
 
 .16 
 
 .21 
 
 Labor included in cost 
 
 
 
 
 of stringing wire 
 
 
 
 (26.08) 
 
 
 (20.95) 
 
 
 Pin 1% X 10> X 1 in 
 
 .06 
 
 .076 
 
 .10 
 
 .48 
 
 .196 
 
 
 
 (16.91) 
 
 
 (12.16) 
 
 
 Pol es 40 ft. 7 in 
 
 16.70 
 
 19.50 
 
 10.26 
 
 12.48 
 
 31.98 
 
 etc. 
 
 
 
 
 
 
COSTS 
 
 17 
 
 (d] Assemblies. 
 
 15 kva. S<A transformer installation 
 
 Pole material 
 
 
 Plus 
 load- 
 ing, 
 per 
 cent 
 
 
 9 No. 3 Porcelain insulators at .16 
 6 Glass insulators at .043 
 
 $ 1.44 
 .258 
 
 31.80 
 31 80 
 
 $ 1.90 
 34 
 
 15 Screw brackets at .22 
 
 3.30 
 
 26.08 
 
 4.16 
 
 2 Six-pin crossarms at .74 
 2 Blocks at .48 
 
 1.48 
 96 
 
 22.64 
 22 64 
 
 1.82 
 1 18 
 
 6 Braces at 1275 
 
 765 
 
 22 64 
 
 94 
 
 10 %-in. bolts at .02 
 4 %-in bolts at 12 
 
 .20 
 
 .48 
 
 22.64 
 22 64 
 
 .245 
 59 
 
 3 Lags at 038 
 
 114 
 
 22 64 
 
 14 
 
 2 Primary fuse boxes at 3.94 
 2 Secondary fuse boxes at 1.00 .... 
 
 7.88 
 2 00 
 
 22.64 
 22 64 
 
 9.67 
 2 45 
 
 2 Lightning arresters 4 86 
 
 9 72 
 
 22 64 
 
 11 91 
 
 Ground-rod, cap and moulding 
 5 ib No e w j re 
 
 1.58 
 1.395 
 
 23.79 
 21 15 
 
 1.95 
 1 69 
 
 6.6 Ib. No. 4 wire 
 
 1.78 
 
 21.15 
 
 2.16 
 
 30 ft.^6~ m - galvanized-iron wire 
 Staples, screws etc 
 
 .525 
 17 
 
 22.64 
 22 64 
 
 .63 
 21 
 
 
 
 
 
 Total 
 Labor 1/3(1^ + ITr + 3L + IGr) = $13.90 
 
 $34.05 
 plus 15.45 
 
 per cent 
 
 $ 41.99 
 16.10 
 
 15-kva. transformer $185.90 plus 16.91 per 
 
 cent 
 
 
 $ 58.09 
 217 00 
 
 
 
 
 
 Total.. 
 
 
 
 $275.09 
 
 Cost per 1,000 ft. of line with 125-ft. span. 
 
 
 Cross- 
 
 
 
 
 Wire, 
 
 Plus 
 
 Plus 
 
 
 arms, 
 
 Pins 
 
 Insula- 
 
 Wire 
 
 pins, 
 
 cross- 
 
 35-ft. 
 
 
 one per 
 
 
 tors 
 
 
 insula- 
 
 arms 
 
 poles 
 
 
 pole 
 
 
 
 
 tor 
 
 
 
 Primary 
 
 
 
 
 
 
 
 
 2 No. 6 
 
 $22 . 80 
 
 $3.14 
 
 $3.36 
 
 $106.80 
 
 $113.30 
 
 $136.10 
 
 $335 22 
 
 2 No. 2 
 
 22.80 
 
 3.14 
 
 3.36 
 
 186.15 
 
 192.65 
 
 215.45 
 
 414.57 
 
 Secondary 
 
 
 
 
 
 
 
 
 3 No. 4 
 
 22.80 
 
 4.70 
 
 1.37 
 
 210.00 
 
 216.87 
 
 239.67 
 
 438.79 
 
18 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 The above are merely examples selected here and there and in 
 no way indicate the complete record. It is easily seen that the 
 compiling of such a record is a matter of considerable labor and 
 investigation. Once obtained, however, in this form a revision is 
 a comparatively simple matter. 
 
 While the costs thus given are average figures, especially for 
 labor, they may be applied in the locality in which they were 
 derived without great error for estimating individual jobs unless 
 some unusual field conditions indicates extraordinary additions 
 to some part of the cost, For economic studies of course average 
 figures are usually desirable. 
 
 ANNUAL COSTS 
 
 The determination of annual costs as used in this work is 
 naturally divided into two parts, that pertaining to the physical 
 property itself such as construction and maintenance costs and 
 that pertaining to the load carried, i.e., cost of energy. 1 
 
 Annual Costs on Physical Property. The items to be con- 
 sidered under the first heading, i.e., annual costs on the physical 
 property, will each be discussed briefly. They include, interest, 
 taxes, insurance, maintenance, repair and depreciation. 
 
 Interest. Whenever money is invested in a piece of property 
 a legitimate rate of interest must be expected as part of the 
 earnings of that property unless it be run at a loss. Interest 
 must be figured on the total investment involved including all 
 material, labor and overhead costs. The rate at which interest 
 should be charged may vary with the problem under considera- 
 tion. Fundamentally it should be the current rate of interest 
 on sound investments or the rate at which the company 
 could borrow money under ordinary conditions. Sometimes the 
 average rate paid on total capitalization may be taken but in 
 case the dividend on capital stock is fairly large, part of it might 
 be considered as a profit in excess of a fair rate of interest. In 
 some cases due to poor financial conditions or for an emergency 
 a company might have to pay a higher rate for money, even 
 on bonds, than the market rate. All such factors should be 
 considered in determining the interest rate to be used. 
 
 Taxes. Taxes are an ever-present charge on any property and 
 usually the definite percentage may be easily determined from 
 the company's accounting records. 
 
 1 For determination of " Energy Cost" see Chapter IV. 
 
COSTS 19 
 
 Insurance. Insurance against loss by fire is the most common 
 form but on some classes of property insurance against theft, 
 storm, etc. is also carried. In any problem the kind and amount 
 of insurance chargeable to each class of property should be investi- 
 gated. In many cases no " Insurance" charge is necessary. 
 
 Maintenance and Repair. Maintenance and repair will be 
 different for each unit considered. No two transformers for 
 example will require the same amount of attention during their 
 life even though similarly located: breaks in a line can rarely be 
 anticipated, etc. A large part of maintenance is occasioned by 
 imperfection in material. Maintenance and repairs due to such 
 causes on any individual piece of equipment cannot be foreseen. 
 In such cases average figures only can be obtained from actual 
 experience over a number of years. Other items of maintenance 
 such as inspection, testing, etc., can be quite definitely determined 
 from payroll and .time reports. 
 
 Depreciation -No detailed discussion of depreciation can be 
 here included. Strictly speaking, depreciation is the percentage 
 by which a piece of property is reduced in value each year of its 
 life (by value is not meant necessarily selling price). From an 
 accounting point of view, on the other hand, a certain amount 
 must be set aside each year to replace the property when worn 
 out. Sometimes the usefulness of the property in service is con- 
 sidered as a measure of its value. All these different viewpoints 
 give rise to different methods of figuring depreciation, and a 
 discussion of these may be found in other works. For the pur- 
 pose of this work however what is known as the straight-line 
 method is probably the simplest and most satisfactory. This 
 method considers a piece of property to have given service for 
 the years of its life for a definite total cost which may be equally 
 divided between the years. The cost is obtained from the total 
 first cost of the property in place including material, labor and 
 overhead charges, less the salvage value at the end of its life, plus 
 the labor cost necessary to salvage it. This cost is divided by the 
 estimated number of years of life and the percentage of deprecia- 
 tion taken as the percentage of the total first cost thus obtained. 
 Naturally this will vary with different classes of property. Some 
 will have little if any salvage value such as crossarms, for 
 example. A pole, when rotted at the base, on the other hand can 
 be sawed off and used again either as a shorter pole or a stub. 
 Bare copper wire will have practically no physical depreciation, 
 
20 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 the cost of stringing and removing together with tie wires and 
 other incidentals making up the depreciation. 
 
 The years of life to use for any unit may depend on other 
 things than its own life. For a transmission line, for example, it 
 is probable that no definite limit can be fixed at which the whole 
 line must be replaced. Poles and crossarms will be replaced 
 from time to time when necessary as long as the line is in service. 
 In such a case it is probably simplest to assume a definite life for 
 the whole line, possibly the assumed life of a pole, and figure 
 depreciation on all units, wire, insulators, etc. on that basis. 
 Thus any class of property may have a different percentage of 
 depreciation depending on where it is used. 
 
 A simple example of a computation for depreciation on 
 insulated wire would be as follows : 
 
 Assuming new wire at 30 cts., scrap copper at 20 cts. per pound. 
 Cost of No. wire per 1,000 ft.-420 Ib. at 30 cents = $126.00 
 
 Labor of stringing (assumed) 10 . 00 
 
 Labor of salvaging (including burning off insulation) 10.00 
 
 Total $146.00 
 
 Salvage value 319 Ib. at 20 cts. per pound 63 . 80 
 
 Net Cost $82.20 
 
 If the assumed life of insulation is 15 years 
 
 82.20 5.47 
 
 = $5 . 47 per year = 4 per cent per year. 
 
 15 136 . 00 
 
 The matter of obsolescence which is sometimes considered as 
 a separate figure may, for this work, be considered as a part of 
 depreciation. Where it is anticipated that materials will become 
 obsolete and require replacement before worn out on account of 
 improvement in design, the assumed life and salvage value used 
 in computing depreciation should be adjusted accordingly. 
 
 As an example of the above, the total percentage of annual 
 charge on a pole line may be taken as 15 per cent, made up as 
 follows : 
 
 PER CENT 
 
 Interest 7 
 
 Taxes 2 
 
 Insurance 
 
 Depreciation 6 
 
 Many special problems arise in figuring annual cost. One 
 which also includes the idea of obsolescence is encountered when 
 
COSTS 21 
 
 the replacement of a serviceable line with one of larger capacity 
 is being considered. The value remaining in the old line must be 
 included in the computations and the total investment repre- 
 sented in the new line must include, in addition to the cost to 
 build it, the present value of the labor necessary to build the old 
 line, i.e., first cost for labor less depreciation for years of age and 
 the labor cost necessary to dismantle the old line. Such prob- 
 lems will be discussed in more detail later. 
 
CHAPTER IV 
 ENERGY COST 
 
 PRINCIPLES AND METHODS INVOLVED IN THE DETERMINATION OF 
 THE COST OF ENERGY AND OF ENERGY LOSSES 
 
 Since any economic study of transmission or distribution is, 
 fundamentally, a consideration of the cost of energy or energy 
 losses as compared with other costs (in general, fixed charges 
 increase as energy loss decreases), it is of the utmost importance 
 that the cost of energy be accurately determined. The value 
 assigned to the unit cost of energy may be the deciding factor in a 
 problem. This value, moreover, may vary considerably according 
 to the assumptions, methods and precision employed in its calcu- 
 lation. It is evident that the cost per kilowatt-hour of the energy 
 used by a 5-h.p. motor running 1 hr. per day, 25 miles north of the 
 generating station may be quite different from the unit cost for 
 residence-lighting load, 5 miles west of the station and both of 
 these may be far from the average unit cost over the whole 
 system. The question of the determination of energy cost 
 offers an extensive field for study and one that generally has been 
 only touched upon. This chapter will give some of the funda- 
 mental principles involved, a few methods of attacking the general 
 problem and suggestions as to conditions governing variations 
 in the general cost as applied to particular uses. 
 
 In studying energy cost, the fundamental difference between 
 the cost of energy for rate making purposes and the cost of energy 
 for economical study must be recognized. ' In determining the 
 cost of energy for the purpose of adopting a rate scale, it may be 
 sufficient to consider the system as a whole and determine the 
 average cost per unit for each of a few general classes of load 
 which have markedly different characteristics. The chief point 
 to be kept in mind is the amount which the customer pays, and 
 that this, on an average should at least equal the expense of the 
 company, plus a reasonable profit. Usually about the same 
 rate must be applied to similar customers within a reasonable 
 area, unless there is a marked difference in the individual cost of 
 serving each. There are also other factors than actual produc- 
 tion and distribution cost which must be considered in rate making, 
 such as public opinion, regulations and franchise agreements, pre- 
 vious practice, competition, etc. On the other hand, in making an 
 
 22 
 
ENERGY COST 23 
 
 economic study, we are interested in the actual amount which the 
 energy delivered to the point under consideration is costing the 
 company and hence, how much money, if any, can be saved 
 by reducing energy losses. If energy costs more per unit 5 miles 
 from the station than it does 1 mile, for example, the amount 
 saved will be correspondingly more important. For this pur- 
 pose, then, it would seem desirable to investigate the cost of 
 energy as fully as is warranted by the amount and accuracy of 
 the information available on costs of construction and operation 
 and on loads carried. 
 
 Even with fairly complete data at hand it is a difficult 
 matter to determine definite figures for energy costs. The cost is 
 affected by many quantities of a variable nature and these limit 
 the extent to which it is practicable to carry the study. At any 
 given point the cost of energy is largely dependent on the fixed 
 charges and operating expenses of the central station and of the 
 distribution system between the station and that point. It is 
 also affected by the size and characteristics of the load at the 
 point in question in relation to all the other loads on the system. 
 Hence, strictly speaking, energy cost may be conceived as having 
 a different value at every point on the system and for every 
 different load at any given point. Loads of the same type may 
 show a different unit cost according to their size and the unit cost 
 may vary at different times during the day or even for different 
 parts of the same load. The length to which the determination 
 of cost of energy might be carried is almost infinite. In practice 
 it will depend not only on the accuracy of the data available but 
 also on how this cost is to be used. 
 
 Classification of Costs of a Central-station System. The 
 various items entering into the total annual cost of a central 
 station may be classified as follows: 
 
 1. Costs dependent on the number of customers. 
 
 2. Costs dependent on the peak load carried or the demand. 
 
 3. Costs dependent on the total output in kilowatt-hours 
 during the year. 
 
 This method of classification which was suggested by Dr. 
 John Hopkinson in England in 1892 and was later enlarged upon 
 by H. L. Doherty and other writers, is quite generally accepted 
 as sufficient for ordinary purposes. It must not be assumed 
 that all expenses come strictly under one of these three classifi- 
 cations. There are a great number of other minor divisions which 
 
24 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 might be made. For example, there are certain operating 
 expenses at the station which are dependent on the efficiency 
 and size of the machines and the relation of the load at any time 
 to the capacity of the generators in use and to the method of 
 operating the station and the system. It does not appear 
 practical however to attempt to include all such classifications. 
 The above three are the ones of most importance and other 
 costs can be included in one of them without great error. 
 
 Consumer's Cost. The first division, consumer's cost, usually 
 need not be considered in determining energy cost for economic 
 study, as it is independent of the amount of load carried. This 
 cost occurs beyond the limits of the lines and need be added 
 only when determining the charge to the customer. Care must 
 be taken however that charges actually belonging to this classi- 
 fication are not included under either of the other heads. Here 
 rightfully belong the greater part of charges for general-office 
 expense, sales expense, metering, billing, collection, etc., part of 
 cost of service wires and a percentage of various other charges 
 according to local conditions. 
 
 Demand Cost and Output Cost. The other two cost divisions, 
 demand cost and output cost, must include all charges which are 
 dependent on the load carried. The demand cost is that part 
 of the total cost which is caused by the fact that the system 
 is built and operated to care for a certain maximum load. If 
 this peak load is 100,000 kw. the station must have a capacity of 
 at least 100,000 kw., with a reasonable amount of reserve, regard- 
 less of the fact that that capacity may be reached for only a 
 short time each day. The output cost or kilowatt-hour cost on 
 the other hand is that part of the total cost which is occasioned 
 by the total output in kilowatt-hours regardless of the rate of 
 that output. 
 
 Some items of the total annual cost clearly belong only to 
 demand cost while others which might seem to depend only on 
 output have some percentage of demand cost included. In 
 the first group come interest, taxes, insurance, depreciation, etc. 
 on the generating-station building, interest, taxes, etc. on boilers, 
 turbines, generators and other equipment and a large part of 
 their depreciation and maintenance. Also fixed charges on lines 
 and substation equipment are here included. Under the second 
 group comes part of the cost of coal, oil and other expendable 
 materials, part of the labor of operating, also part of the energy 
 
ENERGY COST 25 
 
 losses on lines and transformers due to the fact that they are 
 kept energized at all times. The apportioning of such costs is a 
 matter for considerable study. The method of operation may 
 effect the amount chargeable to demand in certain cases. Large 
 machines are less efficient at small loads. Hence if large units 
 are employed and are run far below their most economical load 
 for the greater part of the day, the extra expense due to decreased 
 efficiency may be charged to demand. In a large, efficiently 
 operated station this condition would not occur to any great 
 extent, but it suggests some of the items which must be 
 considered. In fact there must be included in demand cost all 
 charges directly or indirectly occasioned by the total capacity of 
 the system. The remaining annual costs, aside from the con- 
 sumer cost before mentioned, may be considered as dependent on 
 the output. These comprise the kilowatt-hour cost. 
 
 Methods of Making Classification. The actual division of 
 the total annual cost on any part of the system into three classifi- 
 cations will depend largely on local conditions. The relative per- 
 centages will probably be different for each company. Several 
 general methods of attacking the problem are in use. In the 
 generating plant, for example, an empirical analysis can be made 
 of each item of cost, such as charges on building, on steam equip- 
 ment, on electrical equipment, fuel, labor, etc. The proportion 
 belonging to each classification may be estimated from known 
 conditions, keeping in mind the general definitions of demand 
 cost, kilowatt-hour cost and consumer cost. One method of 
 separating demand and kilowatt-hour charges on fuel, lubricants 
 and such items is by comparing costs under a period of light load 
 and one of heavy load a month at different seasons of the year. 
 For. example, 
 
 if CD = demand cost, per unit demand 
 
 / = kilowatt-hour cost, 
 
 D = station demand, 
 
 FI = kilowatt-hours at low period, 
 
 Fz = kilowatt-hours at high period, 
 
 Ci = total cost at low period, 
 
 C 2 = total cost at high period, 
 
 Ci = C D D + c f F l 
 
 C 2 = C D D + C } FZ 
 
 C 2 - C l C l -C / F l 
 
26 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 This should give satisfactory results if there were enough 
 difference between the loads at the two periods to give a good 
 comparison of cost. 
 
 Another somewhat similar method, 1 that may be applied to 
 items such as total station or system costs, which involve all three 
 classifications, considers the total cost over three given periods, 
 three years for example or three different months. Using similar 
 symbols to the above with 
 
 Cc = customers cost, 
 
 N = number of customers, 
 
 Ci = CnD, + C,F! + C C N 1 
 
 If these are solved simultaneously the values of CD, C/ and 
 C c can be determined. 
 
 An example of a typical division of costs between demand and 
 kilowatt-hour of some of the items for the generating station is as 
 follows : 
 
 Demand, Kilowatt- 
 
 per cent hour, per cent 
 Superintendence .................. 100 
 
 Wages ........................... 90 10 
 
 Fuel ............................. 25 75 
 
 Lubricants ....................... 25 75 
 
 Station supplies, etc ............... 100 
 
 These general methods may be adapted to other parts of the 
 system, transmission lines, substations, etc., as well as to the 
 generating plant. In analyzing the cost of a hydro-electric 
 plant, the available supply of water is an important factor in the 
 consideration. It is evident that the kilowatt-hour cost will 
 depend considerably on whether the supply is unlimited or 
 whether storage is resorted to for regulating the flow. 
 
 It appears then that the first step in the study of energy cost 
 is the determination as accurately as possible of the total annual 
 costs on each subdivision of the system and the proportion of 
 these costs, in each case, belonging to demand and to output. 
 Naturally the more detailed the accounting record on various 
 parts of the system, the easier the determination of these costs 
 
 1 This method is more fully explained in ''Central Station Rates in Theory 
 and Practice," by H. E. EISENMENGER, Electrical Review, Vol. 75, Aug. 23, 
 1919, p. 305. 
 
ENERGY COST 27 
 
 will be. If an entirely new system is being considered, the 
 various quantities can be only estimated from data of other 
 similar systems and present prices on construction materials, 
 equipment, etc. For the purpose under consideration, if the 
 study of energy cost is to be carried to any degree of exactness, 
 it is probably better to prepare separate figures on each of the 
 three classifications of costs as related to different parts of the 
 system, generating station, various substations, underground- 
 cable lines, overhead power lines, feeders, etc. Just how much 
 of such detail is necessary will be determined by the method 
 which is to be used in apportioning the costs among the various 
 classes of loads and localities. 
 
 Apportioning of Demand and Output Costs to Various Types 
 of Loads. When the proportions of the total cost chargeable to 
 demand and to output have been determined, there arises the 
 problem of finding what part of that demand cost or of that 
 kilowatt-hour cost belongs to any particular load or type of 
 load. The Kilowatt-hour cost may be simply disposed of for the 
 present by considering that the kilowatt-hour charge at any 
 point in the system is equal to the sum of the kilowatt-hour 
 costs incurred on all parts of the system from that point back to 
 the generating station. Considerable study may be involved in 
 determining the kilowatt-hour costs on such parts as trans- 
 mission lines or power lines, since a great part of the cost depends 
 on the energy losses and the cost of these losses in turn includes 
 the demand and kilowatt-hour charges up to that point. The 
 theory is not complex however. Modification of this method for 
 special purposes will be explained later. The apportioning of 
 the demand cost however presents a more difficult question. 
 
 Demand Cost. If all loads had similar characteristics, i.e., 
 similar load curves, with the peak coming at the same time it is 
 evident that the demand charge for each would be simply pro- 
 portional to its peak load. This assumption is sometimes used 
 in figuring energy cost but is obviously not correct except in 
 cases where all loads are similar or nearly so, such as for a plant 
 serving residence lighting only. On this assumption, the 
 demand charge per kilowatt may be reduced to a figure per 
 kilowatt-hour, inversely proportional to the load factor, and 
 this added to the kilowatt-hour charge determines the total 
 cost of energy. Where both lighting and commercial power 
 or other loads are carried, however, it is evident that the demand 
 
28 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 responsibility of each cannot be accurately obtained in this way 
 as the peaks come at different times and the loads show different 
 characteristics throughout the day. For example the power 
 peak might come at the same time as the station peak at perhaps 
 2 P.M., whereas the lighting peak comes at 8 P.M., the lighting 
 load at 2 P. M. being only 30 per cent of its peak. 
 
 Again the assumption might be made that the demand charge 
 of any load is proportional to its demand at time of station peak. 
 This might sound reasonable inasmuch as the demand cost of 
 the station is figured on the basis of its peak load. The demand 
 responsibility for any customer is sometimes computed on this 
 basis by multiplying his total connected load by his demand fac- 
 tor to get his individual demand, dividing this in turn by the 
 diversity factor on the line for his proportion of the line demand, 
 this by the diversity factor of the substation and so on back to 
 the generating station, thus determining that customer's propor- 
 tion of the station peak. Of course a careful determination of 
 diversity and demand factors is necessary for this. Here again 
 the variable characteristics of the different loads make this 
 assumption erroneous except in special cases. 
 
 Take for a simple example a small station of 1,000 kw. serving 
 two customers A and B. A takes 1,000 kw. for 6 hr. each day. 
 B takes 600 kw. for the remaining 18 hrs. but does not overlap A. 
 On the assumption of demand proportional to individual peak 
 load, A's cost would be x %6 and B's %Q of the total demand 
 cost. On the assumption of demand proportional to load at 
 time of station peak, A's proportion would be the whole station 
 demand and B's nothing (see Fig. 1). 
 
 FIG. 1. 
 
 If, however, the station may be considered as consisting of two 
 parts, one of a capacity of 600 kw. and one of 400 kw., it will be 
 readily seen that the 600 may be assumed to operate 6 hr. for A 
 and 18 for B while the 400 operates only 6 hr. for A (see Fig. 2). 
 
 In this case then A's cost would be 
 
 400 
 1,000 
 
 A v 60 
 
 24 X 1,000 
 
ENERGY COST 
 
 29 
 
 = f 
 
 X 
 
 
 of the total. This theory can be extended to cover any number 
 of loads with variously shaped curves. It may be stated in 
 general that the cost of each unit (kilowatt) of the total demand 
 should be divided in accordance with the length of time or number 
 of hours it is in use, to obtain an accurate apportioning of demand 
 cost. A full explanation of this theory with examples of its 
 application may be found in "Central Station Rates in Theory 
 
 FIG. 2. 
 
 and Practice," by H. E. Eisenmenger, Electrical Review, Vol. 75, 
 Aug. 2 and 9, 1919. Ordinarily sufficient data in regard to the 
 various load curves may not be available or the degree of accuracy 
 desired in the result will not warrant an extensive analysis on this 
 basis. There is no doubt that theoretically it will give a more 
 accurate distribution of the demand cost for most purposes than 
 either of the other methods mentioned and the principles involved 
 may often be used to advantage, even in a more approximate 
 determination. For some special uses, as will be explained later, 
 the second method given above is preferable. 
 
 General Method for Determining Demand Cost at any Point. 
 
 In order to determine the demand cost at any point on the 
 system other than the generating station the following general 
 steps should be followed (see Fig. 3) : 
 
 1. Apportion the generating-station demand cost among the 
 various substations by one of the above methods, in accordance 
 with their loads. 
 
 2. For any substation, add to its portion of the generating 
 station cost the demand costs on its transmission cables and on 
 the substation itself. 
 
 3. The total demand cost for the station can now be distributed 
 among the various lines feeding out from it and by repeating the 
 same method the cost at any point or for any customer may be 
 determined. If more general figures only are desired the total 
 demand cost for the station may be divided among various classes 
 
30 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 of load such as, lighting, commercial power, street lighting, street 
 railway, etc. By assuming average figures on line costs, the cost 
 at any distance from the station for each class of load can be 
 determined. The analysis may be extended in a similar manner 
 
 GENERATING STATION 
 
 n jr j. Distributed accordi'nq 
 Demand Cost to load curves of 
 
 Djvided equally among 
 Operating Cost kilowatt-hours 
 of output 
 
 TRANSMISSION LINES 
 AND CABLES 
 
 SUBSTATION 
 
 Demand Cost on 
 
 Substation 
 + Demand Cost on 
 
 Transmission 
 + Proportion of 
 
 Demand^oston 
 
 Generating Sta. 
 
 Distributed accord- 
 ing to load curves 
 on outgoing loads 
 
 Operating Cost on 
 Substati'on 
 
 ston 
 ansmission 
 
 4- Operating Co 
 Trans mi'ssio' 
 
 Divided equally 
 among kilowatt- 
 hours of output 
 ^-Generating sta- 
 tion kilowatt-hour 
 cost 
 
 Demand Cost on I Mile 
 
 of Power Lme 
 + Proportion of substation 
 
 Demand Cost 
 
 Operating Cpst on I Mile 
 of Power Line divided 
 equally among kilowatt 
 hours of output. 
 
 * Kilowatt hour cost at 
 substation. 
 
 Proportion of substation 
 
 Demand Cost 
 + Demand Cost on /^ mile 
 
 Operating Costpn" I ^Tmile 
 
 of Distribution, 
 
 divided equally among 
 
 kilowatt hours of 
 
 output 
 + kilowatt -hour cost 
 
 at substation 
 
 FIG. 3. Diagram indicating general method of studying energy cost at various 
 
 points on a system. 
 
 to cover suburban transmission lines, substations and distribution 
 even where several substations intervene. 
 
 Any such analysis should give a fairly accurate demand charge 
 at any point. It requires however quite a large amount of 
 accurate data for its accomplishment. Detailed annual costs 
 
ENERGY COST 31 
 
 on each subdivision of the system must be known. Also the load 
 curves throughout the year, for each load considered such as 
 substations, power lines, etc., must be studied to obtain one or 
 more characteristic curves for each as a basis of analysis. Where 
 a load shows an appreciable seasonal variation several curves 
 may be necessary as for high load, low load and average load. 
 In many cases it will be found practicable to determine a general 
 characteristic curve for each type of load, such as residence light- 
 ing, street lighting, . street railway, large power, small power, 
 etc., and refer all loads of that type to it, assuming that the curve 
 will be always proportional to the peak load. The collection, 
 classification and proper selection of data will be found to be a 
 large part of the whole problem of determination of energy cost 
 and requires the application of a high degree of engineering 
 knowledge and judgment. 
 
 In order to make such a detailed analysis as proposed above the 
 following information should be available. 
 
 1. Cost of Generating Station. 
 
 (a) First cost and present value of various parts in sufficient detail so 
 that demand, energy and customer's costs can be separated. 
 
 (6) The proper percentages of interest, taxes, depreciation, etc., 
 chargeable to each part. 
 
 (c) Operating costs in some detail. 
 
 (d) Determination of proper percentage of each item of fixed and 
 operating charges belonging to each of the three divisions of cost. 
 
 2. Cost of Transmission-line Cables. 
 
 (a) First cost and present value of cable lines to each substation with 
 
 proportional cost of tie lines between stations. 
 (6) As above. 
 
 (c) Such operating cost as may be chargeable to cables, repairs, etc. 
 
 (d) As above. 
 
 3. Cost of Substations. 
 
 (a) First cost and present value of each substation in sufficient detail. 
 (6, c, d)As above. 
 
 4. Cost of Overhead Lines. 
 
 (a) First cost and present value of average line of each class, trans- 
 mission, power line, circuit, direct-current feeder, railway feeder, and 
 secondary distribution including transformers, per unit length. 
 
 (6, c, d) As above. 
 
 5. Characteristic Curve for Generating Station. 
 
 (May be taken for seasons or months instead of for year.) 
 
 6. Characteristic Curves for each Substation. 
 
 (Corresponding with generating-station curves.) 
 
 7. Characteristic Curves for Each Class of Load out of Each Substation. 
 
 (Or for each line.) 
 
32 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Total Charge Per Kilowatt-hour. All the formulas which will 
 be developed later include the cost of energy as one total charge 
 per kilowatt-hour. The computations might be made with the 
 demand charge and the kilowatt-hour charge as two separate 
 quantities, but it is found more convenient to use a single 
 charge per kilowatt-hour. If the above analysis is carried out 
 in detail the demand charge for energy will be determined as a 
 different amount for each type of load and for any distance 
 from each substation. The kilowatt-hour, charge will be the 
 same for all loads in any one locality. Since the average load 
 factor for any type of load considered may be determined, the 
 demand charge can be reduced thereby to a charge per kilowatt- 
 hour, since kilowatt hours = kilowatts X load factor X 24 X 365 
 per year. This added to the output or kilowatt-hour charge will 
 give the total charge per kilowatt-hour for that class of load at 
 that point. 
 
 Further Variations in Energy Cost. Up to this point there has 
 been considered, for any type of load and locality, only the 
 average unit cost of the total energy delivered, without attempt- 
 ing to differentiate between costs for large loads and small loads 
 of the same type, high power factors and low power factors, losses 
 and used power, etc. There will now be indicated some of the 
 possible variations in this cost and methods of studying them 
 will be suggested. 
 
 It may be shown that all units of energy even of the same class 
 and locality do not cost the same. For example, energy lost has 
 a higher unit cost than energy used. Again, not only does the 
 total kilovolt-amperes of any load increase as power factor 
 decreases but that increase costs more per unit than the average 
 cost if a 100 per cent power factor were obtained. Further, 
 under some conditions an increase in a load will cost more per 
 unit than the former average. 
 
 The underlying theory involved may be readily seen if we 
 consider the cost of lost or waste energy for example. Most 
 substations are regulated in some way, and if the energy losses 
 between the substation and the customer could be reduced the 
 current in the transmission cable to the substation would be also 
 reduced. Since losses are proportional to the square of the 
 current, the cable losses eliminated by this reduction of the load 
 on the cable would be proportional to the difference in the squares 
 of the currents before and after reduction. Hence it will be 
 
ENERGY COST 33 
 
 seen that reduction in loss per unit reduction in load is greater than 
 the average loss per unit of the total load, after reduction. In 
 other words, the loss in the cable due to the upper part of the 
 load, which may be eliminated and hence may be considered 
 waste, is greater per unit than the loss due to useful load. Since 
 this energy must be supplied at the generating plant, the energy 
 generated will be more per unit for this waste energy than for the 
 remainder of the load or useful load and the cost will hence be 
 more. This may be shown mathematically as follows: 
 
 Suppose a substation B is supplied through cables from a 
 generation station A 
 
 (A) - : - : - _() 
 
 W = load in watts at B, 
 
 P = percentage of W lost beyond B, which may be considered 
 waste power which might be conserved in some way, 
 p = P/100. 
 E = Voltage at B, 
 R = Resistance of cable. 
 
 For simplicity assume single-phase cable, unity power factor. 
 Then 
 
 f /w\ 2 
 PR losses in AB due to total load W = 2 f R, 
 
 I 2 R losses in AB due to power load only = 2/ 
 
 I 2 R losses in AB due to losses beybnd B(=pW) = 
 
 r ~/W\ 2 /W\ 2 /W\ 2 
 
 ^(r) *-(?)*+(?) 
 
 or expressed in percentage of total loss in AB 
 
 = 100(2p - p 2 ) = 2P - per cent (1) 
 
 If the loss or waste power at B is a comparatively small per- 
 centage of W it will be seen that the losses in AB due to that 
 waste are nearly twice that percentage of the total losses. (Ten 
 per cent waste at B means that 19 per cent of losses in A B are due 
 to that waste.) 
 
 If Q = percentage of W lost between A and B, 
 QW 
 
 . = total loss in cable, 
 
 QW 
 100 
 
 (P 2 \ 
 2P -^JQ) Moo loss in cable due to waste beyond B. 
 
34 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 PW 
 
 But waste beyond B = -77^- 
 
 100 
 
 Hence, the loss in the cable due to waste beyond B per unit 
 waste = 
 
 QTF 
 Too 
 
 / P \ 
 
 =/ ioo ( 2 -iS)) 
 
 100 
 
 Since the average total loss in cable per unit total load at 
 B = . If cost of energy per kilowatt at A = C e 
 
 Average cost at A of total energy at B = M + ~^\ C e (3) 
 Average cost at A of energy waste beyond B 
 
 In other words a loss of Q per cent in line A B increases the aver- 
 age cost of total energy delivered at B by Q per cent. But if P per 
 cent of that total energy represents losses which may be assumed to 
 
 be reducible or waste, the average cost of such losses is increased 
 
 / p \ 
 by Q (2 JQQJ per cent. The difference in average unit total 
 
 cost and average unit waste cost is then JQQ ( 1 TQQ) C e , or 
 
 per cent of average cost 
 1 _L _v_ 1 _, __ 
 
 100 100 
 
 at A of total amount of energy delivered at B. ' (5) 
 
 For example, if 1,000 kw. were transmitted over a line for 1 hr. 
 
 with a loss of 6 per cent if 10 per cent of that 1,000 represents 
 
 reducible losses or energy wasted beyond the end of the line 
 
 P = 10 Q = 6 
 
 If energy at A costs .01 per kilowatt-hour the average cost per 
 
 kilowatt-hour delivered at B including losses in AB but not 
 
 fixed charges on line = (l + y^j .01 = .0106 per kilo watt hour 
 
 (from Eq. 3). 
 The average cost of energy waste beyond B is 
 
 1+ 2 - -01 =-01114 (from Eq. 4). 
 
ENERGY COST 35 
 
 The difference is .00054 which is 5.1 per cent of the average. 
 The same percentage of increase would be obtained from Eq. 5. 
 
 li-) 
 
 V 100 / 
 
 10 '100 
 _ 540 
 
 100 + 6 106 
 
 It must be noted that the above deals with the average cost at 
 A of energy delivered at B. In order to determine the average 
 cost at B of energy delivered at B the charges on the line AB must 
 be included. These should be practically proportional to W 
 and hence would average the same per unit for both useful load 
 and for losses. 
 
 The above merely establishes the fact of the increased cost of 
 waste energy and must not be considered as indicative of the 
 actual amount of that increase. In determining this a number of 
 variable quantities must be considered: 
 
 1. The demand charge for waste energy at any point should be 
 figured by the second method given under that heading, i.e., 
 making up the demand charge, for any load, of charges pro- 
 portional to the amount of that load at the time of peak load on 
 station, transmission cable, substation, etc. The reason for 
 this is seen when it is considered that the demand cost on the 
 station, for example, is considered proportional to the station 
 peak. If that peak can be reduced by eliminating waste energy 
 (without reducing the useful load) the demand cost will be 
 likewise reduced in proportion. This line of argument does not 
 apply to useful load since, for this, the company is receiving a 
 monetary return. The actual demand cost should be dis- 
 tributed to such loads by the more equitable third method. In 
 other words, a reduction of losses at off-peak time would not 
 affect the actual demand cost of the station nor the distribution 
 of that cost among the useful loads. A reduction of useful load 
 at off peak would reduce the revenue and hence increase the cost 
 to other loads. 
 
 2. The example above uses the energy charge as a single total 
 charge per kilowatt-hour and the loss at a definite percentage. 
 Since the percentage loss varies as the load varies during the day 
 the increase in cost will not depend on the percentage losses at 
 maximum load. The variation in loss will affect both the 
 demand charge and the kilowatt-hour charge and each in a 
 different proportion. 
 
36 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 3. If several loads of different sizes and characteristics are 
 considered the problem is further complicated. 
 
 4. If lines are operated under the most economical load, the 
 losses may be considered as useful load rather than waste. 
 
 As has been stated before, the theory here explained can also 
 be applied to determination of the additional cost of low power 
 factor, of load increases (in some cases) and other similar prob- 
 lems. It would appear that each unit of energy comprising any 
 load might be conceived as having a different cost. In dealing 
 with increases in useful or saleable load we have the further 
 consideration of economical loading of lines so that it would not 
 be correct to say that an increase in load always costs more per 
 unit than the average unit cost before the increase. In dealing 
 with losses, however, a large percentage of loss may be assumed 
 to cost more per unit than a small percentage. This idea of 
 increased cost of losses and waste energy is an important one to 
 bear in mind in all economical studies, since usually such a study 
 is fundamentally the establishing of the most economical relation 
 between cost of energy loss and other costs. 
 
 This chapter is intended to be more a suggestion as to how the 
 problem of cost of energy and energy losses may be studied 
 rather than a complete solution or a recommended method to 
 cover all cases. It may be easily seen that the study of energy 
 cost might be carried to an almost infinite degree of refinement. 
 Many economical studies are general for any part of a system and 
 are intended to cover a period of time well into the future. For 
 these cases a very detailed determination of energy cost would 
 not seem necessary. If the more accurate costs are once deter- 
 mined however, they can easily be averaged for a more general 
 problem. Lack of time and of the necessary data will, in many 
 cases, limit the study to more or less approximate results. It is 
 hoped, however, that the methods and principles here explained 
 will give a good idea of the nature of the problem. Detailed 
 analyses of energy cost, when data is available, will well repay 
 the effort expended. Even a more approximate study with these 
 theories in mind will indicate the relative costs of energy at 
 various parts of the system, of various" types of loads, and the 
 relation between the cost of waste and useful energy. 
 
CHAPTER V 
 LOAD CHARACTERISTICS 
 
 POWER FACTOR BALANCE FACTOR DEMAND FACTOR DI- 
 VERSITY FACTOR LOAD FACTOR EQUIVALENT HOURS 
 
 The consideration of the cost of energy, in the previous chapter 
 brought forth some of the quantities that are used in analyzing 
 the character of a load and its relation to other loads. It was 
 also indicated that there are many kinds of 'loads and that each 
 one of these can be expressed in terms of its characteristics. It is 
 proposed here to define and briefly comment on some of these 
 terms: power factor, balance factor, load factor, demand factor, 
 diversity factor, and on equivalent hours. When available the 
 definition of each one of these terms will be taken from the 
 " Standardization Rules" of the A. I. E. E. 
 
 Power Factor. "The ratio of the power (cyclic average as 
 defined in No. 26) to the volt-amperes. In the case of sinusoidal 
 current and voltage, the power factor is equal to the cosine of 
 their difference in phase." (Power in an alternating-current 
 circuit. The average value of the products of the coincident 
 instantaneous values of the current and voltage for a complete 
 cycle, as indicated by a wattmeter.) 
 
 Since the date of this issue of the rules there has been con- 
 siderable discussion as to the best definition for power factor. 
 However for the purpose of this book the above general definition 
 is sufficient. From the point of view of the designer of an 
 economical line the power factor is usually determined in advance 
 by the load at the end of the line and it is not within his province 
 to change its value. The interesting point to him is really the 
 kilovolt-amperes his line has to carry, as it is the current of a load 
 that determines, on account of the energy losses, not only the 
 economical loading of an old line, but the economical size of wire 
 for a new line destined to carry a given known load. It is evi- 
 dent, however, that while making studies in economical handling of 
 loads the effects of power factor will be most forcibly brought to 
 view and that the desirability of high power factor and general 
 power-factor improvement, particularly on lines carrying large 
 amounts of power loads, will be shown to be most imperative 
 for increased economy. 
 
 37 
 
38 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Balance Factor. While balance factor has never been posi- 
 tively denned it should be a means of expressing the divergence 
 between an unbalanced load on a polyphase circuit and the same 
 load when perfectly balanced. While it has been assumed 
 throughout this book that loads were balanced, it is evident that 
 many problems will arise where it will be necessary when solving 
 for economical design, to make allowance for such unbalance. 
 Large single-phase loads on polyphase circuits will often result in 
 this necessity. 
 
 Demand Factor. "The ratio of the maximum demand of any 
 system or part of a system, to the total connected load of the 
 system, or of the part of system under consideration." 
 
 (The demand of an Installation or System is the load which is 
 drawn from the source of supply at the receiving terminals 
 averaged over a suitable and specified interval of time. Demand 
 is expressed in kilowatts, kilovolt-amperes, amperes, or other 
 suitable units.) 
 
 (The Maximum Demand of an Installation or System is the 
 greatest of all the demands which have occurred during a given 
 period. It is determined by measurement, according to 
 specifications, over a prescribed time interval.) 
 
 Demand factor is therefore the expression of the relation 
 between apparent load or connected load and the largest actual 
 load that will be expected at any time on an installation. For 
 example, if a house is wired for 30 outlets each using a 40-watt 
 
 lamp and the greatest number of these operating at one time is 
 
 9 
 nine, the demand factor is = .3 or 30 per cent. On the other, 
 
 oU 
 
 hand if a service is wired for a range of 5.5-kw. capacity and if 
 at any time this range is operated with all the elements turned 
 on the demand factory becomes unity. 
 
 Diversity Factor. "The ratio of the sum of the maximum- 
 power demands of the subdivisions of any system or parts of a 
 system to the maximum demand of the whole system or of the 
 part of the system under consideration, measured at the point of 
 supply." 
 
 Here we express a load relation between various loads of the 
 same or of different demand factors and other characteristics. 
 For instance, if we take 10 houses each having the 30 outlets as 
 above and the same demand factor of .3, the total demand will 
 
 10 X .3 X 30 X 40 
 
 not be or 3.6 kw. but some smaller amount 
 
 1000 
 
LOAD CHARACTERISTICS 39 
 
 due to the fact that the maximum demand of all the houses are 
 not simultaneous. Therefore if the maximum load of these 
 
 Q & 
 
 houses taken together is 1.8 kw. the diversity factor is - - =2. 
 
 l.o 
 
 Also taking several electric ranges as above the diversity factor 
 may be .3 while each range at some time or another will be 
 operating at a demand factor of unity. 
 
 Similarly the diversity factor between transformers, between 
 substations, between lines can be obtained. 
 
 Load Factor. "The load factor of a machine, plant or system. 
 The ratio of the average power to the maximum power during a 
 certain period of time. The average power is taken over a certain 
 period of time, such as a day, a month, or a year, and the maxi- 
 mum is taken as the average over a short interval of the 
 maximum load within that period. 
 
 "In each case, the interval of maximum load and the period 
 over which the average is taken should be definitely specified, 
 such as a 'half-hour monthly' load factor. The proper interval 
 and period are usually dependent upon local conditions and upon 
 the purpose for which the load factor is to be used." 
 
 Since in economical design annual costs are generally the basis 
 of analysis it is evident that the period of time to be used will be 
 1 year. In this book equivalent hours have been used for deter- 
 mining energy losses over a line as shown below. 
 
 Equivalent Hours. It is usually convenient in studying a 
 given type of load, such as residence lighting for example, to 
 obtain the energy losses per year in terms of the load carried. 
 By "load carried," the peak load for the year will be meant, 
 since it is for that load that the size of the wire and transformers 
 must be determined. Since the energy loss over a line is depen- 
 dent on the square of the current at any time it is evident that 
 the total loss is not proportional to the load factor since the load 
 factor is determined from the average load and hence involves 
 only the first power of the load at any time. The computation 
 of total energy loss in terms of peak load must be based on the 
 square of the loads at any time. It is therefore convenient for 
 this purpose to determine for each of the various classes of load 
 considered a quantity which has been called the "equivalent 
 hours." 
 
 "Equivalent hours" may be defined as "the average number 
 of hours per day which it would be necessary for the peak load 
 
40 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 of the year to continue in order to give the same total energy 
 loss as that actually given by the variable load throughout the 
 year." It is a quantity which, if multiplied by the loss at peak 
 load on any line gives the average loss per day over the year. If 
 this average daily loss is then multiplied by 365 the total yearly 
 loss in kilowatt-hours is obtained. This times the cost of energy 
 per kilowatt-hour gives the annual cost of energy loss. It is 
 evident that, if the equivalent hours for any load and the peak 
 demand of the year are known, the total yearly cost for losses 
 would be PR X t X 365 X C e . 
 
 where I is the current at peak load, 
 
 R is the resistance of the circuit, 
 
 t is the equivalent hours, 
 
 C e is the cost of energy per kilowatt-hour. 
 
 The quantity, equivalent hours, is also of use in determining 
 the cost of energy per kilowatt-hour previously discussed since 
 the cost of energy losses is a component of that cost. 
 
 If the characteristic curve for any type of load were available 
 for a whole year, the sum of the squares of the current for each 
 hour taken from that curve times the resistance of the conductor, 
 would give the total yearly loss in watt hours. This however 
 would be a tedious computation and in most cases impracticable. 
 It is usually sufficiently accurate to obtain characteristic curves 
 
 TABLE 1 
 
 Circuit, 
 
 Jan. 
 
 Feb. 
 
 Mar. 
 
 Apr. 
 
 May 
 
 June 
 
 July 
 
 Aug. 
 
 Sept. 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 
 number 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 190 
 
 180 
 
 180 
 
 175 
 
 160 
 
 135 
 
 120 
 
 125 
 
 160 
 
 175 
 
 180 
 
 195 
 
 2 
 
 200 
 
 180 
 
 175 
 
 165 
 
 145 
 
 140 
 
 110 
 
 130 
 
 200 
 
 195 
 
 205 
 
 210 
 
 3 
 
 175 
 
 165 
 
 150 
 
 140 
 
 120 
 
 125 
 
 90 
 
 115 
 
 160 
 
 165 
 
 180 
 
 185 
 
 4 
 
 185 
 
 175 
 
 180 
 
 150 
 
 140 
 
 140 
 
 105 
 
 130 
 
 175 
 
 185 
 
 200 
 
 210 
 
 5 
 
 155 
 
 155 
 
 155 
 
 150 
 
 135 
 
 130 
 
 100 
 
 130 
 
 160 
 
 175 
 
 180 
 
 195 
 
 6 
 
 150 
 
 140 
 
 135 
 
 125 
 
 110 
 
 100 
 
 90 
 
 80 
 
 120 
 
 135 
 
 150 
 
 150 
 
 7 
 
 245 
 
 225 
 
 210 
 
 200 
 
 190 
 
 180 
 
 120 
 
 140 
 
 225 
 
 245 
 
 285 
 
 285 
 
 8 
 
 235 
 
 225 
 
 225 
 
 210 
 
 200 
 
 185 
 
 150 
 
 205 
 
 250 
 
 280 
 
 280 
 
 280 
 
 9 
 
 225 
 
 225 
 
 215 
 
 225 
 
 175 
 
 175 
 
 135 
 
 170 
 
 215 
 
 235 
 
 240 
 
 250 
 
 10 
 
 185 
 
 185 
 
 185 
 
 175 
 
 150 
 
 140 
 
 130 
 
 140 
 
 170 
 
 185 
 
 225 
 
 240 
 
 11 
 
 125 
 
 120 
 
 115 
 
 110 
 
 105 
 
 105 
 
 90 
 
 100 
 
 120 
 
 125 
 
 135 
 
 150 
 
 12 
 
 130 
 
 120 
 
 110 
 
 105 
 
 100 
 
 100 
 
 85 
 
 100 
 
 125 
 
 135 
 
 145 
 
 130 
 
 Total 
 
 2,200 
 
 2,095 
 
 2,035 
 
 1,930 
 
 1,730 
 
 1,645 
 
 1,325 
 
 1,565 
 
 2,080 
 
 2,235 
 
 2, 05 
 
 2,480 
 
 Average of 
 
 
 
 
 
 
 
 
 
 
 
 
 
 the 12.... 
 
 183 
 
 175 
 
 170 
 
 161 
 
 144 
 
 137 
 
 111 
 
 130 
 
 173 
 
 186 
 
 200 
 
 207 
 
LOAD CHARACTERISTICS 
 
 41 
 
 for each month or at least a typical characteristic curve applica- 
 ble to any month with allowance for the variations in the peak 
 from month to month. An example of an actual calculation of 
 equivalent hours on residence lighting circuits will indicate a 
 method which can be followed in such computations. 
 
 The characteristic variation of the load on a residence-lighting 
 circuit from month to month was obtained from Table 1 of 
 maximum-current readings for each month during 1 year on 12 
 typical circuits in various districts and with various loads. 
 
 250 
 
 ZZ5 
 
 c 
 '150 
 
 125 
 100 
 
 FIG. 4. Average of monthly maximum loads on twelve typical lighting circuits. 
 
 The curve (Fig. 4) plotted from the above average values indi- 
 cates clearly the variation of the monthly peak loads on a typical 
 circuit. Since the peak for the year occurs in December, the 
 peak for any month may be expressed as a fraction of this yearly 
 peak as follows: 
 
 Jan. 
 0.884 
 
 Feb. 
 0.845 
 
 Mar. 
 0.820 
 
 Apr. 
 
 0.778 
 
 May 
 0.695 
 
 June 
 0.662 
 
 July 
 0.536 
 
 Aug. 
 0.628 
 
 Sept. 
 0.835 
 
 Oct. 
 0.898 
 
 Nov. 
 0.965 
 
 Dec. 
 1.00 
 
 Hence, for example, the maximum current for February equals 
 .845 multiplied by the maximum current for the year, etc. 
 
 For convenience the PR loss for any month may be considered 
 equal to the loss for a typical day in the month times the number 
 of days in the month, since the variation of load from one month 
 to the next is not enough to warrant more detailed computation. 
 
 The PR loss in a feeder for any one day would be very nearly 
 equal to the sum of the squares of the current readings for each 
 
42 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 hour during the day multiplied by R. For a great part of the 
 day, however, the load on a lighting circuit is very light, the heavy 
 load and hence most of the loss occurring within a few hours in 
 the evening. By adding the squares of the hourly current 
 readings throughout a day and dividing by the square of the maxi- 
 mum current for the day a figure is obtained which represents 
 the number of hours for which the peak load for the day would 
 have to be carried steadily to produce the same I 2 R loss. 
 
 For a strictly accurate calculation, enough data should be 
 available to determine the shape of the load curve on a typical 
 circuit for one typical day for each month in the year. If this 
 is not possible, however, a fairly accurate approximation may be 
 arrived at if the load curves for only one or two months are avail- 
 able. If the equivalent number of hours at peak load for a 
 typical day for these months is determined the equivalent hours 
 for the other months may be calculated more or less accurately 
 by making them proportional to the number of hours between 
 sunset and about 10 P. M. allowing a little additional time in 
 the winter months for the morning lighting peak. The figures 
 obtained in the present case for these monthly equivalent hours 
 at peak load will be found in the first column of the accompany- 
 ing table. 
 
 The loss for any day in a month is proportional to the square 
 of the maximum current for the day multiplied by the equivalent 
 hours per day at peak load as obtained above. If now we assume 
 that the average daily peak will be 95 per cent of the peak for 
 the month, the loss for this day is proportional to the square of 
 the maximum current for the month multiplied by .95 2 multi- 
 plied by the equivalent hours per day at daily peak load as deter- 
 mined above. The monthly peak however is equal to a certain 
 fraction of the yearly peak as shown before. Hence the loss 
 for this day in terms of the yearly peak is proportional to the 
 maximum current for the year, squared, multiplied by this 
 fraction for the month, squared, multiplied by .95 2 , multiplied 
 by the equivalent hours per day at daily peak load. If this 
 figure is multiplied by the resistance of the circuit, R, the actual 
 loss is obtained. For example, the equivalent hours per day in 
 terms of daily peak load as determined for February are 4J. 
 The peak for February is .845 of the yearly peak. Hence for a 
 typical day in February the P R loss equals (yearly maximum 
 current) 2 X .845 2 X .95 2 X 4.5 X R. 
 
LOAD CHARACTERISTICS 
 
 43 
 
 The total yearly loss would be the sum of the daily losses 
 thus obtained. An average of the figures for each month would 
 then give the average loss per day. Since the quantity (yearly 
 maximum current) 2 X R is the loss due to the yearly peak load, 
 the average number of hours per day which that load must con- 
 tinue may be obtained by averaging the other factors entering 
 into the computation of average daily loss in terms of yearly 
 peak. These are the quantities (equivalent hours per day at 
 
 daily peak load) X 
 
 X 95' as determined for 
 
 each month. The resulting average is the value of " equivalent 
 hours " for that load. 
 
 The following table shows the data worked out from the above 
 example of residence lighting. 
 
 TABLE 2 
 
 
 1 
 
 
 
 > 
 
 3 
 
 
 
 1 
 
 1 
 
 3 
 
 <u 
 
 ?|i 
 
 
 
 +3 
 
 a 
 
 
 CM 
 
 1 X 3 
 
 
 S 03 
 
 h 
 
 o 
 
 J5* ^ 
 
 
 1 
 
 > 03 
 
 til 
 
 i 
 
 1 
 
 fl ^ 
 
 LH 
 
 
 1 
 
 w^-2 
 
 8^ 
 
 
 v~-^ 
 
 
 January 
 
 5H 
 
 .s 
 
 4 
 
 .781 
 
 4.10 
 
 February 
 
 4;Hz 
 
 ,i 
 
 >45 
 
 .714 
 
 3.21 
 
 March 
 
 sK 
 
 E 
 
 .20 
 
 672 
 
 2 185 
 
 April 
 
 &A 
 
 
 78 
 
 .606 
 
 1.67 
 
 May. ... 
 
 2^^ 
 
 .6 
 
 >95 
 
 483 
 
 1 088 
 
 June 
 
 1 3/ 
 
 e 
 
 >62 
 
 438 
 
 767 
 
 July.. 
 
 2 
 
 i 
 
 >36 
 
 .287 
 
 .574 
 
 August 
 
 
 .e 
 
 >28 
 
 395 
 
 987 
 
 September 
 
 ol/ 
 
 
 
 >35 
 
 697 
 
 2 263 
 
 October 
 
 4 
 
 . 
 
 >98 
 
 .806 
 
 3.224 
 
 November 
 
 5 
 
 .c 
 
 65 
 
 .931 
 
 4.655 
 
 December . 
 
 clZ 
 
 1 C 
 
 
 
 1 00 
 
 5 500 
 
 
 
 
 
 
 
 Total 
 
 
 
 
 
 30 223 
 
 
 
 
 
 
 
 Average 
 
 
 
 
 
 2 519 
 
 
 
 
 
 
 
 2.519 X (.95) 2 = 2.27 eq. hr. per day at peak load. 
 
 The above indicates that for the example used of purely light- 
 
44 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 ing load the total loss for the year will be the same as if the peak 
 load were carried 2.27 hr. per day throughout the year. 
 
 Corrections for Special Conditions. The above figures are sub- 
 ject to correction under certain conditions. Actual loads on 
 circuits were used and no allowance was made for the normal 
 increase which might be expected on lighting load due to addi- 
 tional customers, the increased use of current by old customers, 
 etc. This might be a satisfactory figure for use in many cases 
 but there might be occasions in which a load showing only 
 seasonal variations would be encountered. For example, if 
 distribution transformers are kept loaded nearly to capacity, 
 new installations would care for the yearly increase and each 
 individual transformer would show nearly the same load from 
 year to year. In case the annual rate of increase is known, a 
 correction can be applied to the figures given for each month to 
 reduce it to the same maximum load for the year. In the 
 example given, the average yearly increase over a number of 
 years was found to be 20.3 per cent or 1.69 per cent per month. 
 Assuming December as the yearly peak, the ratio of the peak 
 for each month to the yearly peak was corrected by a proportional 
 part of the yearly increase, i.e., for November it was increased 
 by 1.69 per cent, for October 3.38 per cent, etc. 
 
 TABLE 3 
 
 
 Jan. 
 
 Feb. 
 
 Mar. 
 
 Apr. 
 
 May 
 
 June 
 
 July 
 
 Aug. 
 
 Sept. 
 
 Oct. 
 
 Nov 
 
 Dec. 
 
 Monthly ratio 1 
 
 .884 
 
 .845 
 
 .820 
 
 .778 
 
 .695 
 
 .662 
 
 .536 
 
 .628 
 
 .835 
 
 .898 
 
 .965 
 
 1.00 
 
 Correction fac- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 tor 
 
 1.177 
 
 1.160 
 
 1.143 
 
 1.126 
 
 1.119 
 
 1.102 
 
 1.085 
 
 1.068 
 
 1.051 
 
 1.034 
 
 1.017 
 
 1.00 
 
 Corrected ratio 
 
 1.040 
 
 .981 
 
 .937 
 
 .876 
 
 .778 
 
 .729 
 
 .581 
 
 .571 
 
 .878 
 
 .930 
 
 .980 
 
 1.00 
 
 Monthly Ratio 
 
 peak load for month 
 peak load for year 
 
 unconnected. 
 
 The above indicates that the actual corrected peak on such load 
 is in January but not enough difference will be introduced to 
 necessitate a revision of the figures to that basis. 
 
 If now the equivalent hours are computed on the basis of the 
 above ratios a new figure is obtained for a loading with no yearly 
 increase. In this case the equivalent hours thus corrected are 
 computed to be 2.65 instead of 2.27 as determined for actual 
 loading, with a yearly increase. 
 
 A further correction may be applied if, as in case of transform- 
 ers, the full-load capacity is to be used in studying losses rather 
 
LOAD CHARACTERISTICS 45 
 
 than the actual load carried. If, for example, on the above 
 circuits the transformers were carrying, on an average, 89 per 
 cent of their full-load capacity, the equivalent hours based on 
 connected capacity would be 2.65 X .89 2 = 2.07. That is to 
 say, the year's loss in energy would be equal to the full load 
 current on the transformer carried 2.07 hr. per day throughout 
 the year. 
 
 Other corrections may be necessary to meet particular con- 
 ditions. The above will give an indication of how such 
 corrections should be applied. 
 
 The matter of equivalent hours should be carefully studied and 
 as accurate figures as possible obtained for various classes of load. 
 The values will vary, naturally, for different sections of the 
 country as well as for different localities in the same section or on 
 the same system. The habits of a community, as to hours of 
 rising and retiring, etc., will affect the value for lighting loads. 
 On power loads, of course, the nature of the industry will be a 
 controlling factor. In general for the cases coming within the 
 experience of the writers the values lie within the range given 
 below : 
 
 Power load from to 10 eq. hr. 
 Residence lighting from 2 to 3 eq. hr. 
 Store lighting (small) from 2 to 3 eq. hr. 
 Store lighting (large) from 2 to 5 eq. hr. 
 Street lighting from 5 to 10 eq. hr. 
 
 Relation between Load Factor and Equivalent Hours. It is 
 
 interesting to find what the relation between load factor and 
 equivalent hours is, especially as one or the other quantity may be 
 available in a problem, while the other is necessary for the 
 solution at hand. For our purpose it will be found particularly 
 useful in making an approximate determination of equivalent 
 hours if the load factor is known. 
 
 Limits may be established within which the value of the 
 relation between load factor and equivalent hours will lie in all 
 cases. 
 
 The extreme cases are as follows: 
 
 I. The peak load is on for a short time only. The remainder 
 of the load curve is flat for the rest of the day. See Fig. 5. 
 
 In this case the amount of the continuous load divided by the 
 momentary peak gives the load factor. 
 
46 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 The I Z R loss = I av .*R X 24 = (LF X I max ) 2 R X 24. 
 
 Equivalent hours at peak load = 
 
 X 
 
 X 24 
 
 (LF) 2 X 24. 
 
 II. The peak load is continuous for a part of the day the 
 load is thereafter. See Fig. 6. 
 
 In this case the load factor is the number of hours the load is 
 on, divided by 24. 
 
 ^MAXIMUM LOAD 
 
 
 
 1 
 
 L 
 
 
 5 | 
 
 
 - 24- HOURS " 
 FIG. 5. 
 
 ^.MAXIMUM LOAD 
 
 ^1 
 
 
 
 
 L 
 
 
 -J 
 
 !<- 
 
 FIG. 6. 
 
 
 ^-MAXIMUM LOAD 
 
 24- HOURS 
 FIG. 7. 
 
 The PR loss = I* max R X (LF X 24). 
 
 Equivalent hours = LF X 24. 
 
 III. For any intermediate arrangement of load. See Fig. 7. 
 
 At any time, t nt let the value of / = p n lmax- 
 
 The load factor = 
 
 PR loss = f(p n I ma x) 2 Rdt. 
 Equivalent hours = Sp n *dt. 
 
 (6) 
 
LOAD CHARACTERISTICS 47 
 
 It is evident from the nature of the curves that for any load 
 factor Cases I and II are limiting cases since no greater dis- 
 tribution of load can be obtained than Case I and no greater 
 concentration than Case II. The rule can be stated as follows: 
 
 "For any given load factor, the corresponding value of 
 equivalent hours will be somewhere between the limits of (load 
 factor X 24) and (load factor) 2 X 24. , 
 
CHAPTER VI 
 GENERAL EQUATION 
 
 KELVIN'S LAW GENERAL METHOD OF SOLVING PROBLEMS 
 PRESENTATION OF RESULTS 
 
 The previous chapters have dealt largely with the data neces- 
 sary for the economical study of distribution problems, and the 
 methods of obtaining that data. The cost of material and labor, 
 the annual charges on these items, the unit cost of energy for 
 different loads and the annual cost of energy losses have all been 
 taken up in some detail. Once the data is collected, there still 
 remains the problem of so utilizing it as to obtain the most 
 economical conditions for the line or lines under consideration. 
 Also, means must be found for so exhibiting the results, by 
 equations, graphs, tables, etc., that they will be convenient of 
 application to present problems and to future similar problems 
 and subject to easy revision with changing prices. 
 
 Although every problem of this nature that is considered will 
 present certain characteristics of its own which make it different 
 from all others, there are certain underlying principles and 
 methods of procedure which are applicable to all. A brief dis- 
 cussion of these will be given here. A good understanding of 
 these general methods will simplify the study of their appli- 
 cation to particular problems, which will follow in subsequent 
 chapters. 
 
 Kelvin's Law. To Sir William Thompson (Lord Kelvin) is 
 generally attributed the basic study of economical conduction of 
 electrical currents. In 1881 he expressed the principle that 
 "the most economical size of copper conductor for the trans- 
 mission of electrical energy would be found by comparing the 
 annual interest on the money value of the copper with the money 
 value of the energy lost in it annually in the heat generated in it 
 by the electric current . . . Contrary to a very prevalent 
 impression and belief, the gage to be chosen for the conductor does 
 not depend on the length of it through which the energy is to be 
 transmitted. It depends solely on the strength of the current 
 to be used supposing the cost of the metal and of a unit of energy 
 
GENERAL EQUATION 49 
 
 to be determined." In expressing this mathematically, the total 
 annual cost was expressed as the sum of the fixed charges on the 
 conductor and the cost of energy loss. The size of wire for which 
 this would be a minimum was then determined, being that size 
 for which the two component charges are equal. What is 
 generally known as Kelvin-' s Law has been formulated from this, 
 i.e., that the most economical size of conductor is that for which 
 the annual charge on the investment is equal to the annual cost 
 of energy loss. Under modern conditions with the use of alter- 
 nating currents, wide range of voltages, large variety of wire 
 sizes, with and without insulation, etc., an indiscriminate use of 
 Kelvin's Law as thus stated is liable to lead to considerable error. 
 It will apply strictly only to problems for which the cost of 
 conductor supports can be neglected (or is directly proportional 
 to the size of wire), when the cost of any size of wire is directly 
 proportional to the cross-sectional area of the copper, when no 
 transformers, condensers, or other equipment need be considered 
 and when the cost of energy loss is inversely proportional to the 
 wire size. Needless to say, few problems could be included under 
 the above. In most cases it is necessary to revert to Kelvin's 
 original method which was to determine an expression for the 
 total annual cost and from this to determine the most economical 
 condition desired. This will include not only the investigation 
 of the most economical wire size but also of the most economical 
 voltage, the most economical route, the most economical size 
 and spacing of transformers, and other similar questions. From 
 this can likewise be determined the actual advantage, in dollars, 
 of one installation over another, where it would be economical to 
 change from one type of installation to another, and a great 
 many other extremely useful considerations. 
 
 General Equation. A general expression for total annual 
 cost which will be applicable to most of the problems in electrical 
 distribution lines can be set down. Naturally each of the items 
 included will be somewhat different for different problems and 
 all problems will not include all of these items. The symbol g 
 will be used throughout to indicate percentage of fixed charges 
 on investment. It will vary, of course, with different kinds of 
 property. In this general equation, g will be used as a general 
 symbol, i.e., the expression (g X a quantity) indicates that the 
 annual charges rather than the first cost are considered. Then 
 we have: 
 
50 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Total annual cost = g (cost of right-of-way) 
 
 + g (cost of poles and fixtures or under- 
 ground ducts in place) 
 + g (cost of conductors in place) 
 + g (cost of transformers and transformer 
 
 equipment installed) 
 
 H- g (cost of any special equipment used) 
 + cost of maintenance, inspection, testing, 
 
 etc. 
 
 + cost of annual energy loss on line 
 + cost of annual energy loss on transformers 
 and other equipment. (7) 
 
 Individual cases may produce other charges which must be 
 added but the above is characteristic. 
 
 Units. In some specific problems, such as the comparison of 
 economy between two sizes of conductor for some particular 
 location and definite load, actual values for voltage, resistance, 
 costs of materials and energy and other constants could be intro- 
 duced at once into this equation and the total annual cost of each 
 installation determined in dollars. In the usual case, however, 
 it is desirable to make the study more general, covering a more 
 or less wide variation in conditions, so that it can be utilized to 
 reduce computation on future problems or in the determination 
 of standards for a given class of installations. For this purpose 
 it is advisable to represent as many of the quantities as possible 
 by symbols and to carry these symbols through the computations 
 as far as possible. This also facilitates revision of the formulas, 
 graphs, etc., if a change in prices makes this advisable. For 
 example, the following are some of the symbols most commonly 
 used in this book: 
 
 TABLE 4 
 
 W = load in watts P = resistivity of conductor 
 
 kw. = load in kilowatts material 
 
 E = voltage cos 6 = power factor 
 
 I = amperes A = cross-sectional area of con- 
 
 R = resistance of circuit ductor. 
 
 X = inductance of circuit /l/o'' TC = weight of conductor per 
 
 r = unit resistance of conduc- unit length 
 
 tor T = transformer size in kilovolt- 
 
 x = unit inductance of con- amperes 
 ductor 
 
GENERAL EQUATION 51 
 
 TABLE 4 (Continued) 
 
 Rt = equivalent resistance of trans- g = per cent fixed charges, (interest 
 
 formers taxes, depreciation, etc.) 
 
 C r = cost of right-of-way per unit I = equivalent hours 
 
 Ccu = cost of copper per pound V = per cent voltage drop 
 
 C sr = cost of stringing conductors P = per cent power loss 
 C e = cost of energy (subscript 1, 2, 
 "3, etc., indicate variations) 
 
 In any given case, most of these quantities will be fixed by the 
 conditions of the problem and may be considered as constants 
 so that the general equation for annual cost may usually be 
 reduced, in its final form, to one containing only two or three 
 variables, such as load in kilowatts or amperes, cross-sectional 
 area of wire, power factor, or equivalent hours. Examples of 
 this will be shown later. 
 
 Presentation of Results. The equation for total annual cost 
 once obtained, there still remains the question of getting from it 
 the information desired in the best possible form for convenient 
 use. It is found that there are, in general, three ways in which 
 it is convenient to accomplish this. Conditions of the problem 
 and the results desired will determine which one of these is most 
 applicable in any case. 
 
 1. The actual annual cost in dollars may be plotted in a 
 curve or series of curves. Where the equation for annual cost 
 is expressed in more than one variable, such as size of wire and 
 load, for example, one of these variables must be held constant 
 in plotting any one curve. Sufficient number of such curves 
 must then be plotted to show the required variation in that 
 quantity. In this way a series of curves may be obtained, for 
 example, one curve for each standard size of wire, showing the 
 relation between annual cost and load on a line with that size. 
 An example of this method is shown in Fig. 8. When the expres- 
 sion for annual cost contains more than two such variables, how- 
 ever, this method is usually not applicable, 
 v 2. In some cases with three variables entering into the total 
 annual cost the following method will be founpl convenient. 
 Suppose, for example, the annual cost depends on a variable 
 value for equivalent hours, variable load and variable wire size. 
 If the expression for annual cost for two standard sizes of wire 
 are equated, the resulting equation plotted between load and 
 equivalent hours shows the dividing line between economy for 
 
52 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 one size or the other. The accompanying figure (Fig. 9) shows 
 such a series of curves as determined for three-phase secondary 
 
 15 ZQ 25 30 
 
 Load Density in Kw. per 1000 feet 
 
 FIG. 8. Annual cost per 1,000 ft. of 3 No. 4 secondaries. (Includes fixed charges 
 on wire in place and energy loss.) 
 
 10 W 30 40 50 60 70 80 90 100 HO 120 130 WO 150 160 170 
 K_ Hours per Week 
 
 FIG. 9. Most economical size of wire for three-phase secondaries (small power 
 
 ' loads). 
 
 for a certain type of load. Any point lying between two curves 
 indicates economy for the corresponding size of wire. This 
 method has the disadvantage of not exhibiting quantitative 
 
GENERAL EQUATION 
 
 53 
 
 economy. The results are qualitative only, since the nearer 
 the point lies to either limit of the area the less the relative dif- 
 ference between the cost with the size of wire indicated and the 
 next adjacent size. 
 
 3. A third method gives results that are neither quantitative 
 nor qualitative. Also, its application is usually somewhat 
 limited. In certain cases, however, it is preferable to any other 
 method. Where the expression for annual cost contains several 
 variables, if these can all be reduced to terms of two variables 
 
 
 
 c 
 
 Ib 
 
 Q 
 
 Load Density in Kw.per 1000 feet 
 FIG. 10. Most economical voltage drop on lighting secondaries. 
 
 (as percentage voltage drop and load, for example) and the first v 
 derivative of the resulting expression with respect to one of these 
 variables be set equal to 0, the equation thus obtained will give 
 the most economical relation between the .two variables (as 
 most economical voltage drop for any load under the given con- 
 dition). The accompanying figure gives an example of a curve 
 derived in this manner (Fig. 10). 
 
 As was suggested before, it will be found of considerable 
 advantage to prepare the .equations from which the final curves 
 are plotted in as general terms as possible, i.e., with as many of the 
 constant quantities as possible represented by symbols. It is, 
 then, a comparatively easy matter to revise the curves to meet 
 changing prices of material and labor, or other conditions of 
 load, voltage, power factor or materials of construction than 
 those originally contemplated. 
 
54 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 In some cases it may be found useful to develop the results 
 in numerical tables or sometimes merely by a simple formula. 
 Such cases are not the most usual, however, and the above 
 methods will probably be found sufficient for most purposes. 
 On some more extensive problems, all three methods of exhibiting 
 data will be used. 
 
 In future chapters, a number of problems met with in practice 
 will be taken up and their treatment in accordance with these 
 methods will be described. Some of the details necessarily 
 omitted in the previous discussion of these methods will be 
 brought out in the individual problems and the procedure hereto- 
 fore described in general terms will be actually carried through. 
 

 CHAPTER VII 
 POWER LOSS AND VOLTAGE DROP 
 
 CHARTS FOR SIMPLIFIED SOLUTION FOR POWER Loss AND VOLTAGE 
 
 DROP 
 
 Power Loss. In most of the solutions of problems in economy 
 appearing in this book, the power loss is introduced as a function 
 of the load, the wire size, the equivalent hours, etc. The reason 
 for this is evident from the nature of the methods used and the 
 results desired. There are often cases, however, where the 
 power loss alone is wanted. There will be given in this chapter 
 simple curves, with their derivation, which enable the power 
 loss to be quickly solved in the great majority of problems. 
 The most usual case of power loss is that due to the resistance 
 of the conductor, i.e., the PR loss. In high-tension trans- 
 mission lines, leakage losses and corona losses become important. 
 Charging current also has an effect on PR loss. Such problems, 
 however, are comparatively rare in the work of most engineers 
 and warrant special treatment when encountered. Methods of 
 solving for corona loss, leakage, etc. are given in the handbooks 
 and elsewhere and a discussion of them is beyond the province of 
 this work. In the greater majority of problems, the PR loss is 
 all that need be considered. 
 If D = length of line in feet, 
 W = the load at the receiver end in watts ( = kilowatts X 
 
 1,000), 
 
 E = the receiver voltage, 
 
 A = cross-sectional area of conductor in circular mils, 
 p = resistivity of conductor material in ohms per mil foot, 
 cos 8 = the power factor of the load W. 
 
 PR loss = (=- ) ^- X 2 watts (for single-phase) (8) 
 
 \Hi COS u/ A. 
 
 I W \ 2 oD 
 PR\oss = (-7=^ -) " X 3 watts (for three-phase) (9) 
 
 \\/3Jl/ COS "/ A. 
 
 If P = per cent power loss in terms of power delivered, 
 JP _ 2W 2 P D _1 
 100 (E cos BY A X W 
 
 P = 
 
 (E cos 
 
 \ 55 
 
56 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 _P TPpP 1 
 
 100 (E cos 0) 2 A A F 
 
 For the same load, same voltage between conductors, and the 
 same conductor size the loss with single-phase is twice that with 
 three-phase. 
 
 If it is desired to consider the load, voltage and power factor 
 at the source instead of at the receiver, if 
 
 W f = load at source in watts, 
 E' = voltage at source, 
 cos 6' = power factor at source. 
 
 The line loss in percentage of load at source 
 D/ 20QW'pD ,, . , , 
 
 P =(f rsmgle - phase) 
 
 or three-phase) (13) 
 
 For copper conductor p = 10.8 approximately. If load is 
 expressed in kilowatts (kw), the formula becomes 
 
 2.16Xl0 6 Xfcw.Z>,, , , 
 P = ~ -TCT-T-^TI (for single-phase) (14) 
 
 1.08 X 10 6 Xkw.D 
 
 These formulas are comparatively easy to use. However it is 
 believed the work may be somewhat simplified, especially where 
 a large number of such computations are to be made, by use of 
 the accompanying chart (Fig. 11). The use of this chart reduces 
 the computation to a simple multiplication of round numbers. 
 
 The chart is plotted as follows: 
 
 A series of circular arcs (with the center at 0, 0) are drawn, 
 each representing a given voltage. Voltages range from to 
 150, but as will be shown below, the same arcs may be used for 
 any voltage. Diagonal straight lines are drawn through 0, 
 at various slopes, each representing a given power factor. It 
 is evident that the abscissa of the intersection of any arc with 
 any diagonal gives the corresponding value of E cos 0. 
 
 For each standard wire size considered a curve is now plotted 
 
 j) 
 between r n/i QQQ anc ^ E cos # gi ym g f r anv voltage, power 
 
 factor, and wire size the percentage power loss per kilowatt per 
 
POWER LOSS AND VOLTAGE DROP 
 
 57 
 
 1,000 ft. Curves for other sizes of wire can be easily added if 
 desired. The upper part of these curves was drawn to a con- 
 densed scale to give a greater range of values in the more un- 
 usual cases. The curves here shown are for three-phase. The 
 
 1 
 
 FOR USE IN DETERMINING POWER LOSS IN PERCENT 
 OF DELIVERED POWER ON THREE PHASE LINES 
 For single phase multiply scale readings by 2 
 
 To Use : I. -Reduce voltage to an equivalent secondary 
 voltage within Hie limits of the chart 
 (OtolSOvolts) by dividing by a convenient 
 ratio, 2, 10, 200, etc. 
 2. -Locate intersection of voltage arc with 
 
 power factor diagonal (E cos O) 
 3.- Intersection of this ordinate with curve 
 forgiven wire size gives percent power 
 loss on scale at left. 
 
 4.-D/vide scale read ing by the square of ratio 
 used in 1, and multiply by distance in 
 -thousands of feet and loat 
 
 RESUL T= Total percent power 
 o\o /oss on line 
 
 Ib 50 75 100 
 
 EcosQ 
 
 FIG. 11. Power loss curves. 
 
 \15 
 
 power loss for single-phase would be twice the values shown for 
 three-phase. 
 
 The use of these curves is as follows: 
 
 1. Reduce the voltage considered to any equivalent secondary voltage 
 within the scale of the chart (less than 150 volts) by dividing by some 
 convenient transformation ratio such as 2, 4, 10, 20, 200, 400, 1,000, etc. 
 
58 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 2. Locate intersection of voltage arc and power factor diagonal. 
 
 3. The intersection of the vertical through this point with the curve for 
 the proper wire size gives the value of P on the scale at left. 
 
 4. Divide this scale reading by the square of the transformation ratio 
 used, to give percentage power loss per kilowatt per 1,000 ft. for the given 
 voltage. Multiply by the number of kilowatts and by the length of line 
 in thousands of feet for total percentage power loss in terms of power 
 delivered. (If line is single- phase multiply this quantity by 2.) 
 
 Take for example the following problem : 
 
 Load 1,200 kw. three-phase 
 
 Power factor 75 per cent 
 
 Voltage at receiver 1,400 
 
 Wire size 3 No. 
 
 Distance 8,000 ft. 
 
 4 400 
 ' 4 Q =110 (transformation ratio = 40) 
 
 .38 3 
 
 Power loss = ^ x ^ X WW X 8 =9.12 per cent or 109.4 kw. 
 
 Voltage Drop. In studying the economy of a line, it must not 
 be forgotten that the element of good service is also important. 
 Good service depends largely on good regulation which, in turn 
 depends a great deal on the voltage drop of the line. Often, 
 a line economically loaded will have too great a voltage drop. 
 If it is artificially regulated, the cost of the regulator enters into 
 the consideration of economy. In some cases the use of large 
 conductors without a regulator may be more economical than 
 smaller conductors with a regulator. 
 
 The computation of voltage drop is apt to be a tedious opera- 
 tion if carried out often. Several means of simplifying this have 
 been published, such as the well-known Mershon Diagram and 
 the Dwight chart. 
 
 When relating to high-tension lines, the problem of voltage 
 drop involves consideration of line capacity, etc. These prob- 
 lems are usually of sufficient importance to warrant detailed 
 computation. Formulas for this are given in the handbooks 
 and elsewhere. 
 
 E l = E 2 cosh \/ZY + h \/Z7Y sinh \/ZY 
 
 Ii = 7 2 cosh \ZT + (E 2 /Z/Y) sinh 
 Where EI and E 2 are voltages from phase to neutral at the send- 
 ing and receiving ends respectively; Z is the impedance per wire; 
 Y is the admittance from phase wire to neutral. 
 For medium and low-voltage problems, it is usually sufficient 
 
POWER LOSS AND VOLTAGE DROP 69 
 
 to consider inductive reactance and resistance only. The method 
 and charts for computing voltage drop, given below, are based 
 on these quantities. They are derived from the equation ordi- 
 narily used for such computations, i.e. 
 
 JL - "V(ff cos + VZRID) 2 + (E sin + A/3 X ID) 2 - E 
 100 E 
 
 (for three-phase lines) 
 
 , \ 2 , / . . V3 X IDY t 
 
 cos + ^ - I + (sin + ^ -- ) -1 
 
 Where V = per cent voltage drop in terms of receiver voltage, 
 R = resistance in ohms per foot, 
 X = inductive reactance in ohms per foot, 
 D = distance from source to receiver in feet, 
 I = current per wire. 
 
 But 
 
 100 W D 100 -v/3 RID 
 P = per cent power loss = 77^ -- T^VT = - FJ 
 
 (E cose) 2 A E cos e 
 
 (see above). (17) 
 Substituting 
 
 icfo = \/( 
 
 cos 9 + fo? cos e + sin 
 
 = cos 
 
 7 = B = 10 cos e / / i 4. P \ 2 + /tan g + P X \* - 10 
 
 (19) 
 
 (The above computation is for three-phase but the resulting 
 expression for B is the same for single-phase.) 
 
 B then is a quantity expressing the relation between per cent 
 voltage drop and per cent power loss. It depends on the power 
 
 /X\ 
 factor (cos 6), size and spacing of conductors (-p) and percent 
 
 \ Kit 
 
 power loss (P), but is independent of load or voltage. It is 
 evident that if B is known the per cent voltage drop may be 
 easily obtained from the per cent power loss by the simple relation 
 
 V = BP (20) 
 
 B may be plotted as shown in the accompanying curves (Fig. 
 12) (a) and (b). Since the variation of B with P is not great 
 excepting for high-power factor and large conductors, three 
 
60 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 3.0 
 
 'g'.v/i 
 
 Power Factor^eO, 70,80,90% 
 Percent volts drop -of delivered volts, 1.5 
 
 power 
 
 ," r 
 
 " TO USE '.-Determine value of x /r for si: 
 andspacingofwire used-from 
 
 curves on right. 1 
 
 Determine value of "B ' for this 
 value of x /r from proper curve 
 on left. | | 
 
 Multiply bu per cent power 
 
 lo&s (from other curves) for Q t, 
 
 percent volts drop (V) V-PB 
 
 2.5 
 
 ) 20 W GO 80 
 Spacing in Inches 
 
 FIG. 12a Values of 
 
 "B 
 
 Values of "B"= y / P for 
 
 =30, 95, 100 % 
 
 V=Percent volts drop -of delivered volts 
 P= power hss -of power \. 
 
 TO USE '.-Determine value ofX/rforsne 
 '<o and spacing of wire used- from curves 
 on right 
 
 Determine value of "B "for this value 
 of x/r from proper curve on left: 
 
 Mutt i ply by percent power loss(from 
 er curve st for 
 
 of her curve st for percent volts drop 
 
 1.5 
 "B" 
 FIG. 126. Values of "B " 
 
 20 40 60 80 
 Spacing in Inches 
 
POWER LOSS AND VOLTAGE DROP 
 
 61 
 
 fixed values of P were chosen which cover the usual range of 
 problems, i.e., 1 per cent, 10 per cent and 20 per cent. For 
 each power factor, a curve between B and X/R is plotted for 
 each of these values of P. Other values of P may be interpolated 
 
 Percent of delivered power tost" 
 jper kilowatt per 1000 feet o ver 
 3 phase line ~ 
 For single phase line multiply 
 figure given by curve by 2 
 
 To oMain percent voltage drop, use 
 these curves in conjunction with 
 quantity "B'g/ven by figure tea 
 
 10 80 
 
 Power Factor 
 
 FIG. 13a. Power loss curves. 
 
 if necessary. The value of X/R is shown directly by the curves 
 on the right for standard sizes of conductor and any spacing. 
 Use of Curves. The use of the curves is as follows: Locate 
 the point on the X/R curve for the given wire size and spacing. 
 (For three-phase unequal spacing use the equivalent spacing, S = 
 Pass across the sheet to the left on the horizontal 
 
 through this point until it intersects the curve for the given power 
 
62 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 factor and value of P. The corresponding value of B is found 
 on the scale at the bottom. If this is multiplied by the value of 
 P computed from Fig. 11, the desired value of V (per cent voltage 
 drop) is obtained. 
 
 Example. In the example given above for the determination 
 of Pj let the spacing between the wires be 28 in., 28 in., 56 in. 
 
 L. 
 
 105 
 
 \ 
 
 PERCENT OF DELIVERED POWER LOST PER KILOWATT 
 PER IOOO FEET OVER 3 PHASE' LINE 
 
 For single phase line multiply 
 figure given by curve by 2 
 
 Use curve marked "44OO Volts "for 44, OOO 440O 
 44O J 2^000, 23OO, 220 and 110 Volfs. 
 Use curve marked "4600 Volte "for 4G, OOO. 4GOO. 
 460, 23,000, 2300, 230 and 1/5 Volts. 
 
 Use curve marked "SOOOVolts "forSqOOOjSOOO, ' 
 SCO, 25000, 2SOO, 2SO and I2S Volts, with 
 proper&cale. \ 
 
 To obtain percent voltage drop, use 
 
 these curves in conjunction with 
 
 quantity "B"given by other curves . 
 
 4.8 
 
 0.48 I.J 
 
 10 80 
 
 Power Factor 
 
 FIG. 136. Power loss curves. 
 
 64 
 
 105 
 
 T^i 0.56 2.24 
 
 The equivalent spacing = V28 X 28 X 56 = 35 in. 
 Interpolating between the 70 per cent and 80 per cent power- 
 factor curves 
 
 B = 1.18 
 
 V = 1.18 X 9.12 = 10.75 per cent or 473 volts. 
 
POWER LOSS AND VOLTAGE DROP 
 
 63 
 
 The curves given here are general and may be used for any 
 voltage, load, length of line and spacing between conductors. 
 The range of wire sizes and power factor is more limited but 
 the curves could be easily extended to cover any desired value. 
 For everyday use on the problems arising on any given system less 
 general curves may be derived from these which still further 
 simplify computations. As an example, Fig. 13 (a) and (6) 
 shows a set of curves plotted for work on 4,600-volt lines. Three 
 curves are shown for each wire size giving a range from 4,400 to 
 
 ine 
 
 per 1 
 
 ^ 
 
 ~*y 
 
 0.6 
 
 
 -4600W1--80% Power Factor -Spacmg27"a 
 
 mf 
 
 400 
 
 dOO 
 
 1200 
 
 1600 2000 2400 
 
 Load in Kilowatts 
 
 2800 
 
 3200 3600 
 
 FIG. 14. Curves showing per cent voltage drop (of delivered voltage) per 1,000 
 
 ft. of line. 
 
 5,000 volts. For any power factor (lower scale) the scale on the 
 left gives the per cent power loss per kilowatt per 1,000 ft. 
 Figure 13 (6) shows how scales for other voltages can be added. 
 
 Figure 14 shows another special curve giving per cent voltage 
 drop per 1,000 ft. of line for 4,600 volts, three-phase, 80 per cent 
 power factor, 27-in. spacing. 
 
 Figure 15, called "Load Curve for Power Lines", gives the 
 distance to which any load can be carried on a three-phase, 
 4,600-volt line at 80 per cent power factor with a 10 per cent drop 
 in voltage. 
 
 Figure 16 is a similar curve for single-phase lines with a power 
 factor of 95 per cent. 
 
64 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Figure 17 is a load curve for 220-volt, three-phase secondary 
 with 10 per cent drop. 
 
 Figure 18 is a load curve for 22 ^i2-volt, single-phase second- 
 ary with 3 per cent drop. 
 
 36,000 
 
 32000 
 
 28000 
 
 Load transmitted by 3 phase 
 OL 3wirej46OO volt- lines with" 
 
 ^ IO% voltage drop 
 ^^ Power factor 80% -28 Spacing 
 ^ v Load concentrated at end 
 
 NO TE^The values KW for any , 
 dkfter voltage drop are ven. 
 nearly proportional to 
 
 os^fbr 10 % drop 
 
 Foroth^r voltages, KWis 
 
 proportioned to the square, 
 of the voltae* 
 
 Safe Current Carry ing 
 
 24000 
 
 20000 
 
 16000 
 
 12000 
 
 8000 
 
 4000 
 
 800 1200 
 
 Load Trcmsmittedj 
 
 FIG. 15. Load curves for power lines. 
 
 It is a comparatively simple matter to derive any such special 
 curves desired by use of the general curves for power loss and B 
 given above. 
 
 Approximate Method for Secondaries. For low-voltage 
 problems, such as for secondaries, an approximate determination 
 
POWER LOSS AND VOLTAGE DROP 
 
 65 
 
 is usually as accurate as one more detailed. It will be found 
 that the expression 
 
 -j = volts drop per ampere = R cos 6 + X sin 8 (21) 
 
 while an approximation, is sufficiently accurate for most problems 
 of this class. 
 
 36,000r 
 
 32,000 
 
 28,000 
 
 24,000 
 
 20,000 
 
 9000 
 
 8000 
 
 15,000 
 
 re.ooo 
 
 8000 
 
 4000 
 
 100 200 . 300 400 500 600 
 
 Load in Kilowatts 
 
 FIG. 16. Load curves for single-phase lines. 
 
 As an example for the use of this expression, the curves in Fig. 
 19 (a.b.c) are given. The load considered is residence lighting 
 on three- wire, 11 ?^20~volt secondary. The average per residence 
 is assumed to range from 135 to 200 watts. Two different lengths 
 
66 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 1200 
 
 Horsepower- Load Concentrated at End of Secondary 
 20 40 60 80 100 120 140-160 180 200 220 240 260 280 300 320 340 560 
 
 i i i i i i 
 
 Load Transmitted by 3 Phase, 3 Wire, 22O Volf Secondary 
 with 10 % Vol+a&e Drop. \ 
 Power Factor 80 % -28 "spacing 
 
 Where W= Load in waffs 
 
 A = Cross-sectional area of conductor 
 V = Percentage voltage drop \ \ 
 D - Distance from source to receiver in feef 
 B.and C- Cons tan ts 
 I 
 
 For other va/ues of voltage drop W/s approximately 
 proportional fo V (B varies somewhat with V) f 
 For other vo/f age e W varies with z 
 
 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 
 Horsepower- Load Uniformly pfstributed Transformer at End 
 
 FIQ. 17. Load curves for power lines. 
 
 6000 
 
 Load transmitted by 3 w'ire, .224/nz volt secondaries 
 
 . 
 with 3% voltage drop II 
 
 ower Factor 95 %-Loaduni form/t d 
 
 D= Length of line one way fo. 
 concentrated load, transformer 
 atend 
 
 length of line one-wau L 
 wQr 
 
 distributed had transformer; 
 at end 
 
 ..length of line one way 
 
 distributed had, transformer 
 erf center 
 
 10 12 14 16 18 20 22 24 26 2& 30 32 34 36 
 Kilowatts Transmitted 
 
 FIG. 18. Load curves for secondaries. 
 
POWER LOSS AND VOLTAGE DROP 
 
 67 
 
 of average span are taken as shown. The total voltage drop 
 from the transformer to the end of the secondary is computed 
 by multiplying the number of customers at each pole by the 
 spans between that pole and the transformer and using the sum 
 of these multiplications to find the drop in volts with the proper 
 
 ^(No.Services x No.Sections) 
 10 20 
 
 V 
 
 IT 
 16 
 
 L 
 
 TO USE: 
 
 _ I.- Multiply the number of _ 
 services from any pole by the 
 
 number of sections ft 
 
 rransforme 
 
 actions trorn 
 products} 
 
 2- Add all these _ 
 3. - Curve for appropriate length 
 ~*~of section and average load ~ 
 per service gives total volts 
 drop to end of secondary. 
 
 NOTE:- For 3 #e (load balanced) use bo ' 
 and right hand scales 
 
 For 2*6 use bottom and/efthandscafes. 
 or top and right hand scales. 
 
 10 20 30 40 50 60 10 80 90 100 110 120 
 ^ (No. Serv'icesx No.Sections) 
 
 FIG. 19a. Voltage drop curves for 110-220 volt secondaries (No. 6 wire). 
 
 curve and scale. The scales are so arranged that for three- wire 
 secondary, i.e., 220 volts with load balanced the bottom and 
 right-hand scales should be used. For two-wire secondary, 110 
 volts, the bottom and left-hand scales or the top and right-hand 
 scales give the required result. 
 
 For example consider a secondary as shown below: 
 
68 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Sections 150 ft., average load 135 watts 
 
 First pole 1X1=1 
 
 Second pole 2X2=4 
 Third pole 
 
 Fourth pole 1X4=4 
 Fifth pole 3 X 5 = 15 
 Sixth pole 1X6=6 
 
 ^(No.Services x No.Sections) 
 
 20 36 40 50 
 
 Obtained from The 
 approximate formula :- 
 Volts drop per ampere 
 =RcosG+XsinQ 
 
 LL 
 
 UL 
 
 LL 
 
 7 
 
 Z 
 
 services from any pole 
 -by the number of sections 
 from transformer 
 
 2- Acfdallthese products 
 
 3- Curve tor appropriate 
 
 length of section and average 
 load per service gives ratal - 
 voltc- ' - 
 seconc/dru 
 
 nd^t 
 
 ed)cise 
 
 nd scales. 
 
 NO TE: - For 3 #4 (load balanced) use 
 bottom and righ t hands cales. 
 For 2 #4 use bottom and le ft 
 . hand scales or top and right . 
 hand scales 
 
 50 
 
 100 . 150 200 
 
 ^. ( No . Services x No. Sections) 
 
 FIG. 196. Voltage drop curves for 110-220 volt secondaries (No. 4 wire), 
 
 Seventh pole 
 Eighth pole 2 X 8 = 16 
 Ninth pole 3 X 9 = 27 
 Tenth pole 
 
 Eleventh pole 2 X 11 = 22 
 Total 95 
 
POWER LOSS AND VOLTAGE DROP 
 
 69 
 
 Voltage drop from curve 3.93 volts. Which is higher than it 
 should be, 3^ volts being about the limit to be used. In case 
 the drop to the eighth pole is desired. 
 
 ^ (No. Services * No Sections) 
 20 30 40 50 
 
 
 
 VOLTAGE DROP CURVES 
 16 1 Obtained from -Hie 
 
 approximate formula:- 
 Volfs drop per ampere 
 ~ Rcos O+X&ih B 
 
 Multiply thenutnbei 
 ^ ofservices from any 
 pole by the number of 
 sections from trans - 
 former [ 
 
 2. Add all these products 
 3. Curve for appropriate length 
 of sect ion and aver age load per 
 service gives total volts drop to 
 end of secondary 
 
 A 'OTE:- For 3#2(load balanced) use bo from 
 and right hand scales. 
 
 For 2#Z use bottom and left hand 
 scales or top and right hand scales - 
 
 50 100 150 200 
 
 ^ (No. Services x No. Sections) 
 
 FIG. 19c. Voltage drop curves for 110-220 volt secondaries (No. 2 wire). 
 TRANSFORMER 
 
 SMER X 3 2 
 
 -"---!-"--- ^ ---- 
 
 FIG. 20. 
 
 Sum of first seven above = 30 
 Eighth pole, 7X8 = 56 
 
 86 
 Voltage drop, 3.56 volts. 
 
PART II 
 
 CHAPTER VIII 
 TRANSMISSION-LINE PROBLEMS 
 
 METHOD OF DETERMINING MOST ECONOMICAL DESIGN FOR MAIN 
 
 OR " BACKBONE" TRANSMISSION LINES 
 
 "BACKBONE" TRANSMISSION LINES 
 
 In considering the economical design of transmission lines two 
 general types of lines are encountered. These may be called for 
 convenience : 
 
 1. " Backbone" transmission lines. 
 
 2. Secondary transmission lines. 
 
 To the first classification belongs that line or system of lines 
 which forms the ''backbone" of any system large or small. It 
 is usually a line which transmits a comparatively large load 
 (relative to the total load on the system), to some considerable 
 distance, at a relatively high voltage. Secondary transmission 
 lines on the other hand partake more of the nature of distribution 
 lines. They distribute the energy from the central feeding 
 points, served by the " backbone" line, to auxiliary stations from 
 which it may be distributed by the ordinary distribution lines. 
 For example, in a large system there may be several generating 
 stations tied together by a 110,000- volt line, "the backbone." 
 At various points along this line, substations are located, stepping 
 the voltage down to 22,000 volts, the secondary transmission. 
 These run to the various communities to be served where the 
 voltage is transformed, in local substations, to ordinary 
 distribution voltage. 
 
 Each of these types of lines present certain distinguishing 
 characteristics which affect the method used in determining the 
 most economical design. The problem of a backbone line is 
 usually a specific one, i.e., a definite load of known characteristics 
 is to be transmitted from one given point to another. Future 
 small extensions are not anticipated, these being cared for by 
 the secondary transmission lines. On the other hand certain 
 
 71 
 
72 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 conditions such as voltage, wire size, and span are not limited 
 except by considerations of economy and good operating 
 conditions. 
 
 Secondary transmission lines on the other hand must care for 
 a variety of loads and distances. They are subject to short 
 extensions to care for additional loads. They are apt to be tied 
 together in a network. All these points must be considered and 
 the best voltage, span, etc. to fit the average conditions must be 
 adopted as a standard. Wire sizes may also be standardized to 
 one or two sizes. Further problems are then limited to consider- 
 ations of the economical load for any line, number of lines for 
 any load, most economical routes, etc. The problems of under- 
 ground transmission lines fall mostly under this classification but 
 will be touched on separately with other underground problems. 
 
 Naturally the above division is somewhat elastic. In some 
 systems there are no true backbone lines, the high-voltage 
 lines being so extensive as to be similar to secondary transmission 
 lines. On the other hand, in small systems there may be no 
 secondary transmission lines, one or two lines comprising the 
 whole transmission system. In such cases the backbone lines 
 are of such a voltage, load, etc. that on a large system they would 
 probably be classed as secondary transmission or even distribu- 
 tion. They are backbone lines, however, in relation to the small 
 system, and their voltage, wire size, etc. may usually be deter- 
 mined by economy without any local limiting conditions. In 
 this chapter, the problem of the backbone line will be taken up 
 and the various elements affecting its solution will be discussed. 
 A line of relatively high voltage and heavy load will be assumed 
 but the same principles might be applied to any backbone line. 
 
 Having given the load to be carried, with its probable future 
 increase, the points from and to which the load is to be trans- 
 mitted, and the characteristics of that load, i.e., its variations 
 during the day and seasons, and the cost per kilowatt-hour at 
 the generating station, the problem remains to determine the 
 most economical route, the most economical voltage, the most 
 economical wire size and the most economical span and arrange- 
 ment of supporting structures. 
 
 It may be stated at once that no method of solving for any of 
 these unknown quantities has yet been presented which is simple 
 and at the same time accurate enough for the basis of a final 
 design. Several writers have published approximate methods 
 
TRANSMISSION-LINE PROBLEMS 73 
 
 of making such determinations. In these, however, the variable 
 quantities are so numerous that it is necessary to make certain 
 assumptions for simplicity and thus the results obtained are 
 subject to considerable question. It is believed that these are 
 valuable in establishing the limits of a problem but that the 
 actual design, especially for a line of any considerable importance, 
 should be checked up by a summation of actual cost figures as 
 compared with the cost of several other possible alternatives. 
 
 Some of the approximate methods are given in the references 
 below. 1 
 
 In the paper on " Problems of 220 kv. Power Transmission" by 
 A. E. Silver in the A. I.E. E. Proceedings, June, 1919, is given one 
 of the most careful and complete studies of a high-tension trans- 
 mission line, from an economic point of view, yet published. In 
 that paper, the voltage is assumed, the economical wire size is 
 determined from consideration of fixed charges and losses, and 
 the most economical span determined by a complete comparison 
 of cost figures on actual tower designs. 
 
 General Equation. In this problem, as is the case with most 
 economic problems, the design sought for is, in general, that one 
 for which the total annual cost will be a minimum. The total 
 annual cost may be expressed as follows, letting "g X (any 
 quantity)" indicate that the annual cost is to be used. 
 
 Total annual cost = g (cost of right-of-way) 
 
 + g (cost of towers and foundations in 
 
 place) 
 
 + g (cost of insulators in place) 
 -f- g (cost of conductor and ground wire in 
 
 place) 
 
 + g (cost of transformers in place) 
 + g (cost of lightning arresters in place) 
 + g (cost of switches in place) 
 
 1 "Economic Voltage of Long Transmission Lines," by HENRY H. PLUMB, 
 Journal A. I. E. E., April, 1920. 
 
 "Transmission Line Design," by F. K. KIRSTEN, A. I. E. E. Proceedings, 
 Vol. 37, 1917, p. 685. 
 
 "Electric Power Transmission," by A. E. STILL, p. 64, approximate 
 
 economical voltage = A/distance + - - an empirical formula. 
 
 "Notes on the Calculation of Transmission Lines for Maximum Econ- 
 omy," by E. BATICLE, Revue Generate de L'Electricite, Oct. 30, 1920. 
 
74 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 + g (cost of special apparatus, regulators, 
 
 condensers, etc. in place) 
 + g (cost of substation structures) 
 + annual cost of energy loss on the line 
 + annual cost of energy loss on the trans- 
 formers 
 
 + annual cost of patroling, testing and other 
 maintenance. (22) 
 
 Elements Affecting Costs. Each of these subdivisions of cost 
 is affected by one or more of the variable quantities whose 
 solution is being sought, some in an extremely complicated 
 manner. The manner in which the costs are affected will be 
 discussed briefly. 
 
 (a) Cost of Right-of-way. Right-of-way may have a definite 
 cost for the whole line regardless of tower size or spacing, in 
 which case it is affected only by the route chosen. Again, it 
 may vary with the number of towers only, regardless of their 
 size. In other places, it may vary both with the number and 
 base dimensions of the towers. The locality through which the 
 line will pass and the value of land will determine which of the 
 above is applicable to the problem in hand. 
 
 (6) Cost of Towers. The cost of towers is affected by a great 
 number of conditions and its relation to the economy of a line 
 is a difficult problem. The type of tower to be used, wide base 
 or narrow, one, two or more circuits, etc., is of primary impor- 
 tance. The type of base, in turn, depends somewhat on the right 
 of-way available. The number of circuits depends on the 
 economical size of conductor, the limits for corona voltage on 
 conductor and the expected increase in load. The height of the 
 tower is also a determining factor. This depends on the minimum 
 clearance requirements for the span, and the sag in the conductor 
 (which in turn depends on the span, wire size and the assumptions 
 for heaviest loading). It also is affected by the vertical spacing 
 between conductors, which is a function of the voltage. The 
 horizontal spacing (which varies with the voltage) likewise affects 
 the cost, since the weight and torsional stresses are increased by 
 longer arms. The wire size influences the cost of the tower 
 since in most cases the tower design depends, to a considerable 
 extent, on the maximum stress allowable on the conductor, which 
 is proportional to its cross-sectional area. The span is also a 
 
TRANSMISSION-LINE PROBLEMS 75 
 
 factor, since the longer the span, the greater the lateral wind 
 pressure on the conductors, and this is an important factor in 
 the design of the straight line towers. It appears to be practi- 
 cally impossible to develop any simple formula which would indi- 
 cate the variation in tower cost with all these variable elements. 
 
 (c) Cost of Insulators. The cost of the insulators increases 
 with the voltage but not in direct ratio on account of the de- 
 creased efficiency per unit in long strings of insulators. 
 
 (d) Cost of Conductor. The cost of conductor in place will be 
 very nearly proportional to its cross-sectional area. It will, 
 of course, depend also on the material used. It may vary some- 
 what with the number of towers. The cost of ground wire will 
 in most cases be practically a constant for the line, depending 
 only on its length, and number, size and material of ground 
 wires decided upon. 
 
 (e) Cost of Transformers. The transformer cost will increase 
 as the voltage increases (but not in direct ratio) and also depends 
 on the size of unit to be used and the type (air-cooled or water- 
 cooled; secondary- voltage; etc.). In most problems of this kind 
 the secondary-voltage requirements will be established and the 
 type and size of transformer will be indicated by the load and type 
 of substation to be used. The cost will then vary with the 
 voltage only. 
 
 (/) Cost of Lightning Arresters. If a given type of arrester is 
 chosen, the cost will vary with the voltage. 
 
 (g) Cost of Switches. If the type and size of switch is deter- 
 mined by the load, the cost will also vary with the voltage. 
 
 (h) Cost of Special Apparatus. If special apparatus is needed 
 to maintain proper voltage regulation, its cost must be included 
 in determining economy, since it might be eliminated if large 
 enough conductor or enough lines were used. The cost will 
 vary with the size required, which depends upon the amount of 
 regulation, and also with the voltage. 
 
 (i) Cost of Substation Structures. For a given size trans- 
 former, the cost of the terminal station pertaining to the trans- 
 mission line will be practically constant, varying somewhat with 
 the voltage on account of the increased spacing of conductors. 
 If regulators or condensers are used the cost of the substation 
 space occupied by them must be included. 
 
 (j) Cost of Energy Loss. The energy lost on a transmission 
 line consists of PR loss due to the resistance of the conductor, 
 
76 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 leakage, and corona loss. If the insulators are well designed 
 and the conductor is larger than the corona limit for the voltage 
 used, the latter two will be comparatively small and in most 
 cases may be neglected in determining economy. The I 2 R loss 
 naturally depends on the cross-sectional area of conductor, on 
 the material used, and on the current transmitted. The current 
 is a function of the voltage, if the load, power factor, equivalent 
 hours, etc. are fixed (neglecting charging current). If charging 
 current must be considered, the average between current at 
 source and at receiver will be sufficiently close approximation in 
 most cases. The cost of the PR loss can be computed from the 
 PR loss at peak load, equivalent hours, and cost per unit of 
 energy loss. The latter two are usually fixed by local condi- 
 tions and are not variables for the problem. 
 
 The energy losses on transformers consist of core losses and 
 copper losses. Since transformer efficiencies do not vary greatly 
 with size or voltage, the variations in the losses will not be con- 
 siderable. In general these losses decrease somewhat as the size 
 of transformer increases and increase slightly as the voltage 
 increases. The exact amount of variation must be deter- 
 mined for the particular transformers to be used. Other energy 
 losses will be encountered if special apparatus such as regulators 
 or condensers are used and the cost of this loss must be included. 
 
 (k) Cost of Patroling, etc. The cost of testing insulators will 
 depend somewhat on the number of units used in a string and 
 hence on the voltage. The cost of patroling will vary with 
 the length of the line and the nature of the country to be 
 covered. Other maintenance and repair costs can only be esti- 
 mated and will probably be roughly proportional to the length 
 of the line. 
 
 Solution by Equation Impracticable. The above is an indica- 
 tion of the various number of elements which affect the cost of 
 each of the factors that go to make up the total annual cost on 
 a transmission line. It is evident, that, with so many variable 
 quantities, there is no simple solution for minimum annual cost, 
 which will be general for all cases. It is true that, for any given 
 problem, when all the conditions are fixed excepting the four 
 chief variables mentioned above, i.e., voltage, wire size, tower 
 spacing, and route, equations can be written for each of the ele- 
 ments of cost in terms of these variables, which will approximate 
 quite closely that cost under any condition. For example, the 
 
TRANSMISSION-LINE PROBLEMS 77 
 
 cost of a given type of transformer can be represented with suffi- 
 cient accuracy by the expression 
 
 Cost = Ki + K 2 E n 
 
 where KI, K 2 and n are constants and E is the voltage. Some of 
 the costs are very difficult to represent, the cost of towers for 
 example. If all these expressions are determined, however, 
 and are combined into the general equation of annual cost, 
 it will be found that this equation is so complicated and contains 
 the variables in so many different powers, that a solution is 
 impossible except by trial. There is then no apparent advantage 
 over the method of assuming a number of alternative designs, 
 using definite values for each of the variables, and computing 
 the actual annual cost of each of these designs. A comparison 
 of these costs will indicate the most economical. 
 
 Method of Solution Outlined. In order to systematize the 
 computation and to facilitate the determination of the effect 
 on the total cost, of changes or additions to any design, the 
 following method is suggested. 
 
 1. Fixed Quantities. As many as possible of the elements 
 affecting the problem should be fixed. These will be: 
 
 (a) The maximum load. The expected increase should also be deter- 
 mined. In case of more than one feeding point each load must be considered 
 separately. 
 
 (6) The load factor and equivalent hours of the load. 
 
 (c) The power factor of load. 
 
 (d) The cost of energy per kilowatt hour. 
 
 (e) The length and cost of right-of-way (at least approximately) for the 
 various possible routes. 
 
 (/) The type of tower to be used for each route. 
 
 (g) The per cent annual charges applicable to the various kinds of property 
 entering into a transmission line. 
 
 2. Limits of Problem. It is essential to discover approxi- 
 mately the limits of the problem so that time will not be wasted 
 in considering voltages and wire sizes which are far from the final 
 result. In case the designer's experience is not sufficient to tell 
 him this, the approximate formulas found in the references given 
 above will be found useful. 
 
 3. Cost Data. The necessary cost data, quotations, etc., 
 must be collected. These will be: 
 
 (a) Cost of transformers, of the size determined by the load for various 
 voltages within the range of the problem (more than one size may be neces- 
 
78 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 sary especially if there are several feeding points with different sized loads.) 
 Core loss and copper loss should also be ascertained. 
 (6) Cost of switches, of proper size, for various voltages. 
 
 (c) Cost of lightning arresters, for various voltages. 
 
 (d) Cost of insulators, for various voltages. The limit in mechanical load 
 for any string of insulators should also be determined. In cases where pin 
 insulators are considered in comparison to suspension type, the cost of 
 both must be obtained. The cost of placing insulators should be included. 
 
 (e) The cost per pound of conductors of different materials copper, 
 copper-clad steel, and aluminum steel. In case the price per pound varies 
 considerably with the size, the cost for a variety of sizes within the range of 
 the problem should be obtained. The cost of stringing the conductor should 
 be estimated or determined from previous experience. 
 
 (/) The cost of various sizes of tower for each type. If possible sufficient 
 costs on towers should be obtained that the variation of cost with span, wire 
 size and voltage may be determined. This could be accomplished if three 
 voltages, three wire sizes, and three different spans, covering the probable 
 range of the problem, are considered. By plotting curves, the intermediate 
 values could be determined with sufficient accuracy. The cost of foundation 
 should be included with each tower. Anchor towers and semi-anchor towers 
 should also be considered and similar costs obtained. 
 
 (g) The cost of terminal substations of the required size. The variation 
 in this cost with voltage should be determined, if possible. In case other 
 special terminal apparatus, such as condensers or regulators are considered, 
 the additional substation cost for these must be estimated. 
 
 (h) Quotations on special apparatus considered, regulators, condensers, 
 etc., should be obtained with variations in this cost with size and voltage. 
 
 (i) Annual cost of maintenance, testing, patroling, etc. must be estimated. 
 
 4. Arrangement of Cost Data. After the data is collected, it 
 should be arranged for convenient use. This probably can best 
 be done by means of curves. Such curves are illustrated in the 
 accompanying figures. They should give: 
 
 (a) Annual cost of transformers (in place) in terms of voltage (Fig. 21). 
 (6) Annual cost of switches (in place) in terms of voltage (Fig. 22). 
 
 (c) Annual cost of lightning arresters (in place) in terms of voltage (Fig. 
 22). 
 
 (d) Annual cost of insulators per string in terms of voltage. 
 
 (e) Annual cost per mile of conductors of different materials in place in 
 terms of cross-sectional area. 
 
 (/) Annual cost of towers in place. These can be arranged (for example) 
 as a series of curves for each standard voltage showing for each standard wire 
 size (and material) the variation of cost with span. 
 
 (g) Annual cost on terminal substation in terms of voltage. 
 
 (h) Annual cost of transformer energy losses in terms of voltage. 
 
 (i) Annual cost of line energy losses per mile for each material of conductor 
 considered, in terms of cross-sectional area. 
 
TRANSMISSION-LINE PROBLEMS 
 
 79 
 
 5. Most Economical Design. It now remains to apply these 
 costs to determine the most economical design. The route is 
 usually the most limited of all the variables, there ordinarily 
 being no more than two or three possible routes at most. Like- 
 
 24,000 
 
 65 10 15 80 85 90 95 100 105 110 115 120 125 I2>0 135 140 145 150 155 
 
 Voltages in K.Y. 
 
 FIG. 21. Cost of single-phase 60-cycle transformers. 
 
 16,000 
 
 12,000 
 
 65,000 15, 00 85,000 
 
 Switches Cost- 
 
 105,000 115,000 <Z5,000 Ii5,000 145,000 
 Voltage 
 
 5,000 
 
 FIG. 22. Cost of lightning arresters and switches showing variation of cost with 
 
 voltage. 
 
 wise, on any route, the layout is somewhat limited. Locations 
 of anchor towers and semi-anchor towers are usually more or 
 less fixed. On straight runs the span can often be varied. If 
 two or three possible layouts are made for each route, varying 
 the span where possible, a sufficient comparison will be obtained. 
 
80 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 If, now, for each of these layouts, a sheet of curves is prepared, 
 showing for each standard conductor considered, a curve, giving 
 the variation in total annual cost with the voltage, a complete 
 and accurate study of the economy may be made. Not only will 
 the most economical voltage, wire size, route and layout be ob- 
 tained, but the comparative effect of variation in these quantities 
 may be studied. Curves may be added to show comparison of 
 two or more circuits with a single circuit. If a design, chosen 
 tentatively, proves, on further investigation, to be impracticable 
 on account of corona limit, too great regulation, etc., the cost of 
 correcting the fault by a change in design, addition of regulator, 
 etc., as compared with one of the other designs not having that 
 fault, may be easily determined. 
 
 Space does not permit dwelling on the subject of " backbone" 
 transmission lines more in detail. Any problem of this kind is 
 one which will bear an almost infinite amount of study. Natu- 
 rally the detail to which such a study should be carried will 
 depend somewhat on the size and nature of the project. As a 
 rule, however, time spent on an economical determination is well 
 repaid. In many cases some of the variables are limited voltage 
 may be determined by other than economical considerations, 
 route and location for towers may be fixed, etc. This simplifies 
 the problem but in any case some such study as that outlined 
 above is essential to an accurate determination. Variations in 
 the method will, of course, be found to accommodate special 
 conditions. In any case, it is to be emphasized that the problem 
 is a complicated one and usually does not bear solution by any 
 easy, approximate method. 
 
CHAPTER IX 
 
 TRANSMISSION-LINE PROBLEMS 
 SECONDARY TRANSMISSION LINES 
 
 DETERMINATION OF MOST ECONOMICAL STANDARDS OF CON- 
 STRUCTION, CONDUCTOR SIZE, LOADING, ROUTE, ETC. ON 
 LESSER OR SECONDARY TRANSMISSION LINES 
 
 In the preceding chapter there were discussed the points of 
 distinction between " backbone" transmission lines and " second- 
 ary" transmission lines. It is purposed here to consider the 
 problems arising in connection with " secondary" transmission 
 lines and to indicate their economic solution. 
 
 The " secondary" transmission lines of any system must be 
 considered as a class instead of as one or more specific problems. 
 Their service is varied. They are called upon to feed loads of 
 various sizes, and various power factors and load factors. They 
 may have a considerable diversity in lengths, types of route, 
 etc. It is obviously impracticable to consider each line as a 
 special problem. It is necessary to adopt certain standards 
 which will best fit all cases on an average, allowing some variation 
 if necessary in wire sizes, etc. There then remains the problem 
 of how best to adapt these standards to fit any given condition. 
 
 In this study, therefore, there are two separate divisions. The 
 first is of a somewhat similar nature to that of the " backbone" 
 transmission line, excepting that the problem is general rather 
 than specific. It deals with the establishment of standards 
 which will best fit average conditions, such as a standard voltage, 
 one or more standard wire sizes (the variety depending somewhat 
 on the range of loads handled) and a standard span. The 
 second division includes problems relating to the proper use and 
 combination of these standards. Such problems as the most 
 economical route, the economical division of load among several 
 lines, the load at which wire size should be changed or an addi- 
 tional line run, and reconstruction problems are in this class. 
 
 It would be impossible in a reasonable amount of space to 
 attempt to cover in any detail all the problems which might arise 
 in connection with secondary transmission lines. The field of 
 study is very large and new problems are constantly appearing. 
 Some of the problems commonly met with under each of the two 
 
 6 81 
 
82 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 divisions will be discussed, however, and the elements affecting 
 their solution will be indicated. 
 
 The general method of attacking any problem of this kind is to 
 determine an expression for annual cost and then to discover 
 under what conditions that cost will be a minimum. Since the 
 results are to be of general application, it is necessary that the 
 data used be general, for the whole system. Average costs of 
 pole line per mile should be used, for example, rather than an 
 estimated cost for a certain line over a given route. Similarly, 
 the results obtained should be displayed in formulas, curves or 
 tables which may be of general application. 
 
 Data Necessary. The data necessary for solving such prob- 
 lems on secondary transmission lines include the following: 
 
 (a) Range of sizes of loads considered. 
 
 (6) Characteristics of various types of load carried, load factor, power 
 factor, equivalent hours, etc. If the characteristics of the general types of 
 load are thoroughly understood any individual load can be studied as a 
 combination of several general types in different amounts. A load may 
 consist of part lighting, part power, part street railway, etc. and its individual 
 characteristics are a combination of the characteristics of all these types. 
 
 (c) Average material and labor costs on standard construction; average 
 right-of-way costs if such figures are obtainable. 
 
 (d) Cost of energy for various types of load in various parts of the system. 
 
 Standard Voltage. The solution of the first class of problems 
 mentioned above is very often limited by other considerations 
 than strict economy. Extensive systems rarely spring into being 
 suddenly. Usually they are developments from comparatively 
 small beginnings. Hence, what was yesterday the backbone 
 transmission line of the small system, becomes today part of the 
 secondary transmission network of the large system. It happens, 
 therefore, that the secondary transmission voltage is rarely 
 chosen as such, but is rather a development from past practice. 
 There is always the question of the economy of changing the 
 standard voltage, usually to one higher. This can only be 
 determined by a careful study of present and probable future 
 conditions, taking into consideration the additional cost for 
 transformers and station equipment and the cost of making the 
 change-over, as compared with the saving in line losses, the smaller 
 conductor used, and the increase in capacity in the system with 
 the proposed higher voltage . Such a change can sometimes be made 
 economically, where single-phase transformers are used to a large ex- 
 
TRANSMISSION-LINE PROBLEMS 83 
 
 tent, by changing from a delta to a star connected system. In cases 
 where the voltage can be chosen in advance on the basis of 
 economy, an analysis similar to that suggested for " backbone" 
 transmission lines will be necessary. 
 
 Standard Span. The standard span to be chosen will depend 
 largely on the type of construction used. Conditions may vary 
 from the use of a steel-tower line on private right-of-way, to 
 that of wood-pole lines along the highway, carrying distribution 
 in addition. For the first, probably the most advantageous span 
 can be chosen somewhat as outlined for " backbone" lines and 
 need not necessarily be uniform nor the same for all lines. For 
 the latter, the span will be determined partly by the strength 
 of the poles and partly by the needs of the distribution lines. 
 These spans can be fairly uniform, not exceeding a definite maxi- 
 mum of 150 to 250 ft. ordinarily. 
 
 Conductor Size. The question of standardization of conductor 
 size will depend largely on the kind of loads carried and their 
 distribution over the system. In some cases it will be found ad- 
 vantageous to standardize on one or two sizes and care for 
 additional load by more lines. In others, it may be better to use 
 a wider variety of conductors, accommodating the size to the 
 load carried. A method of studying conductor economy is given 
 below. By its means, the most economical wire size for any load 
 may be chosen and, under proper conditions, standard sizes may 
 be fixed. This method of study also aids in the solution of 
 several of the problems of the second class as will be pointed out 
 later. 
 
 Example of Study of Conductor Economy. Let a standardized 
 type of construction with given voltage and standard span be 
 assumed. The total annual cost of such a transmission line is 
 composed of: 
 
 1. Annual cost on construction exclusive of conductor. 
 
 2. Annual cost of conductor in place. 
 
 3. Annual cost of energy loss. 
 
 For this example, a wood-pole line will be assumed with a 
 maximum span of 175 ft. to accommodate distribution. 
 
 The annual cost of poles and fixtures per 1,000 ft. of line will, 
 of course, vary with the wire size, if the line is properly designed 
 for safety, probably after some such formulas as KI + K 2 A, 
 where KI and K 2 are constants and A is the cross-sectional area 
 of the conductor. 
 
84 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 The annual cost of the conductor per 1,000 ft. will vary approxi- 
 mately with A as also will the cost of stringing, i.e., cost in place = 
 K 3 + K*A. 
 
 The annual cost of energy loss per 1,000 ft. will vary with the 
 square of the load, the first power of the equivalent hours, and 
 the cost of energy, and inversely with the cross-sectional area. 
 
 Then the total annual cost per 1,000 ft. of line 
 
 Y = K, +K 3 + (K 2 + K 4 )A + K b (23) 
 
 Where t = equivalent hours, C e cost of energy loss per kilowatt- 
 hour, kw = load in kilowatts 
 Simplified 
 
 Y = K a + K b A + K c (24) 
 
 Where K a = K, +-K> and K b = K 2 + K*, and K c = K b 
 
 Most Economical Conductor Size. The most economical 
 conductor size will be that for which the value of Y is a minimum. 
 This can be obtained by setting the first derivative of Y with 
 respect to A equal to 
 
 dY kwHC. 
 
 dA = Kb Ac A 2 
 
 A = kw J^ t C e (25) 
 
 Most Economical Load. If it is desired to determine the most 
 economical load for any line, the derivative of Y must be taken 
 with respect to the load, kw. It is evident that, with the equa- 
 tion in the above form, the result would be kw = 0, since the 
 minimum line loss is obtained with no load. However the line 
 must be considered as a working unit. Hence, if the above 
 expression is changed to represent the annual cost per kilowatt 
 transmitted, and the minimum value of that quantity found, the 
 most economical load for the line will be discovered. 
 If Y' = the annual cost per kilowatt transmitted 
 ^K a +K b A K c kwtC e 
 kw A 
 
 K, 
 
 / a ,\ 
 dkw ~ \ kw 2 J ' A 
 
 - (27) 
 
TRANSMISSION-LINE PROBLEMS 85 
 
 Numerical Example. The values of the constants must be 
 obtained to accord with local conditions. K c is evaluated as 
 follows : 
 Y e = annual cost of energy loss per 1,000 ft. = 
 
 Where / = current per wire 
 
 R = resistance per 1,000 ft., one wire, 
 
 R = - - -r 1 Where p = resistivity per mil foot 
 
 A. 
 kw X 1,000 
 
 = E cos e 
 
 Where K < - x " x 365 
 
 if in s K - 3 ' 940 x 1Q6 
 
 If " " K ~ (E cos ey 
 
 Let us assume for example a 22,000-volt line. 
 For which K l = 39 
 12.2 
 
 K 3 =1.715 
 
 (2.21 + 147.5C CU \ ,, T , n _ cost of conduc- 
 ~ \ ~10^~ / Ccu = tor per pound 
 
 = If C cu = 0.20 
 
 5 
 
 K, = 11.3 
 K a = 40.72 
 
 43.9 
 
 ^ 6= TOT 
 K c = 11.3 
 
 (From Eq. 24) 7 = 40.72 + ^A + 11.3 
 
 V1 1 Q V 1 5 /^ 
 "T^V^ = 160. 
 4o.y 
 
 40 q 
 
 ^ 
 
 T ll.O 
 
 (From Eq. 27)fc, - ^40.72 
 
86 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 The foregoing are all simple equations in terms of the load 
 carried, the cross-sectional area of conductor, the equivalent 
 hours and the cost of energy. The latter two quantities are 
 more or less inter-related. A study of energy cost will show 
 that the cost per kilowatt-hour varies, among other things, with 
 the load factor and hence with the equivalent hours. The value 
 of tC e may then be obtained approximately for any value of t 
 with any type of load. 
 
 O.H 
 
 0.04- 
 
 4000 6000 &000 
 
 Load in Kilowatts 
 
 10,000 
 
 FIG. 23. Annual cost per kilowatt per 1,000 ft. of 22, 000- volt transmission line 
 (single circuit tC e = .06;' cos 6 = .85). 
 
 It will be found advantageous for study to plot curves of all 
 the equations given above. If different values of tC e are chosen, 
 as tC e = .02, tC e = .04, tC e = .06, tC e = .08 and tC e = .10 covering 
 the desired range of values and a sheet made up for each, giving a 
 curve, plotted between annual cost, Y, and load kw, for each wire 
 size, the first equation (24) is well displayed; similarly Eq. 26. 
 For Eqs. 25 and 27, a curve, plotted between A and kw for each 
 desired value of tC e , may be obtained. 
 
 For a numerical example the value of equivalent hours is taken 
 
TRANSMISSION-LINE PROBLEMS 
 
 87 
 
 as 6 and the corresponding value of tC e assumed as .06. The 
 curves for Y', the annual cost per kilowatt per 1,000 ft., are 
 plotted as shown in Fig. 23. The curves for Y, the total annual 
 cost per 1,000 ft. are easily obtained if desired but are not shown 
 
 2000 4000 . 6000 8000 10,000 12,000 
 
 Load in Kilowatts 
 
 FIG. 24. Most economical conductor size for 22,000-volt transmission line. 
 
 here. These curves for Y' are interesting in that they show defi- 
 nitely the point of minimum cost or the load of greatest economy 
 for any given conductor size. Likewise, they show, for any load, 
 the most economical conductor and just how much cheaper one 
 size is than another for any load. 
 
 Figures 24 and 25 are also plotted from Eqs. 25 and 27, using 
 
88 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 various values for tC e . Figure 24 gives the most economical con- 
 ductor size for any load and Fig. 25 the most economical load 
 for any wire size. The curves for tC e = .06 might have been 
 derived directly from Fig. 23, Fig. 24 being a bounding curve 
 for all the curves shown and Fig. 25 the locus of all the minimum 
 points. It is interesting to note that the most economical size 
 of wire for any load is not the same as the size for which that is 
 the most economical load. The reason for this is easily seen 
 from Fig. 23, for while at 2,500 kw. No. wire appears to 
 
 12,000 
 
 06 * 9Z #0 *00 000 
 
 Conductor Site in Circular Mils 
 
 FIG. 25. Most economical loading for 22,000-volt transmission line. 
 
 be most economical, with an annual cost of .051 per kilowatt 
 per 1,000 ft., if a No. line is loaded to 3,800 kw., the annual 
 cost will be only .0473 per kilowatt per 1,000 ft. 
 
 Use of Curves. By means of such curves as these, a number of 
 the problems of secondary transmission lines may be readily 
 solved. If the size and characteristics of a load are known, the 
 most economical conductor may be found from the proper 
 curve. If some other size than the most economical is used, 
 its additional cost is given by the curves. If standard sizes are 
 to be determined, the range of loads, and their characteristics, 
 to be handled may be studied in connection with the curves 
 and the most economical size or sizes to fit the majority of cases 
 may be chosen. If the problem is one of changing conductor 
 sizes, the load at which the annual cost after the change (includ- 
 ing the annual charges on the cost of changes) will be less than 
 the annual cost with the present size, may be taken from such 
 
TRANSMISSION-LINE PROBLEMS 89 
 
 curves as Fig. 23. Similarly, if curves for two (or more) circuit 
 lines are prepared, the economy of adding a second circuit, of 
 building an additional line, etc., may be studied. (This problem 
 usually involves considerations of reconstruction, see Chap. X.) 
 If several lines of different sizes are feeding one station, the 
 economical division of load is shown by curves similar to Fig. 25. 
 
 Most Economical Route. Another problem which is encoun- 
 tered in the design of nearly all transmission lines is the choice 
 of the best route. It is seldom possible to follow anything like a 
 direct "air line" route on account of the difficulty in obtaining 
 right-of-way. If wooden poles or steel structures with small 
 base are used, a route is ordinarily chosen which follows the 
 highway in such a way as to arrive at the destination with the 
 least possible length of line and the fewest possible difficulties in 
 construction. It often occurs, however, that this choice is not a 
 simple matter. There may be two or more routes, each of which 
 has advantages and disadvantages which more or less offset each 
 other, and there is no self evident choice between them. It is 
 then necessary to make a careful cost analysis to determine the 
 most economical route. The relative economy of the various 
 routes can be weighed against any other features which are not 
 subject to a tangible cost comparison, and the most practicable 
 route will usually be made evident. 
 
 Probably one of the most usual and simple problems of this 
 kind is the choice between two routes, one of which is shorter 
 than the other but necessitates the purchase of a certain amount 
 of right-of-way, while the other is longer but follows the highway 
 and is not subject to right-of-way charges. It is the purpose 
 here to show how the cost of two such routes can be readily 
 compared and a general equation derived which will cover all 
 such problems on any particular system. This equation can be 
 used to determine curves that give a more tangible method of 
 choosing the best solution. 
 
 As a corollary to this problem we have the condition where 
 there is a choice of a longer route, clear of trees and other obstruc- 
 tions, and a shorter one which will require extra high structures, 
 additional guying, etc. In fact the problem resolves itself into 
 the question of "How much can we afford to spend in addition to 
 the normal cost of construction in order to shorten a route?" 
 
 The basis for such a cost comparison will naturally be the 
 annual cost of the two routes. If we can determine what will be 
 
90 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 the saving in annual cost of the shorter route over the longer 
 when only normal line construction cost is considered, we can 
 then tell what is the maximum amount which we would be 
 justified in expending for right-of-way or additional material, 
 etc., in order to use that shorter route. If the longer route should 
 also require some extra construction expenses these should natu- 
 rally be included in the comparison. 
 
 The annual cost of any route is made up of the annual charges 
 against the construction itself and the annual charges for the 
 energy loss, the latter depending on the load carried and the size 
 of wire used. It is first necessary to determine as accurately as 
 possible what the unit annual charges will be on a normal line, 
 i.e. one without exceptional difficulties. 
 
 Determination of Normal Line Cost. The annual charges 
 against the construction will depend upon the standard type of 
 construction used and on the cost of materials and labor in the 
 locality under consideration. There must be obtained the total 
 cost per mile of the line in place including both material and labor 
 costs on poles, crossarms, pins, insulators, wire, grounds, guying, 
 etc. In doing this, the sizes and quantities of the various 
 materials which would constitute an average normal mile of line 
 must first be determined. The annual cost may then be obtained 
 by figuring interest, taxes and depreciation on this first cost 
 allowing for the difference in depreciation between the different 
 materials used. A certain amount per year .should also be added 
 for maintenance, patrolling, etc. The resulting annual cost will 
 be a constant for any condition of loading, providing the size of 
 wire is fixed, and may be represented by K & . 
 
 Usually when a shorter route is selected a number of corners 
 are eliminated. The cost of a corner construction, especially on 
 high-tension lines, may be quite an appreciable addition to the 
 normal cost of the line. It can be figured however in the same 
 manner as the normal line cost. This annual cost per corner 
 may be represented by K 8 . 
 
 Annual Cost of Energy Losses. The annual charges due to 
 loss of energy may be determined as shown above in the study of 
 conductor economy. 
 Then, 
 
 Annual cost of energy loss = K 7 kw 2 " per mile. 
 
 ,_, v 365,000 rtC e 
 Where X 7 =- 
 
TRANSMISSION-LINE PROBLEMS 91 
 
 Where r = resistance of conductor in ohms per mile. 
 
 When these constants have been evaluated for the particular 
 conditions of the problem in hand, it is then possible to determine 
 the total annual cost per mile with normal line construction and 
 the total annual cost for any particular length including the 
 corners. 
 
 Determination of General Equation. If we let L represent the 
 difference in length between the shorter route and the longer, 
 the savings in annual cost of the shorter route over the longer 
 would be L times the annual cost per mile if normal line construc- 
 tion only is considered. This, then, represents the maximum 
 amount which it would be economical to spend in addition to the 
 normal cost of the line in order to utilize the shorter route. If 
 the additional cost is for right-of-way, this annual amount is the 
 maximum rent which we could afford to pay for it or if the right- 
 of-way is bought outright, this amount represents the interest 
 and taxes on the maximum amount which could be paid for the 
 necessary land. In the case where the additional cost would be 
 for extra poles and other line material, the above savings would 
 represent the interest, taxes, and depreciation on the maximum 
 amounts which could be so expended. 
 
 If L = the length in miles which would be saved by 
 
 using the shorter route, 
 
 Ci = the maximum extra expenditure allowable in 
 order to use the shorter route, 
 
 C 
 
 C = the extra expenditure per mile saved = y*> 
 
 N = the number of corners saved by using the shorter 
 
 route, 
 g = per cent annual charges on additional expenditure 
 
 (interest, taxes, depreciation), 
 KG = the annual cost of normal line construction per 
 
 mile, 
 K 7 kw 2 = annual cost of energy loss per mile, 
 
 KS = annual cost of a corner in addition to normal line 
 
 cost, 
 Then 
 
 The annual charges on the additional expenditure for right-of- 
 way, extra poles, etc. would be gd 
 
 gCi = L(K 6 + K 7 kw*) + K B N (28) 
 
92 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 which is a general equation and can be applied to any such case 
 if the proper values for the various constants are obtained. If 
 then the contemplated shorter route can be used with a less 
 expenditure than C\ it would be an economical proposition to use 
 it. 
 
 The above equation can be made even more general and be 
 shown graphically by means of a curve if we reduce it to terms of 
 C or the maximum allowable expenditure per mile of length saved, 
 omitting for the present the amount saved on corners. 
 
 r _ & 7 w ( . 
 
 ~17i6o~ 
 
 The value of g as indicated above depends upon the nature of 
 the additional expenditure. If it is for right-of-way, interest and 
 taxes must be charged but no depreciation need be considered. 
 We may take interest at 6 per cent., taxes at 2 per cent or g 
 8 per cent. This represents yearly rental. If extra line material 
 is to be purchased depreciation also must be added. Considering 
 the life of such material as 20 years, depreciation = 5 per cent 
 and g = 13 per cent. 
 
 Data for Curves. The accompanying curves, Fig. 26, were 
 plotted assuming values for K G , K 7 and K 8 as follows: 
 
 K 6 $350 (wood-pole construction) 
 r = . 539 for No. wire 
 t = 6 hr. 
 E = 46,000 volts 
 cos = .90 
 
 C e = .01 per kilowatt-hour 
 365,000 rtCe 0.69 
 
 7 
 
 (E cos BY 10 5 ' 
 8 = $40 
 g = 8 and 13 
 
 .69 kw* .69 kw* 
 
 " -- 
 
 C = - T^B - and C = 
 
 .08 .13 
 
 These equations are plotted for various values of the load, kw. 
 The lower curve A is plotted for g = 8 per cent or when the 
 additional expenditures to be made in using the shorter route will 
 be for such property as right-of-way which has no depreciation. 
 
TRANSMISSION-LINE PROBLEMS 
 
 93 
 
 In case the right-of-way is obtained on a yearly rental basis, 
 8 per cent of the amount shown by the curve would be the allow- 
 able yearly rent. The upper curve is to be used when the extra 
 cost will be for extra line material such as poles, stubs, guys, etc. 
 for which the life was assumed to be 20 years and g = 13 per cent. 
 If any corners will be saved by using the shorter route, the cost 
 of these may be added to the amount shown by the curve or 
 
 10,000 
 8000 
 6000 
 4000 
 
 200O 
 
 
 
 / 
 
 / 
 
 
 
 
 
 
 / 
 
 / 
 
 /" 
 
 
 / 
 
 2 
 
 
 
 / 
 
 / 
 
 /A - When expenditure is for) 
 having no depreciation 
 
 B - When expenditure is fbrpr 
 having 20 year life 
 
 property 
 vperry 
 
 / 
 
 / 
 
 If arty corner 
 shown by at 
 for each co> 
 
 s are elimina 
 rve^ $500 fot 
 'ner 
 
 ted } ae(dto& 
 -A } $308 fo 
 
 mount- 
 rB; 
 
 o 
 
 
 / 
 
 
 
 
 
 '2000 4000 6000 8000 10,000 12,000 14,000 
 
 Maximum Economical Expenditure per Mile Saved, dollars 
 
 FIG. 26. Curves showing maximum economical expenditure in addition to 
 normal line cost in order to shorten a route. 
 
 TTOO ^ or eac ^ corner - ^ ^ 8 = $40, this amoun ts to $500 for 
 g = 8 per cent and $308 for g = 13 per cent. 
 
 Discussion of Curves. These curves, then, show the amount 
 which could be spent in constructing a line in addition to the 
 normal cost of construction under average conditions in order to 
 shorten the length of the route 1 mile. The value of C when the 
 load is represents annual charges on the normal construction 
 cost of 1 mile of line capitalized at the percentage g, which is the 
 percentage of annual charge against the additional property 
 
94 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 acquired and depends on its nature. As the load increases the 
 value of C increases due to the increasing cost of the line loss. 
 
 The slope of the line away from the vertical illustrates graph- 
 ically the importance of knowing as accurately as possible the load 
 and the load factor that it is intended to carry on the line, in 
 order to correctly make the choice of routes. Though an accu- 
 rate determination may be practically impossible, it is usually 
 possible to work between some assumed limits that will represent 
 a wide range of conditions and still lend themselves to analysis 
 by means of the curves. 
 
 Furthermore, as the load increases, the point which stands out 
 plainly is the great economy which will ordinarily be achieved 
 by the use of a shorter route. For example if right-of-way can 
 be obtained for a yearly rental of $1 per pole, at 8 per cent this 
 would represent $12.50 per pole or at 35 poles per mile, $437.50 
 per mile. With a load of 4,000 kw. the curves show that we 
 could spend, according to curve A, $5,300 for each mile saved, 
 which would purchase right-of-way, at the above rate for 12 
 miles. Or, from curve B we could expend $3,300 for additional 
 material in order to save 1 mile. If any corners were eliminated 
 the amount would be still more. Hence, if the actual expenditure 
 necessary is any less than the above figures, the difference would 
 represent a real saving and it would be economy to use the 
 shorter route. 
 
 Naturally the curves here given could not be used for any but 
 the particular type of construction and voltage for which they 
 were computed. It is a simple matter, however, to develop 
 similar curves to fit any other case from the equations given. 
 The economy shown by such a curve, however, cannot be used 
 as an absolute criterion in the choice of a route. There are other 
 factors which do not lend themselves to such an exact cost 
 analysis. The matter of patrolling a line and making repairs 
 might be considerably more expensive along private right-of-way 
 than along a main highway. The proximity to a railroad might 
 considerably affect the cost of erection. The matter of protec- 
 tion against severe storms must be considered. Many other 
 details will, in any construction project, present themselves for 
 analysis. If, however, we have a concrete comparison of the 
 relative economy of one route over another, we have gone a long 
 way toward an exact determination of which will be the most 
 advantageous under all conditions. 
 
TRANSMISSION-LINE PROBLEMS 95 
 
 As brought out at the beginning of this discussion, only one 
 specific point in economies of routes has been covered here, that 
 is, where the choice between two routes is to be made, the shorter 
 one requiring purchase of right-of-way or installation of higher 
 structures and special reinforcements. The many other prob- 
 lems in the choosing of the economical route for a transmission 
 line could be treated in a very similar manner. The question of 
 the economy of diverging from the shortest route in order to 
 utilize old poles already in place is taken up in the next chapter 
 under " Reconstruction Problems." 
 
 The above will give an idea of the kind of problems in con- 
 nection with a secondary transmission system which require a 
 solution for economy. It must always be kept in mind, of 
 course, that economy is not the only consideration in designing 
 a line. Mechanical strength is an important feature. Good 
 regulation is essential. While the economical conductor size is 
 usually independent of the length of the line, regulation depends 
 on the length. Often, on a long line, a larger size than the most 
 economical must be used to give good voltage. In case artificial 
 means of improving regulation are considered, their cost will 
 tend to offset the economy of the smaller conductor and a study 
 of the line as a whole including all such items is necessary. The 
 problem is still one of economy. In any case the determination 
 of the most economical conductor size, voltage, span, route, etc. 
 will serve as a starting point for the study of the most 
 advantageous design for final adoption. 
 
CHAPTER X 
 RECONSTRUCTION PROBLEMS 
 
 PRINCIPLES INVOLVED IN THE SOLUTION OF PROBLEMS DEALING 
 
 WITH THE ALTERATION OR RECONSTRUCTION OF LINES 
 
 ALREADY BUILT METHOD OF INCLUDING VALUE 
 
 OF SALVAGED MATERIAL IN COST STUDY 
 
 A great number of the problems in transmission and dis- 
 tribution lines, involve the consideration of reconstruction, that is 
 the improvement or enlargement of a system already in oper- 
 ation. For example, it may be desired to determine the advis- 
 ability of shortening a line already in operation by rebuilding 
 part of it over a new route. Or, it may be necessary to meet an 
 unforseen increase in the load carried, by rebuilding an old line, 
 increasing the size of the wire, adding an additional line on the 
 same poles, or changing the voltage. While this chapter deals 
 primarily with transmission line reconstruction, the principles 
 enumerated may be applied to such problems on any part of the 
 system. 
 
 For convenience, the term "reconstructed line" will be used to 
 designate the line after the change is made. In any such case, 
 it is necessary to consider the fact that the old line has more or 
 less of its serviceable life left and the value of this must be added 
 to the cost of the new construction in determining the total 
 investment represented by the finished line. This total is the 
 amount upon which the reconstructed line must pay a return 
 and should be justified by the increased capacity secured. Or, 
 in a problem of economy, such as shortening a line, it is the 
 amount upon which the annual cost of the new installation must 
 be figured. This annual cost must be less than that of the old 
 line if there is to be any advantage in the change. 
 
 Total Investment Represented in Reconstructed Line. In 
 order to determine the total annual cost of the reconstructed 
 line or section of line, it is easier to first consider the factors which 
 affect the total investment involved, either to increase or decrease 
 it. It will at first be assumed that the new line is to be entirely 
 of new construction and the old line is to be salvaged. Later, 
 the use of old material in the new line will be discussed. It is 
 evident that, for the purpose of this analysis, the amount of 
 investment represented by the old line as it stands must be 
 computed on the principle of the cost to reproduce it in its 
 
 96 
 
RECONSTRUCTION PROBLEMS 97 
 
 present condition, at present prices. It is to be replaced by new 
 construction at present prices and the comparison of annual costs 
 of old and new must be on the same basis. This amount may 
 be obtained by figuring the detailed cost of such a line, including 
 material and labor, at present prices and subtracting from this a 
 percentage of the whole equal to the fraction of the total assumed 
 life of the line which has already elapsed. The value of the 
 elapsed life is assumed to have been paid for in the return from 
 the operation of the line up to the present time. If, therefore, 
 a new line is built to replace the old and the old line is removed, 
 there is added to the total investment the cost of all the new 
 material used and the labor involved in the change. There is 
 subtracted from the total investment the salvage value of the 
 material recovered. The investment represented by the old 
 line as it stands is thus increased by: 
 
 1. The cost of the material used in the reconstructed line. 
 
 2. The cost of the labor necessary in the building of the 
 reconstructed line. 
 
 3. The cost of the labor necessary in dismantling and removing 
 to the warehouse the material in the old line. 
 
 The investment is decreased by: 
 
 The present value of the material in the old line, (i.e., its cost 
 if new, less a percentage of depreciation for age). 
 Hence, the 
 
 Net increase in investment = Cost of material in reconstructed line 
 + Cost of labor in reconstructed line 
 + Cost of labor removing old line 
 Present value of material in old line. 
 As was explained above, 
 
 Investment represented by the old line = Present value of 
 
 material in old line 
 + Present value of 
 labor in old line. 
 
 It is evident that the last item in "net increase in investment" 
 cancels the first item in " investment represented by the old 
 line," hence this quantity " present value of material in old 
 line" does not enter into the " total investment represented." 
 Therefore, 
 Total investment .represented = Present value of labor in old line 
 
 by the new line + Cost of material in reconstructed line 
 
 -|- Cost of labor in reconstructed line (30) 
 + Cost of labor removing old line. 
 
98 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 In other words, a new line, built to replace an old one which 
 has not outlived its usefulness, represents an investment including 
 the present value of the labor of erecting the old line and the cost 
 of labor in removing the old line in addition to the total cost of 
 material and labor in erecting the new line. 
 
 Annual Cost of Reconstructed Line. The subject of annual 
 cost in general has been discussed in Chap. Ill and the forma- 
 tion of the general equation in Chap. VI. The application to the 
 special problem of the reconstructed line will be taken up here. 
 
 Investment Charges. In order to obtain the yearly cost charge- 
 able to the reconstructed line which is the quantity which must 
 be used in considering its economy, it is necessary to spread this 
 total investment over a period of time estimated for the new 
 construction. The percentage to be charged annually must 
 include interest, taxes, insurance and depreciation. Ordinarily, 
 at least two different percentages will be used in actual computa- 
 tion. One will apply to labor investment which has no salvage 
 value, and to materials such as poles, crossarms, pins, etc., which 
 have a comparatively short life, and whose scrap value at the 
 end of the useful life is probably about equal to the cost of labor 
 necessary for salvaging. Another figure will be used on such 
 material as bare copper wire which has practically no physical 
 depreciation. Its value at the end of the life of the line is as 
 great as at present, providing prices do not decrease, and the 
 only expense incurred will be the labor cost of salvaging. The 
 market value of copper has varied so widely during the past few 
 years that a great element of uncertainty is introduced into this 
 item. For this discussion, however, it will be assumed that the 
 price of copper will remain constant. For the present both these 
 percentages will be represented by the letter g. 
 
 g = per cent interest, taxes, insurance and depreciation 
 chargeable annually against the construction. The expression 
 11 g X any quantity" merely indicates that it is the annual charge 
 against that quantity which is being considered. 
 
 Operating Charges. Another element of annual cost is the 
 operating expense. The chief item under this head will be that 
 of energy loss. If the reconstructed line is of higher voltage or 
 of larger wire size than the old, or if the length is shortened, 
 the energy loss may be materially reduced. This will tend to 
 reduce the annual cost and is an important item when relative 
 economy is to be considered. Here, again, considerable uncer- 
 
RECONSTRUCTION PROBLEMS 99 
 
 tainty is introduced on account of the variation in load factor 
 and load at present and in the future. The value of these items 
 must be selected for the particular problem by careful study 
 of the local conditions, since here, as in any engineering problem, 
 the solution must be based on certain definite assumed values 
 for all variable quantities. The success or failure of the solution 
 will depend upon the good judgment used in such selection. 
 The operating expense also includes the cost of superintending, 
 repairing, patrolling, etc., which will, in most cases, probably 
 not be materially different on the reconstructed line from what 
 it was on the old line. In case the difference is marked this point 
 must be taken into consideration in the final decision. 
 
 General Equation for Annual Cost of Reconstructed Line. 
 A general equation may now be formulated for the annual cost 
 of the reconstructed line by combining these factors. 
 
 Annual cost of ^ esen * value .* labor in ld line : v 
 
 reconstructed +Cost of material in reconstructed line. 
 
 +Cost of labor in reconstructed line. i ,.,,. 
 
 +Cost of labor removing old line. 
 
 line = 9 x 
 
 + Cost of superintendence, repairs, patrolling, etc. 
 +Cost of energy losss on recontructed line. 
 Similarly, 
 
 Value of material in old line. 
 
 The annual cost on 
 the old line = gX 
 
 + Value of labor in old line. 
 
 +Cost of superintendence, repairs, 
 
 patrolling, etc. 
 +Cost of energy loss on old line. 
 
 (32) 
 
 By proper selection and application of the percentage g, and 
 computation of the various costs involved, in accordance with 
 local conditions, these equations may be applied to any such 
 problems of reconstruction such as determining the relative 
 economy of replacing a present line with one of higher voltage 
 or larger wire, or of shortening a line by rebuilding part of it, as 
 will be shown later. 
 
 Utilization of Old Materials in Reconstructed Line. In the 
 above discussion it was assumed that all the old material would 
 be salvaged. It very often occurs, however, in such cases as an 
 increase in voltage, that the new line may to advantage follow 
 the old route for part of the distance at least, using the old poles 
 and, in some cases, the old crossarms, insulators, etc. The 
 general equation still holds good for this condition if its deriva- 
 
100 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 tion is kept in mind and the various items adjusted to fit the con- 
 ditions. The "cost of material in reconstructed line" must 
 include the " present value" of the old material used, since, it 
 will be remembered, this originally entered the equation as a 
 credit (to be salvaged) and was cancelled out. The "cost of 
 labor in reconstructed line" must be reduced due to the fact 
 that the old material used is already in place but it must also 
 include any labor cost necessary to adapt the old material to the 
 new conditions. The "cost of labor of removing old line" will, 
 of course, be reduced by the amount that would be required 
 for the material not removed. The value of g must also be 
 selected to take into account the fact that the present life of the 
 old material will not be as great as the new and hence the labor 
 necessary in adapting it will have a greater yearly depreciation. 
 With these conditions clearly in mind, however, it is evident 
 that the general equation may be safely applied to any such 
 problem of reconstruction, even when part of the old construc- 
 tion is to be utilized. This would include also the case when a 
 second line was to be added on the same poles. 
 
 So far the discussion has been entirely of a general nature. It 
 has been shown, how, in any problems involving reconstruction 
 of a line still serviceable, the investment cost and annual charges 
 pertaining to the old line must be included in the costs chargeable 
 to the new line. These costs will be increased or decreased by the 
 various extra expenditures or savings belonging to the new line. 
 It is now proposed to show how the general method can be applied 
 to specific cases, and how in some cases "short-cuts" may be 
 introduced to greatly simplify the problem. 
 
 Economy of Shortening a Line already in Service. Let us 
 assume, for example, a line several years old which is in good 
 condition and of sufficient capacity to care for the predicted load 
 up to the probable length of its life. Since its construction, how- 
 ever, conditions have so changed that it is now possible to reduce 
 its length considerably by using a more direct route in some places. 
 The question arises as to whether or not it would be economical 
 to rebuild those portions of the line by the shorter routes. In 
 order to determine this it is necessary to compare the annual 
 cost of the present installation with the total annual cost charge- 
 able to the reconstructed installation, if the change is made. 
 These annual costs may be determined by use of the general 
 equations previously developed. 
 
RECONSTRUCTION PROBLEMS 101 
 
 The present annual cost charged to the section of the line under 
 consideration is given by Eq. 32. 
 
 The annual cost in case the old line is replaced by the new 
 section would be as shown by Eq. 31. 
 
 It would be economical, then, to undertake the new construc- 
 tion only if the annual cost of the reconstructed line, thus com- 
 puted, is less than that of the old line. The added construction 
 cost must be offset by the reduced energy loss. The limiting 
 case is when the two annual costs are equal: 
 
 Annual cost of- reconstructed line = Annual cost of old line. 
 It is evident, however, that several of the items in both annual 
 costs are practically the same, so for the purpose of this compari- 
 son they may be eliminated. Since the reconstructed section is 
 but a small part of the line as a whole, its useful life must be 
 assumed to be that remaining in the old line. Hence the annual 
 charges against the " present value of labor in old line," which is 
 part of the "annual cost of the reconstructed line," will be prac- 
 tically equal to the annual charges against "cost of labor in old 
 line," which is part of the "annual cost of old line." The cost 
 of "superintendence, repairs, patrolling, etc.," will be practically 
 the same for both, unless the difference in length is large. The 
 equation then reduces to 
 
 Cost of material in recon- 
 
 structed line + 
 Cost of labor in recon- 
 structed line + 
 Cost of labor removing 
 
 g (Cost of material in old 
 
 line) 
 
 + Cost of energy loss on old 
 
 line (33) 
 
 old line 
 + Cost of energy loss on re- 
 constructed line 
 
 The cost of material and labor in the new line will of course 
 include the cost of any private right-of-way which it is necessary 
 to purchase and of any other extra expense connected with the 
 construction. 
 
 A system of symbols will be adopted, conforming to those 
 previously used. 
 
 Let L 2 = length in miles of the new section to be built, 
 Le = length in miles of the old section to be replaced, 
 kw = load in kilowatts, 
 
 L 4 = the length in miles of right-of-way to be purchased, 
 N = number of new corners necessary in reconstructed line, 
 C r = cost per mile for right-of-way. 
 
102 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 The details of the computation of unit costs used here need not 
 be given as they would apply to only one particular case. The 
 figures used are based on a line at 46,000 volts, single construction, 
 on wooden poles, with No. bare copper wire, energy costing 
 1 ct. (.01) per kilowatt-hour. The annual cost on such new 
 construction introduced into an old line will vary somewhat with 
 the age of the line as explained above. However, since such a 
 reconstruction would probably not be considered except during 
 the early life of the line, and of course, the shorter the remaining 
 useful life assumed, the greater the salvage value of the new 
 material at the end of that life, this may be assumed to be 
 constant for any age. This assumption would also be favored 
 by the probability that this new section would be used, at least 
 in part, beyond the life of the rest of the line. 
 
 The unit costs are then assumed to be as follows : 
 
 Annual charges on material and labor in recon- 
 
 structed line = $360 . 00 per mile 
 
 Annual charges on cost of labor removing old line = 30 . 00 per mile 
 Annual charges on cost of material on old line = 225 . 00 per mile 
 
 Annual cost of energy loss = . 69 kw 2 per mile 
 
 10 5 
 
 Annual cost of new corner = 40.00 
 
 Annual cost of right-of-way = . 08 C r per mile 
 
 Substituting these values, the equation for determining economy becomes 
 
 360L 2 + 40AT + 0.08C r L 4 + 30L 6 + kw 2 L 2 = 225L 6 
 
 165L 2 + 40N + 0.08C r L 4 = (L 6 - L 2 ) jg* 2 + 19$ (34) 
 
 There are too many variables to exhibit this equation as a curve. 
 The limiting values, however, can be so expressed. If no right- 
 of-way need be purchased and no new corners are added, the 
 greatest length of a new line, L 2 , in order to effect a saving of 
 1 mile (Le L 2 = 1) would be 
 
 L^L, - 65- ~W~ kW + L18 
 
 This is plotted in curve A (Fig. 26 a) . This curve shows the great- 
 est length of new construction which could be economically built 
 under the most favorable conditions, that is, normal line con- 
 struction cost with no corners and no other extra expense for 
 right-of-way, etc., for each mile subtracted thereby from the 
 
RECONSTRUCTION PROBLEMS 
 
 103 
 
 total length of the line. If such other extra expense is necessary 
 
 the value of j f- will be less, as may be seen from the equa- 
 te Lt2 
 
 tion (34) . Hence this curve may be used as a test. If the length 
 of new line under consideration is greater than that shown by the 
 curve, it will not be economical to build it. If it is less, the exact 
 economy must be determined from the equation (34). For 
 
 12,000 
 
 10,000 
 
 8000 
 
 4000 
 
 2000 
 
 A - New construction art normal line 
 cost only, with no extra expense for 
 
 B -New construction on private right, 
 of way at $35 per mile per year 
 
 If 'extra corners are necessary^ubtracf 
 as shown ty curve 
 
 from value 
 
 0.2 
 
 f r each corner 
 
 ( e 3 4 5 6 7 
 
 Length of New Construction per Mile of Length Saved= Jd Miles 
 
 FIG. 26o. Curve showing maximum economical length of new construction, 
 replacing old line, per mile of length eliminated. 
 
 example, as a rather extreme case, suppose there are extra corners 
 necessary and the total distance, L 2 , will be on private right-of- 
 way costing $35 per mile per year. 
 
 T L * = *^kw* + 0.975 - T ' 2N T (36) 
 
 *-/6 -L/2 -L/6 JL/2 
 
 If the last term is omitted this equation may be plotted as shown 
 by the curve B, Fig. 26, and gives the economical values of 
 
 T j- (new construction per mile saved) when the extra 
 
 LQ LZ 
 
 2 
 construction cost is fairly large. For extra corners 7 7- miles 
 
 Ltfy JL/2 
 
104 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 must be subtracted from the value shown by the curve for each 
 additional corner. Of course there might be other extra costs 
 for more poles, higher poles, etc., which have not been especially 
 considered here. These could be included in the same manner 
 however. 
 
 The economy for any particular loading on such a line may now 
 be readily discovered from the curves and equations. If, for ex- 
 ample, under the conditions assumed, the load is 6,000 kw. for no 
 corners and no cost of right-of-way, according to curve A, the 
 new construction could be built for 2.7 miles for each mile saved 
 thereby, or say, 8.1 miles in order ta save 3 miles. In case the 
 new construction requires private right-of-way, as per curve B, 
 2.2 miles may be built for each mile saved or 6.6 to save three 
 miles. If two extra corners were necessary, the figure would be 
 1.8 miles for only one mile saved, or for three miles, 2.067 per 
 mile or 6.2 miles and so forth. 
 
 There is thus displayed a tangible method of exhibiting the 
 economy of shortening an old line by reconstruction. For any 
 particular system, of course, the costs must be derived inde- 
 pendently. From them, similar curves can be drawn and applied 
 to the solution of such problems. 
 
 Economy of Using Part of Old Line Equipment in its Original 
 Location for Reconstructed Line. Another characteristic prob- 
 lem of this general nature is the following. In building a new 
 line to replace an old one, which is not worn out, it may be 
 possible to use a considerable length of the old line poles or other 
 materials in place. However some shortening of the length of 
 the line might be accomplished by using a more direct route on 
 entirely new construction. How far from the direct route may 
 the line diverge economically in order to use this old line material? 
 
 The solution is, again, reached by a comparison of the annual 
 costs on the two routes using the general equation already 
 developed. This comparison, however, may be considerably 
 simplified, as in the preceeding problem, by making use of 
 certain short cuts which reduce the number of terms to be con- 
 sidered. The "present value of labor in old line" is a cost 
 chargeable to the reconstructed line as a whole. It will be the 
 same no matter whether the old material is used in place or not. 
 Hence, it may be omitted in the comparison of economy of the 
 two methods of construction on any section of line where the use 
 of old material is considered. The "cost of labor removing the 
 
RECONSTRUCTION PROBLEMS 105 
 
 old line " is also chargeable to the new line as a whole and it might 
 be somewhat difficult to properly apportion the amount to be 
 charged to any particular section. This may be avoided if it is 
 considered, for the comparative costs, that this item is the same 
 amount for both alternative constructions and hence may be 
 omitted from both. Since, however, some of the old material 
 left in place will reduce the actual amount of this item in the one 
 case, the cost represented by the old materials used must be 
 adjusted by subtracting from their present value the amount 
 which it would have cost to remove them. It is seen that this 
 method is equivalent to 'charging both alternatives with the 
 proper share of the "cost of removing old line" and then sub- 
 tracting from one the removal cost eliminated by the use of the 
 old material in place. It is also equivalent to considering that 
 old material in place should be charged at its warehouse value 
 which is its present value as material, if available, less the cost of 
 making it available or removing it to the warehouse. The 
 adjusted annual costs of the reconstructed line on old poles, and 
 of new construction, per mile, may be thus obtained and are 
 convenient figures to use in all such economic comparisons. It 
 must always be borne in mind however that these are adjusted 
 costs and are not the true annual costs chargeable to the line. 
 
 The comparison of the annual costs of the two routes may now 
 be made by comparing the values obtained from the expression 
 
 g (cost of material and labor (adjusted cost)) + cost of energy 
 loss; with the proper selection of g, for both alternatives. 
 
 The following symbols have been used: 
 
 LI = length in miles of the old pole line which can be used, 
 L 2 = length in miles of the alternative new line, 
 L 3 = length in miles of the new construction necessary to supple- 
 ment the old pole line, i.e., bringing the line from the new route 
 to the old if necessary, 
 
 N = the number of new 90 corner constructions saved by 
 
 using the more direct route, 
 kw = load carried in kilowatts, 
 a = age of old-pole line in years, 
 C e = cost of energy per kilowatt hour, 
 Z/4 = length in miles, of right-of-way to be purchased, 
 C r = purchase price per mile of right-of-way. 
 
 The figures here given are based on an old line on untreated 
 
106 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 wood poles, new line on treated poles, voltage of reconstructed 
 line 46,000 volts necessitating new crossarms and insulators 
 throughout, wire No. bare copper on both old and reconstructed 
 lines. 
 
 The annual charges obtained under these conditions are as 
 follows: Annual charges on new construction per 
 
 mile = $350.00 
 Annual charges on reconstruction on old 
 
 , ., 3,350 - 260a + 3.2a 2 
 
 poles per mile = - 
 
 15 a 
 
 Annual charges on energy loss 
 
 Annual charges on corner construction = $38 . 00 
 It is now a question of whether the annual cost on LI -f- L 3 is 
 greater or less than on L 2 using the above figures. It would be 
 advantageous to divert to the old pole line when 
 '3,350 - 260a + 3.2a 2 
 
 , e 
 
 -a ~ro~; 
 
 Ls (350 + g^) + 38AT/Z, (350 
 
 .(L 2 - L 3 ) (350 + 69 fr* Ce ) - 387V + .08L 4 CV 
 T / _ : _ ir _ _ ___ 
 
 \ 3,350 - 260a + 3.2a 2 69kw*C e 
 
 15-a 10 5 
 
 An example of the use of this equation would be in such a case 
 as that of a line following a private right-of-way instead of a 
 highway thereby eliminating four corners and considerable 
 distance. If we assume the old line to be 5 years old (a = 5) 
 and C e .01 and right-of-way at $35 per mile per year the 
 equation would become for that particular case with L 4 = L 2 ; 
 L 3 = 0; # = 4. 
 
 (350 + ~^- 2 ) - 152 + 35L 2 
 
 Ll 
 
 3,350 - 1,300 + 80 .69/cw 2 
 10 10 5 
 
 (385 + ^rgr-J " 152 
 
 (38) 
 
 : + 213 
 
 which is easily solved for known values of LI, L 2 and the load and 
 the relative economy thus determined. If LI is greater than the 
 
RECONSTRUCTION PROBLEMS 
 
 107 
 
 second member, the economy evidently lies in the new route, if 
 less, in the old one. No curve could be plotted which would be 
 of any great value in this instance since there are so many 
 variables in the expression which are fixed for only one particular 
 problem or condition. 
 
 Application of Above Method of Choice of Transmission 
 Route. There is a very useful application of the principles of 
 this last problem in the choice of a route for a transmission line 
 as a whole. When there are several alternative routes a condi- 
 tion similar to the following concrete example is very often 
 encountered. Power was to be transmitted to a distance of 
 approximately 30 miles from the central station. There were 
 three routes available for the new line no one of which had any 
 evident marked advantage over the others. Each one included 
 a different amount of private right-of-way, of old pole line, of new 
 construction, of corners, and of distance over which the construc- 
 tion cost would be excessive on account of high trees and other 
 obstructions. The comparative economy of the three routes was 
 displayed, by use of the adjusted annual costs explained above, 
 and tabulated as follows: 
 
 TABLE 5. ADJUSTED ANNUAL COSTS 
 
 Item 
 
 Route A 
 
 Route B 
 
 Route C 
 
 Miles 
 
 
 Miles 
 
 
 Miles 
 
 
 New construction at $350 
 per mile 
 
 21 
 
 8 
 
 6 
 9 
 
 29 
 3 
 
 $ 7,360 
 1,710 
 
 210 
 342 
 
 3,190 
 
 19 
 
 12 
 
 4 
 
 7 
 
 31 
 2H 
 
 $ 6,650 
 2,560 
 
 140 
 266 
 
 3,410 
 
 22 
 6 
 
 10 
 
 7 
 
 28 
 3 
 
 $ 7,700 
 1,280 
 
 350 
 266 
 
 3,080 
 
 Reconstruction on old poles 
 at $213 per mile 
 Right-of-way at $35 per 
 mile 
 
 Corners at $38 each 
 Energy loss, 4,000 kw. load 
 at $110 per mile 
 Difficult construction 
 
 Total 
 
 
 
 $12,812 
 
 
 $13,026 
 
 
 
 $12,676 
 
 
 If is evident that Route C is the most economical even though 
 it requires the. purchase of more private right-of-way and uses 
 less of the old pole line, since it is somewhat shorter and some 
 corners are eliminated. The extra cost of difficult construction is 
 
108 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 rather hard to arrive at without a detailed layout of the line. 
 It can be estimated however. In the above case, the amount of 
 extra construction was so nearly equal in each case that this was 
 not thought necessary. 
 
 As has been explained in the previous chapter there are some 
 other features of a line that cannot be included in such a cost 
 analysis and which might have considerable weight in the 
 determination of a route. One route might be more accessible 
 for patrolling and repairs than the others. The proximity to a 
 railroad might affect the construction cost considerably. The 
 shelter afforded the line against severe storms might be a large 
 factor for consideration. These points can however be weighed 
 against the advantage in cost alone and the most advantageous 
 route will usually be evident. 
 
 In the foregoing discussion, there has been treated only a few 
 of the many problems involving reconstruction which are en- 
 countered by the engineer. However it has been attempted to 
 set forth the underlying principles of the analysis of costs on such 
 problems so that they can be applied to any similar problems 
 with modification to suit the particular case. The curves and 
 figures given herewith are not intended for use under any other 
 conditions than those for which they were derived. They 
 merely serve as an example of how the methods used may be 
 applied to a given case. The outstanding feature, which is 
 recognized after the application of these methods to a few specific 
 problems, especially problems of relative economy, is the great 
 advantage ordinarily gained by shortening a line. As a rule any 
 ordinary extra construction cost is justifiable if the line is notice- 
 ably shortened thereby. This, therefore, points to the advantage 
 of greater care in the selection of the routes for new lines in new 
 territory even though at the time it may seem advisable, on 
 account of light load, to keep the construction cost a minimum. 
 
CHAPTER XI 
 POWER CIRCUITS 
 
 PROBLEMS RELATING TO LINES CARRYING POWER LOAD CHIEFLY 
 VOLTAGE ECONOMICAL CONDUCTOR SIZE USE OF Two 
 LINES IN PLACE OF ONE DISTRIBUTION OF LOAD 
 OVER SEVERAL LINES 
 
 The study of primary lines will be divided into two parts, i.e., 
 the consideration of circuits carrying power chiefly and of those 
 devoted largely to lighting load. It is realized that on most 
 systems, there is no sharp division between these two classes. 
 In most cases, where the power load is comparatively small, 
 power and lighting are both carried on the same lines. When 
 larger power loads come on, the difficulties of regulation usually 
 call for a separation of circuits, even though the load factor on 
 the lines is thereby reduced somewhat. The facts that lighting 
 load requires a closer regulation than power, and that, when 
 large power loads are considered, lighting and power load usually 
 overlap considerably during the heavy loading season, justify 
 such a practice. 
 
 On some systems, the power and lighting circuits are kept 
 entirely distinct. On others, where three- or two-phase circuits 
 are used for lighting, small- and medium-sized power loads are 
 taken on the same circuits, while separate lines are run for large 
 power loads. In the first case, it is very often the practice to run 
 a three-phase line to some central feeding point and there sepa- 
 rate the phases into individual single-phase circuits for lighting. 
 In the latter case, all branches carrying lighting only are ordina- 
 rily single-phase. It appears, therefore, that the problems relating 
 to power only, to power and lighting combined and to lighting 
 only may be very similar up to a certain point, differing only in 
 load factor. On the other hand, if lighting only is considered, 
 the quite definite load factor simplifies the study somewhat. 
 Also the single-phase lines are problems in themselves. This 
 chapter will be devoted to the problems of power circuits. 
 
 Kind of Problems Encountered. The questions arising in 
 connection with power circuits have to do mostly with voltage 
 and conductor size. Such lines are nearly always run on roads, 
 streets, alleys or lot lines and the pole spacing and location is 
 limited by the mechanical strength of the pole, by the arrange- 
 
 109 
 
110 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 ment of street and lot lines, by provisions for future extensions, 
 etc. Pole heights are governed by standard practice, by city 
 ordinances and by heights necessary to clear obstructions. 
 Occasionally there may arise questions as to the economy of using 
 private right-of-way instead of the public highway for short 
 distances but such problems are usually small ones and are easily 
 solved by a comparison of the annual cost of the two alternatives. 
 Voltage. The voltage for use on power circuits is usually a 
 development from past practice, although it is often found 
 economical to increase the voltage when the load increases 
 beyond a certain amount. The voltages in common use have 
 been pretty well standardized at 2,200, 4,400, 6,600 and 11,000 
 volts. There is a tendency at present to go even higher for 
 power lines with heavy loads, but such lines partake more of the 
 nature of transmission lines. On any line, the higher the voltage 
 the less the line losses and the larger the loads which can be 
 carried on the line with a given regulation. On the other hand, 
 the cost of insulation of the line and the cost of transformers is 
 increased. The additional precautions which it is necessary 
 for construction men to take in working with a higher voltage is 
 also a factor to be considered. To give a comparison, 6, 600- volt 
 transformers can be obtained for about 18 or 20 per cent more 
 than 2,200-volt. Thereby the voltage is multiplied by three, 
 hence the line loss is divided by nine for any given load and wire 
 size. Or, for the same per cent voltage drop and conductor, nine 
 times the load can be carried. Where loads are not heavy, such 
 an increase in capacity may not be desirable as compared with 
 the increased cost of construction. Where heavy loads are 
 handled, however, a voltage of 6,600 or 11,000 may often be 
 found very advantageous. When the problem is one of chang- 
 ing the voltage of a system already in operation to a higher volt- 
 age, the cost of making the change must be taken into account. 
 The old line transformers must be disposed of, or the change 
 made gradually, using the old transformers in certain districts 
 until they are worn out. Station transformers and other appa- 
 ratus, suitable for the higher voltage, must be provided. Often, 
 cables must be replaced with those of higher rating. The increase 
 in annual charges due to all these items must be carefully studied 
 in connection with the value of the increased capacity and reduc- 
 tion in losses achieved, in order to determine the economy of any 
 such alteration. In this connection, the probable increase of 
 
POWER CIRCUITS 111 
 
 load on the system for some time in the future must be esti- 
 mated and what further changes will have to be made eventually 
 to care for probable future conditions. 
 
 Voltage Drop. The problem of voltage drop is an important 
 one to consider in connection with power circuits. The allow- 
 able regulation at the customer is more or less fixed by consid- 
 erations of good service or by contract. The substation bus 
 voltage may be kept within certain known limits. The question 
 is then one of whether to serve the customer by a circuit of small 
 or medium-sized conductor with a regulator, or of large-sized 
 conductor without a regulator. The annual cost of the instal- 
 lation as a whole, including cost of energy losses, will be the 
 criterion. On a large system with a steadily increasing load it is 
 often the practice to standardize on one or two conductor sizes. 
 A new line is built of standard size and allowed to operate at low 
 loads unregulated. Load is added from time to time until, 
 when it becomes too heavy, a regulator is added. From a 
 practical standpoint this method has its advantages. A study of 
 the economy of the installation, however, will still be of real 
 advantage in indicating the standard sizes to use, when the lines 
 become loaded beyond the economical limit, etc. 
 
 Power-factor Improvement. The question of power factor is 
 a prominent one at present. Aside from the proposition of 
 inducing the customer to improve his power factor by using that 
 as a basis for rates, there is a further interesting problem for the 
 central station. Poor power factor means increased losses for 
 the same delivered load in kilowatts. There will be, then, a 
 point at which it will be economical to install static or synchron- 
 ous condensers in order to reduce these losses by improving the 
 power factor. This point can be determined by a comparison of 
 the annual cost of the condenser in place, with the value of the 
 energy conserved, considering also the improvement in regulation. 
 
 Most Economical Conductor Size. The determination of the 
 most economical size of wire for any load, is closely connected 
 with the considerations of voltage and voltage drop, as is evident 
 from the above paragraphs. If the most economical conductor 
 size, considering the line only, is known, however, it serves as a 
 starting point for the further study of the economy of the circuit 
 as a whole, including regulators, etc. An example will be given 
 here of the method of attacking this problem of most economical 
 wire size for power circuits. 
 
112 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Annual Cost Equation. For a three-phase line the annual cost 
 per 1,000 ft. of line 
 
 = g (cost of wire + cost of stringing) 
 + (cost of poles, fixtures, guys, etc.) 
 -f cost of energy loss. 
 
 Where g = per cent interest, taxes, depreciation, etc. 
 (Cost of right-of-way is omitted as it is not always present on 
 such lines, and, in any case, is the same for all sizes of conductor.) 
 
 In this case, instead of developing the above equation in terms 
 of the cross-sectional area of conductor and then, by means of the 
 first derivative, determining the most economical size, it has been 
 found more useful to investigate the range of loads for which 
 any given stock wire size is more economical than any other. 
 The method used is as follows: 
 
 The annual cost per 1,000 ft. of line for each standard conduc- 
 tor size is obtained in terms of the cost of copper, the load, 
 voltage, power factor, equivalent hours, and cost of energy. 
 This equation is of the form 
 
 C. (39) 
 
 Where Y = annual cost per 1,000 ft. of line, 
 C cu = cost per pound for copper, 
 kw = load in kilowatts, 
 
 E = voltage, 
 Cos = power factor, 
 
 t = equivalent hours, 
 C e = cost of energy per kilowatt-hour, 
 KI, K 2 , K s = constants. 
 
 Combined Equation. If the equation for any stock size of wire 
 is combined with that of the next adjacent size by equating the 
 annual costs, another equation is obtained which expresses the 
 conditions under which there is no choice in economy between 
 the two. If, for example, in the above equation E, and C cu are 
 fixed, the combined expression would give, for any value of 
 tC e , the load at which the economy changes from one size of 
 conductor to the next. If such expressions are determined and 
 plotted for No. 6 to No. 4 and for No. 4 to No. 2, for example, 
 the values of kw/cos 6 between the two curves, for any value of 
 tC e , indicates the range of loads for which No. 4 wire is more 
 
POWER CIRCUITS 
 
 113 
 
 economical then either adjacent size. For smaller loads, No. 6 
 is more economical, for larger loads, No. 2 (see Fig. 27). 
 
 Value of Constants. In the above equation (39) the constants 
 KI and K% depend on wire size, local costs for stringing wire, 
 and local standards of construction and costs for use with each 
 wire size. It may be assumed in this case that the poles will 
 be the same for any size of conductor, and hence their cost will 
 cancel out when the equations for two wire sizes are combined. 
 
 Drawn far46OO Volis. 
 
 Maybe used for ZZOOvoffs if had 
 iven is mulfipl/ecf by 2 
 opper af 3O cerrhs per pound 
 For values ofiCe see Table H 
 
 200 400 GOO 800 1000 1200 1400 1600 1800 
 
 FIG. 27. Economical wire size for three-phase primary. 
 
 The cost of pole fixtures (crossarms, etc) does not increase pro- 
 portionally with wire size but will be the same for several sizes 
 and then change abruptly for the next larger group. The cost 
 of conductor in place, including incidental material and labor 
 cost of stringing, will follow such an expression as K a + Kb C cu , 
 K a and K b being separately determined for each size of wire. 
 These constants, properly combined give the values of KI and K 2 . 
 Cost of Energy Loss. The cost of energy loss is determined 
 from the equation: Annual charge for energy loss per 1,000 ft. = 
 
 C I kw \ 2 
 
 37 2 r X 365 X t X = 365,000 rtC e (40) 
 
 Where r = resistance of conductor per 1,000 ft. 
 
 8 
 
114 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Then K 3 (in Eq. 39) = 365,000 r, for any size of conductor. 
 
 Numerical Example. The following equations of total annual 
 cost were obtained in a specific instance, with all constants 
 evaluated. 
 
 TABLE 6. EQUATIONS OF TOTAL ANNUAL COST 
 
 SIZE OF 
 WIRE 
 
 (Icin \ 2 
 *p g J tC e 
 
 4 8.41+ 80.0Ccu+ 92,500( J Y tC e 
 
 \Hi COS u/ 
 
 2 11.30 + 126.0C CU + 58,100 %C e 
 
 12 . 43 + 204 . OC CM + 37 , 300 ( -, kw \ *tC e 
 
 \Mi COS v/ 
 
 00 14 . 62 + 251 . OCcu + 29 ,500 (-, fcM? J) \C e 
 
 \Ei COS P/ 
 
 000 16 . 43 + 313 . OCcu + 23 , 400 ( -, kw } * tC e 
 
 \Jti cos 6/ 
 
 0000 17 . 32 + 382 . OC CM + 18 , 600 ( * w \ * tC e 
 
 Equating the expressions for cost for each adjacent pair of 
 wire sizes, the following expressions are obtained. 
 
 TABLE 7 
 
 SIZE OF SIZE OF 
 
 WIRE WIRE 
 
 6 to 4 54,60o( F fc ^ Y tC e = 23.8 C cu + 1.71 
 
 \.Cf COS (// 
 
 4 to 2 34,400^^ \ 2 tC e = 46.1C CM + 2.91 
 
 2 to 20,800 (-, kw .) 2 tC e = 78.0C + 1.13 
 
 \is cos oi 
 
 to 00 7,8( 
 
 00 to 000 6>100 Vgcosg/ <Ce = 61.2C CM + 1.81 
 
 000 to 0000 4,800^^ g \ * tC e = 69.0C CM + 0.89 
 
 to 0000 18 , 700 ( kw ) * tC e = 178 . OC CU + 4 . 89 
 
 \/i COS U/ 
 
 Assuming the voltage of 4,600 and two prices for copper, 30 cts. 
 and 20 cts., which represent a good range of values, the equations 
 become 
 
POWER CIRCUITS 115 
 
 TABLE 8. FOR 4,600 VOLTS 
 
 SIZE OF SIZE OF 
 
 WIRE WIRE 
 
 6 to 4 
 
 4 to 2 
 
 2 to 
 
 to 00 
 
 00 to 000 
 
 000 to 0000 
 
 to 0000 
 
 30-c 
 
 tC e = 
 iC 1 - 
 
 T. COPPER 30-CT. COPPER 
 3,430 2,510 
 
 (kw/cos 0)2 
 10,290 
 
 (kw/cos 0) 2 
 7,460 
 
 tC 1 
 
 (kw/cos 0)2 
 24,900 
 
 (kw/cos 0) 2 
 17,100 
 
 tC e = 
 tC e = 
 
 fC 
 
 (kw/cos 0)2 
 41,800 
 
 (kw/cos 0) 2 
 28,900 
 
 (kw/cos 0) 2 
 70,000 
 
 (kw/cos 0)2 
 48,750 
 
 (kw/cos 0) 2 
 95,200 
 
 (kw/cos 0) 2 
 64,750 
 
 tC e = 
 
 (kw/cos 0) 2 
 66,000 
 
 (kw/cos 0)2 
 45,800 
 
 (kw/cos 0) 2 
 
 (kw/cos 0) 2 
 
 For 2,300 volts the numerators of the above expressions for 
 4,600 volts should be divided by 4 
 
 (as No. 6 to No. 4. tt7. - 
 
 The curves plotted for 4,600 volts can be used for 2,300 volts if 
 the given load for 2,300 volts is multiplied by 2 before applying 
 curve. 
 
 Plotting Results in Curves. As was mentioned before in 
 Chap. IX on " Secondary Transmission Lines," the cost of energy 
 losses per kilowatt-hour, C ej and the equivalent hours, t, are inter- 
 related. For any type of load, such as that on a typical power 
 circuit, the value of C e corresponding to any value of t, may be 
 determined approximately. For power circuits in heavily 
 loaded districts, such as the industrial areas in a large city, the 
 load factor, equivalent hours, and power factor may be practically 
 the same for nearly all lines. In that case, the problem is simpli- 
 fied and the range of loads for which any wire size is most eco- 
 nomical is more simply defined. For the general case of power 
 circuits, however, the load may vary from a single motor to a 
 large manufacturing plant load, from a few hours per week oper- 
 ation to continuous 24 hr. per day. Naturally the load factor 
 will vary through a large range, and the equivalent hours for 
 the same load factor will be different for different loads, depending 
 on the operation. 
 
 The above equations therefore have been plotted using tC e 
 as one coordinate, Figs. 27 and 28. The method of determining 
 
116 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 the value of tC e for any load will be explained later. In order to 
 make the curves applicable to loads of various power factors, the 
 
 other coordinate was made -- -. The computation of - 
 
 cos 6 cos 6 
 
 for any value of either quantity can be made graphically by use 
 of the curves on the lower half of the figure. The intersection of 
 the curve for any load with the horizontal for the desired power 
 
 factor gives the value of 
 
 COS u 
 
 on the scale below. 
 
 Drawn for4600Vo!rs. 
 May be used for 23 OO vo Its if load 
 giver is multiplied bu 2 
 Copper afzo cents per pound 
 For values offC e see fable H 
 
 1600 1800 
 
 FIG. 28. Economical wire size for three-phase primary. 
 
 Use of Curves. Since the equations were so developed as to 
 show the points where economy changes from one size of wire to 
 another, it follows that the area between two curves is the locus 
 of all points for which the size of wire shown is most economical. 
 
 Jci/j 
 
 Thus, on Fig. 28, for ( ^ = 600 and tC e = 0.06, No. primary 
 
 is more economical than No. 2 or No. 00. Similarly it is more 
 economical than No. 2 for all values of tC e greater than .045 and 
 is more economical than No. 00 for all values of tC e less than .08. 
 The dotted curve shows the division between No. and No. 0000 
 which can be used in case No. 00 and No. 000 are not used as 
 standards for such lines. Two sets of curves are given, one for 
 
POWER CIRCUITS 
 
 117 
 
 20 ct. copper and one for 30-ct. Intermediate values can be 
 interpolated. 
 
 The use of the curves, then, is as follows : 
 
 1. Locate the intersection of the curve for the load in kilowatts with the 
 horizontal of its power factor. 
 
 2. Locate the intersection of the ordinate through this point .with the 
 proper value of tC e (scale on left). 
 
 3. The area in which this point lies indicates the most economical wire 
 size. The distance of the point from the curve dividing that area from the 
 next adjacent area is an indication of the amount of economical advantage 
 of the one size over the other. 
 
 Determination of tC e . The proper value of tC e to use for any 
 load may be determined as follows: It was shown in Chap. V 
 in the discussion of equivalent hours, that the value of equivalent 
 hours corresponding to any load factor may vary between certain 
 limits. The upper limit (load factor X 24) would be correct 
 only for a load, such as a single motor, which has a constant value 
 for its whole time of operation. The minimum value would be 
 (load factor) 2 X 24 for a load with a momentary peak and the 
 remainder of the day's curve flat. With power loads, it is prob- 
 able that t varies from somewhere near the first quantity for 
 small loads, such as one or two motors, to somewhere near the 
 average between the two for large loads with a number of motors 
 not running simultaneously. Probably, for most loads encoun- 
 tered, t will be nearer the latter figure. 
 
 The limits of t would be as follows: 
 
 TABLE 9 
 
 Load factor 
 
 Continuous load 
 
 Momentary peak 
 
 Average 
 
 
 
 
 
 
 
 
 
 .10 
 
 2.4 
 
 .24 
 
 1.32 
 
 .20 
 
 4.8 
 
 .96 
 
 2.88 
 
 .30 
 
 7.2 
 
 2.16 
 
 4.68 
 
 .40 
 
 9.6 
 
 3.84 
 
 6.72 
 
 .50 
 
 12.0 
 
 6.0 
 
 9.00 
 
 .60 
 
 14.4 
 
 8.64 
 
 11.52 
 
 .70 
 
 16.8 
 
 11.76 
 
 14.28 
 
 .80 
 
 19.2 
 
 15.36 
 
 17.28 
 
 .90 
 
 21.6 
 
 19.44 
 
 20.52 
 
 1.00 
 
 24.0 
 
 24.00 
 
 24.00 
 
118 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 An approximate determination of the variation of the cost of 
 energy losses with load factor can be made quite easily as indi- 
 cated in "Appendix A." Of course, if more accurate cost figures 
 have been determined they are preferable. The following indi- 
 cates such a characteristic variation. 
 
 TABLE 10 
 
 LOAD FACTOK 
 
 .10 
 .20 
 .30 
 .40 
 .50 
 .60 
 .70 
 .80 
 .90 
 1.00 
 
 COST OF ENERGY Loss 
 PER KILOWATT-HOUR 
 
 .0278 
 .0184 
 .0150 
 .0133 
 .0121 
 .0113 
 .0107 
 .0101 
 .0096 
 .0092 
 
 0.24 
 0.22 
 0.20 
 0.18 
 0.16 
 0.14 
 
 o.io 
 
 0.08 
 006 
 0.04 
 0.02 
 
 
 Average Hours per Day (Including Sundays) 
 3 6 9 12 15 18 21 24 
 
 CURVE A -For loads which are constarfr-during time ' 
 of operation as one motor with steady hadftimiting case) 
 
 CURVE B - For loads which have a short peak with the 
 remainder of curve flat. (Limiting case} ' 
 
 <7 
 
 CUR 
 
 VEC- Average of A and i 
 of average loads 
 
 3. Fits 
 
 \ 
 
 most 
 
 cases 
 
 
 
 (/ t 
 
 
 
 
 
 
 
 \ 
 
 )/ 
 
 / \ 
 
 / 
 
 / 
 
 
 
 
 
 
 
 
 V 
 
 :/ 
 
 ty 
 
 ' 
 
 
 
 
 
 dSSi 
 
 <&' 
 
 \ / 
 
 ^4 
 
 f 
 
 
 
 
 
 
 7\ 
 -M* 
 
 ^rai 
 
 v 
 
 W] 
 
 / 
 
 
 
 
 
 
 i 
 
 
 
 
 
 / 
 
 / 
 
 I 
 
 \ 
 
 
 
 
 / 
 
 ^ 
 
 7 
 
 i 
 
 ' y 
 
 \ 
 
 ! 
 
 
 
 
 / 
 
 
 
 
 x ! 
 
 
 I 
 
 i 
 
 
 
 
 (~* 
 
 ^ 
 
 
 ; 
 
 
 ! 
 
 i 
 
 
 
 
 0.10 0.20 O.JO 0.40 0.50 0.60 0.70 0.80 0.90 1.00 
 
 Load Factor 
 FIG. 29. Curves showing values of tC z for power loads. 
 
 Using these figures in connection with the table given above 
 the curves shown on Fig. 29 are plotted which show the values 
 
POWER CIRCUITS 119 
 
 of tC e corresponding to any load factor. A second scale is shown 
 at the bottom giving the average number of hours per day of 
 peak operation, corresponding to any load factor, which is useful, 
 especially in connection with small loads. If the load factor of 
 a load is known and the approximate shape of its typical curve, 
 the value of tC e may be selected. For a single motor averaging 
 3 hr. a day, for example, tC e would be about .072. For a larger 
 load with a definite peak and an average load curve, with a load 
 factor of 30 per cent, the value of tC e would be somewhere near 
 .08. In general it may be said that: 
 
 TABLE 11 
 
 Small power 
 
 .LOAD 
 
 FACTOR 
 
 Oto 20 
 
 LOAD CURVES 
 Continuous to average . . 
 
 tCe 
 
 . . 04 to . 06 
 
 Medium power 
 
 10 to 30 
 
 Near average 
 
 . 0.04 to 0.07 
 
 Large power 
 
 20 to 40 
 
 Near average 
 
 . . 05 to . 09 
 
 These figures are indicated on the curves, Figs. 27, 28, and by 
 the brackets on the left. For lines carrying lighting only, the 
 value of tC e would be between .03 and .05, being nearer the former 
 for residence lighting only, and approaching the latter figure for 
 heavy store lighting, etc., with a fair average of about .04. 
 
 For a load which combines power and lighting the value of 
 tC e will depend on the proportion of each. Where the power 
 predominates, the lighting load will have the effect of increasing 
 the load factor. Where lighting predominates the power load 
 will have the same effect. In either case a higher value of tC e 
 should be used than would be assumed for the predominating 
 type of load alone. The amount of increase must be estimated 
 from a consideration of the probable load factor for the particular 
 case. 
 
 Similar Curves Should be Derived Locally. Curves such as 
 those illustrated here, may be worked out for any system, using 
 local cost figures. (The examples given here must not be 
 considered applicable to any but the system for which they were 
 derived.) By their use the most economical wire size for any 
 load may be determined and, from that point on, the problem 
 becomes one of obtaining proper regulation in the most econom- 
 ical way. 
 
 Power Circuits with t Constant. As was indicated above, for 
 power circuits in heavily loaded manufacturing districts where 
 
120 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 there is considerable diversity of load on each line, but the general 
 characteristics of all loads are somewhat similar, the value of 
 equivalent hours will be nearly the same for all lines. If an 
 average value of t is determined, the problem of economical 
 conductor size may be simplified and it is possible to study the 
 effect of variations in the cost of copper and of energy to better 
 advantage. 
 
 Determination of t. The determination of average equivalent 
 hours for a number of circuits was explained in Chap. V with an 
 example of lighting circuits. For power circuits the method is 
 similar. In a specific instance, bi-monthly curves were taken 
 for a number of power circuits for 13 months and the equivalent 
 hours for each such curve in terms of the curve's peak was 
 determined. This was assumed to be the average for the half 
 month covered. Each of these figures was reduced to a value of 
 equivalent hours in terms of the year's peak by multiplying it by 
 the square of the ratio between the peak for the day for which 
 the figure was derived and the year's peak. The 26 values thus 
 obtained were averaged and the result was assumed to be a fairly 
 accurate figure for the equivalent hours for the whole year, 
 considering days of operation only, exclusive of Sundays and 
 holidays. In the example taken, the value of equivalent hours 
 was found to be 9.69. This is high for ordinary purposes, being 
 obtained at a time of high production during the war period, but 
 will serve as an illustration of the method. 
 
 Equations for Annual Cost. The formulas for annual cost with 
 any size of conductor were altered somewhat for convenience, 
 and to bring out another method of representing the results. 
 It was assumed, as an approximation, that all charges on con- 
 struction, which are not proportional to the cost of copper, are the 
 same for all sizes, and hence, cancel out when the equations for 
 two sizes are combined. The error thus introduced is small for 
 any two sizes near together, such as No. 6 and No. 4, but becomes 
 greater for such combinations as No. and No. 0000. In any 
 case, however, it can be kept in mind and compensated for when 
 using the resulting curves. 
 
 Then, the annual charge per 1,000 ft. on the above basis 
 
 Y = ?L (3,OOOwiC Ctt ) + I 2 X -f X 3,000 X t X 300 X 
 
 + 900/ 2 - tC e (41) 
 
POWER CIRCUITS 
 
 121 
 
 for a conductor weighting Wi pounds per foot and of cross- 
 sectional area A i, using 300 working days per year to correspond 
 to t as derived above. 
 
 If this expression is combined with a similar one for a conductor 
 of weight Wz and area A 2) the resulting equation may be reduced 
 to 
 
 ^cu == ~t 7 T~ ~~~\ 1 t (42) 
 
 If p, t, and g are considered constant and known for any pair 
 of conductor sizes, this equation may be reduced to 
 
 r 
 
 cu _ If '[ 2 
 
 c7 : 
 
 Curves for Economical Wire Size. The curves shown in 
 Fig. 30 have been plotted from this equation. Their use is 
 
 (43) 
 
 Sec 
 
 IU 
 
 4? 60 
 
 6 20 
 
 FIG. 30. Economical wire size for any ratio of copper cost to energy cost for 
 3< power circuits, equivalent hours = 9.69. 
 
 similar to that of Figs. 27 and 28, the areas between the curves 
 belonging to the conductor sizes indicated. It is clearly evident 
 that, as the ratio between copper price and energy cost increases, 
 by copper price increasing or energy cost decreasing, the smaller 
 wire becomes more economical. As was explained above, in 
 using the curve between No. and No. 0000, the excess cost of 
 stringing the larger size must be kept in mind. This would 
 have the same effect as an increase in price of copper, i.e.j to raise 
 the curve somewhat. 
 
 Use of Two Lines in Place of One. The study of economical 
 conductor size may be further extended, in a similar manner, to 
 
122 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 the consideration of the economy of the use of two lines in place 
 of one. Assuming that, for mechanical reasons, No. 0000 wire 
 is the largest size which is used as a standard on a given overhead 
 system, there is a load at which it becomes economical to use 
 more than one No. 0000 circuit. Another circuit of any size of 
 conductor might be added, according to the load. For simplicity, 
 the problem here will be limited to that of discovering at what 
 load two No. 0000 lines are more economical than one, assuming 
 the same conditions of load as in the previous example, with 
 t constant. 
 
 JJ_A $ 
 
 -* 
 
 i l 
 
 < 
 
 3<f> Power 
 Line 
 
 3 
 
 
 8 
 
 3< 'Power 
 Line 
 
 > 
 
 i i 
 
 - 
 
 s,**~0 
 
 
 5 &-S4 Primary 
 
 fl 
 
 FIG. 31. Arrangement of wires showing average line conditions. 
 
 In this case, the charges on construction cannot be assumed to 
 be the same for both conditions since twice as much pole space is 
 occupied by two circuits as by one. In some cases, this would 
 simply cause the addition of an extra crossarm. It very often 
 occurs, however, that, in districts where such large loads are 
 found, the poles are heavily loaded and all available pin positions 
 are valuable. In such a case, the circuit should be charged with 
 that proportion of the total cost of poles and fixtures equal to 
 the proportion of the total available pole space which it occupies. 
 If, for example, such an arrangement as shown on Fig. 31 is 
 assumed to represent average line conditions, one of the three- 
 phase lines occupies one-fourth the pole space. If the average 
 cost of a pole and four crossarms with other fittings is $60 the 
 cost of pole space for the line in question is $15 per pole, or 
 about $150 per 1,000 ft. 
 
POWER CIRCUITS 
 
 123 
 
 If Cp = cost of pole space per 1,000 ft. 
 
 and C c = cost of stringing, conductor, insulators and pins per 
 1,000 ft. 
 The total annual cost for one circuit 
 
 Fi = j-^(3,OOOu;C ctt + C p + C c ) + 900 7 2 jj- tC e (44) 
 and for two circuits 
 
 + C p + C c ) + 2 X 900 } 2 ^ *C e (45) 
 
 The investment cost is doubled while the energy loss is halved. 
 Equating YI and Y* 
 
 C 
 
 c) = 4507 2 
 
 (46) 
 
 In order to put the curves in the same form as those for single 
 
 C 
 lines, this equation must be reduced to an evaluation for -^r* 
 
 Tf I- = 1 -I- p 
 
 * 3,OOOwC Ctt 
 
 fc) = 4507 2 ^ tC e 
 
 C e gwkA 
 
 7 2 
 
 (47) 
 
 of 
 GJ 
 
 rwo#oooo 
 
 100 120 140 160 180 ZOO 2tO Z40 260 
 Maximum Load in Amperes 
 
 300 3aO WO 360 
 
 FIG. 32. Economical circuits for any ratio of copper cost to energy cost for 
 
 three-phase power. 
 
 If p, t, g, w and A are constant and k is evaluated for several 
 values of C CU} this equation can be plotted in a series of curves as 
 
124 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 shown on Fig. 32, one curve for each value of C cu . This could 
 have been added to Fig. 30, if desired, as it is of the same form. 
 
 Improvement of Regulation by Two Circuits. An interesting 
 point arises in connection with the use of two circuits instead 
 of one when a low power factor is encountered. While the resist- 
 ance drop of a conductor decreases proportionally with the in- 
 crease in size, the inductance drop decreases slowly. Hence, for 
 power factors below a certain value, little improvement in regu- 
 lation is accomplished by increasing the size of conductor. On 
 the other hand, the addition of a second circuit materially 
 reduces the inductance drop and hence the regulation. 
 
 As a concrete example, compare the voltage drop on a No. 0000 
 circuit carrying 3,000 kw., 7,500 ft., with two No. circuits, both 
 at 4,600 volts, three-phase, with 28-in. spacing. The resistance 
 of No. wire being about twice that of No. 0000, the equivalent 
 resistance of both installations is about the same. The power loss 
 and voltage drop, at different power factors, as figured by the 
 charts in Chap. VII, are as follows: 
 
 TABLE 12 
 
 Power factor 
 
 Power loss 
 
 Voltage drop 
 
 One No. 0000 
 
 Two No. 
 
 One No. 0000 
 
 Two No. 
 
 Per cent 
 
 Per cent 
 
 Per cent 
 
 50 
 85 
 95 
 
 14.6 
 5.03 
 4.05 
 
 14.25 
 4.87 
 3.97 
 
 18.05 
 8.78 
 6.64 
 
 10.97 
 6.04 
 5.05 
 
 It is very evident, from the above, that, at a low power factor, 
 considerable advantage in regulation is gained by using two 
 circuits of small conductor rather than one circuit of twice the 
 size. This would be economical if the improvement in regulation 
 is worth more than the increased cost of construction. At high 
 power factors, the advantage disappears since the voltage drop is 
 practically equal to the resistance drop. 
 
 Economical Distribution of Load over Several Lines. One 
 more typical problem will be included in this discussion of power 
 circuits. In cases where there are several different lines, of 
 different lengths and conductor size, feeding a large load, it is 
 
POWER CIRCUITS 125 
 
 desirable to determine the most economical division of the load 
 among those lines. For simplicity consider two circuits only. 
 If 7 = the total load current, 
 
 I a = the economical current on line a, 
 Ib = the economical current on line 6, 
 7 = I a -f h (approximately). 
 
 Since the lines are already in place, the annual charges on 
 construction will be constant for each 
 
 K a = annual charges on construction for line a. 
 Kb = annual charges on construction for line b. 
 The annual cost of energy loss on each circuit will vary with 
 7 2 and C e , and with the resistance, t being fixed. 
 K c I a 2 C e R a = cost of energy loss in line a. 
 K c I b 2 C e Rb = cost of energy loss in line b. 
 Then the total annual charges on line a, 
 
 FTT _L ~V T ir< E> fAQ\ 
 
 a = J^a + Ac/a^Ce/Ia (48) 
 
 and on line 6, 
 
 7jf | Jf T 2/^ E> //in\ 
 
 6 = J^-b T~ J^cib^eKb V*yJ 
 
 and on the total installation, 
 
 7-V _L_ V TT _I_7T _l_JT/^/'72E> -LT^E^ 
 
 J- a \ J- b J^a "t" -L*-l ~T J^c^e(J-a tia T IbKb) 
 
 Substituting for 7 b its value 7 7 , 
 
 Y = K a + Kb + K c C e (la 2 R a + # & (7 2 ~ 27J + 7 2 ) (50) 
 
 and the annual cost per ampere carried 
 
 - 2I + TJ) < 51 > 
 
 The condition of maximum economy is reached when y- be- 
 comes a minimum. The value of I a to accomplish this may 
 
 Y 
 
 be determined by taking the first derivative of -=- with respect to 
 
 7 a , since 7 is constant 
 
 dY/I /2Ra _ ,2R_b T \ 
 
 dl a 6 M 7 la " 7 V : 
 
 7 - Rb I (vn 
 
 *a r> _i_ r> ^ \O*) 
 
 Ra -\-tib 
 
 Similarly 
 
 to = p^ETff ' () 
 
 /l a T ^6 
 
 and 
 
 laRa ~ IbRb (54) 
 
126 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 In case more than two lines are considered, two or more lines 
 can be represented together as one equivalent circuit and the 
 same mathematics apply. 
 
 From the above, therefore, the rule can be formulated that: 
 
 The most economical distribution of a load over several cir- 
 cuits is effected by making the load on each circuit inversely 
 proportional to its resistance, i.e., by making the IR drop equal 
 on all lines. 
 
 This is what might have been expected if it is considered that 
 the total annual charges on construction will be the same, regard- 
 less of the division of the load, and hence, the object is to make 
 the total energy losses a minimum. The accompanying Fig. 
 33 indicates how the total cost of energy loss on two circuits 
 
 f Energy Losses in dollars 
 
 
 
 
 Tofat L 
 Lima 
 
 octd ** 400 
 -400 ft of 4 
 -. 400 ft of* 
 Loss at * 
 lent Hour 
 er Voltage 
 Factorof 
 
 1/fw. 
 
 SOMCMcai 
 50 MC Meat 
 O.OIperkn 
 ,=/0 
 = 4600v. 
 'oad*80% 
 
 >le+SOOOt 
 le +5000 f 
 -hr 
 
 t.of#000 
 >-.OftO 
 
 ? 
 
 \ 
 
 
 , 
 
 Line b 
 &*!, 
 tguiva 
 fleceiv 
 Power 
 
 
 \ 
 
 
 | 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 < 
 
 \ 
 
 
 1 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 I 
 
 finnn 
 
 
 V 
 
 J 
 
 [ 
 
 ^^ 
 
 -"" 
 
 
 
 
 
 06 0.8 1.0 
 
 Ratio of Load on Two Lines in Ampres= ^ 
 
 FIG. 33. Total annual cost of energy losses for various methods of dividing a 
 load between two lines. 
 
 with different divisions of load can be displayed graphically. 
 The point of minimum cost is clearly defined. 
 
 Division of Load Between Lines in Parallel. For direct- 
 current circuits the load would naturally divide itself in prac- 
 tically the most economical proportions if the lines are paralleled. 
 For alternating current, however, the inductive reactance has 
 an effect, and the natural division of current would be somewhat 
 different. The currents would divide so that the ratio of the 
 current in any circuit to the load current is the same as the ratio 
 of the admittance of that circuit to the combined equivalent 
 admittances of all the circuits. For example with a No. and a 
 No. 0000 overhead circuit, of the same length, spacing, etc. 
 
 For No. 0, RQ = .539 ohm per mile 
 No. 0000, #0000 = .269 ohm per mile 
 
POWER CIRCUITS' 127 
 
 The most economical division of current would then be 
 9fiQ 
 
 7 " 0.589 +0.269 f = 
 
 
 Q 539 + Q 269 
 
 The natural division, if the circuits were paralleled would be 
 
 7 ooo = l = <667/ r % the total current 
 
 7 = ^ 7 = 0.4447 or 44.4 per cent of the load current 
 
 Y total 
 
 7oooo = ^r^ 30 7 = 0.5667 or 56.6 per cent of the load current 
 
 Y total 
 
 The fact that the arithmetical sum of 7 and 7 ooo is not equal 
 to 7 is due to the fact that they are not in phase with each other, 
 nor with 7. 
 
 Large Economies Possible on Power Circuits. The field of 
 study of the economics of power circuits will be found quite 
 extensive and very fruitful of real results. Such circuits in 
 manufacturing districts, usually handle loads many times as 
 large as those on the ordinary lighting circuits, single loads 
 amounting to several thousand kilowatts in some places. The 
 load factor is also comparatively high in most cases. The cost 
 of pole space, on the other hand, is often considerably less per 
 circuit on account of the fact that power circuits are usually 
 run on poles which would be set for lighting circuits in any case. 
 It is evident, therefore, that the saving of a few per cent in line 
 loss on such a line may mean the saving of a considerable sum of 
 money during the year. Hence, the operation under the most 
 economical conditions possible is probably productive of more real 
 saving in money than on any other part of the system. A study 
 of the economics of any type of installation is always beneficial, 
 but in the case of power circuits it is most imperative. The 
 problems exemplified in this chapter indicate the type of question 
 of this kind which will arise most often and illustrate methods 
 which have been found useful for attacking their solution. 
 
CHAPTER XII 
 LIGHTING CIRCUITS 
 
 ECONOMICAL STUDIES ON CIRCUITS CARRYING LIGHTING ONLY 
 
 PREDICTION OF LOAD CONDUCTOR SIZE INCREASING 
 
 CAPACITY OF OVERLOADED SYSTEMS 
 
 The preceding chapter indicated that in general there were 
 three classes of primary circuits, those carrying lighting load 
 only, those carrying power loads only and those carrying loads 
 made up of a combination of the two. It is here planned to deal 
 with circuits carrying lighting load only particularly in reference 
 to residence lighting. 
 
 The problems encountered in general can be divided into three 
 classes. In the first class are those pertaining to the design of a 
 new system to handle a given or predicted load, such as would be 
 found where it is planned to build a distribution system in a 
 town where no electric service has been furnished before. In 
 this case we meet a problem somewhat similar in characteristics 
 to the one encountered under " backbone" transmission lines. 
 Here our purpose is to design the most economical system possible 
 with few of the factors of design previously established. The 
 only limiting features would be accepted practice, equipment 
 obtainable, and the general knowledge of the subject as recorded 
 in other work of the same class. The problem would there- 
 fore resolve itself into one of compiling costs on materials and 
 labor, and working out, as pointed out previously, comparative 
 costs for several different alternative designs using different 
 voltages, different types of primaries, such as single-phase, 
 three-phase, four-wire, etc. The method of determining an 
 economical conductor size will be discussed later on in this 
 chapter. It is a matter of finding the lowest annual charges 
 (investment and energy losses) within the limits of the quality of 
 service desired. 
 
 The second class of problems would be that of operating and 
 extending an existing system of primary lighting lines in the 
 most economical manner. This class of problems would be the 
 most commonly encountered in practice. Here certain limita- 
 tions are found prescribed by the characteristics of the system 
 at hand which would, in general, prevent any considerable change 
 from the practice laid out at the inception of the project. 
 
 128 
 
LIGHTING CIRCUITS 129 
 
 However, it is advisable to be prepared always to contemplate 
 the possibility of a radical change in such a system. This brings 
 in the third class of problems. They call for a study of the 
 advisability of making such changes as raising the voltage, 
 changing from S to 3 0, three- wire or four- wire, etc. 
 
 Predicting Load on Residence Lighting Circuits. The problem 
 of predicting load on circuits does not involve economics. How- 
 ever, a correct estimate of what should be expected in the next 
 following years is essential as a basis for economical design. 
 
 If the demands for electric service which are made each year 
 are to be met economically, they must be anticipated far enough 
 in advance to enable provisions for serving to be made them when 
 it is most economical to do so. About the only way to make 
 intelligent estimates of future requirements is to couple an analysis 
 of past rates of load increase with a far-sighted judgment which 
 will take into account the effect on future conditions of the past 
 rate of growth. 
 
 Predicting the load on circuits carrying a load consisting of 
 residence and store lighting is usually simplified by the fact that 
 the growth is relatively constant and that the other factors that 
 come in to modify the estimates can be analyzed and allowance 
 can be made for them. 
 
 In the following it will be shown how an analysis of the con- 
 ditions causing the increase has been attempted for a number of 
 single-phase circuits fed by 200,000-circ. mil. underground cables 
 and No. or No. 0000 overhead wires. 
 
 The data available for such a study consist of a set of curves, 
 one for each circuit, showing the monthly maxima over a period 
 of 4 years. These curves are not all of the same outline, yet 
 there are certain characteristics appertaining to all which could 
 be expressed in one curve to be considered typical. A typical 
 curve would serve as a fairly firm foundation on which to base 
 estimates for the future. 
 
 The changes in circuit loads, shown by an inspection of the 
 monthly maxima, are caused by a number of easily defined 
 factors. The first is a seasonal condition due to the change in 
 the length of days and the result of cold weather keeping people 
 indoors; the second is an increase in the use of electricity by each 
 customer, caused by the greater appreciation of the uses of 
 electrical energy in the home; the third is due to increase of 
 population within the boundaries of each circuit; the fourth is due 
 
130 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 to the embracing of new territory by the circuit or to the loss of 
 old territory caused by switching of load on account of overload 
 or other operating necessities. 
 
 The first two factors are constants in their own particular 
 sense that is to say, the rate of increase due to these factors is a 
 fairly constant value and the third is partly so, for it may be 
 subdivided into two, the factor of normal increase in population 
 and that of abnormal increase. Normal increase will be encount- 
 ered in practically every section of the city, abnormal only in 
 sections not yet closely built up and to which people are attracted 
 by real estate activity or the circumstances of industrial develop- 
 ment. The fourth factor is a result of the preceding three and 
 niust be considered as indefinite, as it appears both for and 
 against increases of load. It is a factor, however, for which 
 correction may be made, since its use lies within the control of 
 the operating company. 
 
 Ordincrtes of Curve I 
 i oo to o r- 
 
 ,.i-r,T 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r> i o> o o = 
 
 Ordincrres of CurveTI 
 
 
 ^x 
 
 ^"*" 
 
 
 *** 
 
 
 
 >-., 
 
 ; 
 
 
 
 
 
 X 
 
 X 
 
 ^^ 
 
 
 ^ 
 
 
 
 
 > 
 
 x v 
 
 
 
 ~7 
 
 / t 
 
 / 
 
 X 
 
 
 
 
 
 
 ^ 
 
 N 
 
 N 
 N 
 
 s 
 s 
 
 / 
 
 
 
 
 
 
 
 
 
 x 
 
 "\ 
 
 'A 
 
 
 --->OzdiDsoe > *2 > & *^~ 
 b C . Q M < o- < z Li ^ Q- 
 
 Sozo<3i<2:52< 
 
 FIG. 34. Typical curves to show variation in monthly maxima. 
 
 The ordinates of curve I, Fig. 34 are proportionate to the 
 sums of the maximum monthly loads on the circuits, averaged 
 over 3 years. It was considered that the indeterminate factor of 
 changes in territory would not affect the curve, inasmuch as they 
 appeared in different circuits at different times of the year and 
 their effect would be reduced to a comparatively small value 
 with an average equal effect on all the ordinates of the curve, in 
 view of the long period of time covered. This assumption is 
 borne out by the comparison of curve I with curve II. This 
 last curve covers a period of only one year and was corrected 
 for changes in territory. The corrections were made by deduct- 
 ing from the totals of each month the amount representing the 
 load in the same territory appearing in two circuits in the same 
 
LIGHTING CIRCUITS 131 
 
 month. This condition of duplication of load is due to the 
 fact that when a change in boundary for the relief of any 
 circuit is made, the load on the section cut off appears in the 
 maxima of both the circuits relieved and relieving. It would 
 be possible naturally to take a well-built-up district including 
 several circuits and establish a curve showing the growth, per 
 year, of that nature of load, leaving the new circuits, with abnor- 
 mal growth, for a special study. This refinement is hardly 
 necessary, as the percentage of growth would not be much 
 diminished. In the method used here there is introduced a small 
 factor of safety. 
 
 In order to make the curve more easily applicable it has been 
 reduced to a table of percentages. Table 13, in which the load 
 for each month in the year appears as a percentage of every other 
 month in the year. It is possible by the use of this table to take 
 the load on any circuit for any given month and predict on that 
 circuit the load for any future date, always considering that the 
 boundaries of the circuit remain unchanged and that there are no 
 particular conditions in that circuit which will cause an abnormal 
 increase or decrease in the load. 
 
 It is not the intention that this curve, or the percentage 
 table developed from it, shall figure as an absolute method for 
 estimating future loads, but merely as a basis on which are to be 
 imposed the particular conditions obtaining for each territory 
 under consideration. The table contains no allowances for 
 abnormal conditions, and the results derived from it may, in 
 some cases, have to be considerably modified by such conditions 
 as changing of circuit boundaries, rapid increase in rate of settle- 
 ment, and others. However, the table is sufficiently accurate 
 to serve as a basis in estimates on the necessity for future 
 work, unless the speed of growth of the city is greatly diminished 
 or increased from the average rate maintained for the years 
 considered. 
 
 The table gives the load on any circuit for any month in terms 
 of any other month, and by correct selection of factors it is 
 possible to predict with reasonable accuracy the future maximum 
 load in amperes on any circuit if a present or previous reading on 
 that circuit is given. For example, the November load on a 
 circuit is 223 amp., and it is desired to estimate its load for 
 January, two years later. Taking the November load as 1 in 
 the table, the load for the following September is .993 of the 
 
132 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 November load; then taking this September load as 1, the load 
 for the following January is 1.23 of this calculated September 
 load. In this case, therefore, the January maximum load of 
 this circuit will be 223 X .993 X 1.23 = 272 amp. This table 
 should be read downward for predicting future load, upward if 
 past loads are to be determined. 
 
 TABLE 13. RATIO OF EACH MONTH'S LOAD TO THE TOTAL LOAD FOR THE 
 REMAINING MONTHS OF THE YEAR 
 
 
 Sept. 
 
 Oct. 
 
 Nov. 
 
 Dec. 
 
 Jan. 
 
 Feb. 
 
 Mar. 
 
 Apr. 
 
 May 
 
 June 
 
 July 
 
 Aug. 
 
 Sept. 
 
 Sept.... 
 
 1.000 
 
 0.892 
 
 0.825 
 
 0.815 
 
 0.812 
 
 0.831 
 
 0.836 
 
 0.862 
 
 0.911 
 
 1.003 
 
 1.098 
 
 0.988 
 
 0.831 
 
 Oct 
 
 1.113 
 
 1.000 
 
 0.913 
 
 0.907 
 
 0.904 
 
 0.925 
 
 0.931 
 
 0.958 
 
 1.013 
 
 1.117 
 
 1.223 
 
 1.100 
 
 0.925 
 
 Nov.... 
 
 1 210 
 
 1.087 
 
 1.000 
 
 0.986 
 
 0.983 
 
 1.006 
 
 1.012 
 
 042 
 
 1 102 
 
 1 216 
 
 1 330 
 
 1 196 
 
 1.006 
 
 Dec. . . . 
 
 1.226 
 
 1.101 
 
 1.012 
 
 1.000 
 
 0.996 
 
 1.018 
 
 1.026 
 
 .057 
 
 1.117 
 
 1.231 
 
 1.348 
 
 1.212 
 
 1.018 
 
 Jan 
 
 1.230 
 
 1.105 
 
 1.015 
 
 1.003 
 
 1.000 
 
 1.022 
 
 1.029 
 
 .061 
 
 1.121 
 
 1.236 
 
 1.352 
 
 1.216 
 
 1.022 
 
 Feb 
 
 1 . 203 
 
 1.081 
 
 0.993 
 
 0.980 
 
 0.977 
 
 1.000 
 
 1.006 
 
 .037 
 
 1.096 
 
 1.208 
 
 1.332 
 
 1.188 
 
 1.000 
 
 Mar. . . . 
 
 1.195 
 
 1.073 
 
 0.986 
 
 0.974 
 
 0.972 
 
 0.993 
 
 1.000 
 
 .030 
 
 1.088 
 
 1.200 
 
 1.312 
 
 1.182 
 
 0.993 
 
 Apr .... 
 
 1.159 
 
 1.041 
 
 0.9.57 
 
 0.945 
 
 0.943 
 
 0.964 
 
 0.970 
 
 .000 
 
 1.056 
 
 1.163 
 
 1.273 
 
 1.146 
 
 0.964 
 
 May . . . 
 
 1.096 
 
 0.986 
 
 0.906 
 
 0.895 
 
 0.892 
 
 0.912 
 
 0.918 
 
 0.946 
 
 1.000 
 
 1.100 
 
 1.204 
 
 1.083 
 
 0.912 
 
 June. . . 
 
 0.995 
 
 0.894 
 
 0.822 
 
 0.812 
 
 0.809 
 
 0.827 
 
 0.833 
 
 0.858 
 
 0.906 
 
 1.000 
 
 1.093 
 
 0.984 
 
 0.827 
 
 July.... 
 
 0.910 
 
 0.818 
 
 0.752 
 
 0.742 
 
 0.739 
 
 0.756 
 
 0.761 
 
 0.784 
 
 0.829 
 
 0.914 
 
 1.000 
 
 0.900 
 
 0.756 
 
 Aug. . . . 
 
 1.010 
 
 0.908 
 
 0.835 
 
 0.825 
 
 0.822 
 
 0.841 
 
 0.846 
 
 0.872 
 
 0.922 
 
 1.014 
 
 1.110 
 
 1.000 
 
 0.841 
 
 Sept.... 
 
 1.203 
 
 1.081 
 
 0.993 
 
 0.980 
 
 0.977 
 
 1.000 
 
 1.006 
 
 1.037 
 
 1.096 
 
 1.208 
 
 1.322 
 
 1.188 
 
 1.000 
 
 Having established a table for predicting loads it is applied 
 to the planning of the necessary new equipment required to take 
 care of the expected loads. It will serve not only as a basis 
 for designing new overhead or underground feeders, but also for 
 substation, transmission and power line requirements. It is 
 evident that proper prediction of load is of great importance in 
 the work of economics of distribution as it affects the growth 
 of the system as a whole. Proper care in compilation of data and 
 considerable study of local conditions will be well repaid in 
 bringing about the possible economies of the system. 
 
 Economical Wire Size for Single -phase Lighting Primaries. 
 In extending an existing system the problem of the proper size 
 for lighting primaries will present itself. Loads to be handled will 
 be estimated as shown above. A method of obtaining the equiva- 
 lent hours for those loads was given in Chap. V. With these two 
 factors known we can proceed, as an example, to determine the 
 economical wire size for single-phase, 4,600-volt primaries. In 
 the preceding chapter on power circuits a complete analysis of a 
 
LIGHTING CIRCUITS 133 
 
 similar problem was given. Here it is proposed briefly to give 
 the equations and the constants that apply particularly to 
 lighting circuits. 
 
 For single-phase lines the cost of conductor will be two-thirds 
 that for three-phase. The cost for pole fixtures will be more 
 than two-thirds on account of the fact that the crossarm cost is 
 included. 
 
 The following equations were obtained in a specific instance. 
 
 TABLE 14. EQUATIONS OF TOTAL ANNUAL COST 
 
 SIZE OF 
 WIRE 
 
 6 5.70 + 37.5C CM + 294,000 ( -, kw \ * tC e 
 
 \rj COS a/ 
 
 4 6.17 + 53.3CW + 185,000 ( kw ) ' tC e 
 
 \J1/ COS (// 
 
 2 8.10+ 84.0C +116,200 ( kw V tC e 
 
 \E cos 6 / 
 
 (Icill \ 2 
 E s J tC e 
 
 00 10.58 + 167.4C CU + 59,000 
 
 000 11.80 + 208. 5C Ctt + 46,800 ( kw }*lC e 
 
 ] 
 
 E cos 
 
 kw 
 \E cos d 
 
 0000 
 
 12 . 39 + 255 . OC CU + 37 , 200 ( * \ * tC e 
 
 Equating the expressions for cost for each adjacent pair of 
 wire sizes the following equations are obtained. 
 
 TABLE 15 
 
 SIZE OF SIZE OB 1 
 WIRE WIRE 
 
 6 to 4 109,000 (0 c^ 0) ^ = 15.8C CM + 0.47 
 
 4 to 2 68,800 ( "^ j tC e = 30 . 7C C + 1 . 93 
 
 2 to 41,600 (., kw y tC e = 52.0C CW + 0.76 
 
 \Jbj cos u / 
 
 to 00 15,600 (_, kw Y tC e = 31.4C CU + 1.72 
 
 \xi COS "/ 
 
 / Jem \ 2 
 
 00 to 000 12,200 ( J tC e = 41.1C eu + 1.22 
 
 \A cos 0/ 
 
 000 to 0000 9,600 (_ kw Y tC e = 46.5C CW + 0.59 
 
 \ii COS J 
 
 to 0000 37,400 (^ kw \ * tC e = 119.0 C cu + 3.53 
 
 \E cos 0/ 
 
134 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Assuming the voltage at 4,600 volts and copper at 30 cts. and 
 20 cts. per pound, the equations become 
 
 TABLE 16. FOR 4,600 VOLTS 
 
 SIZE OF 
 WIRE 
 
 SIZE OF 
 WIRE 
 
 6 
 
 to 
 
 4 
 
 4 
 
 to 
 
 2 
 
 2 
 
 to 
 
 
 
 
 
 to 
 
 00 
 
 00 
 
 to 
 
 000 
 
 000 
 
 to 
 
 0000 
 
 
 
 to 
 
 0000 
 
 tC 1 - 
 
 30-CT. COPPER 
 1,010 
 
 20-cx. COPPER 
 
 705 
 
 ll^e 
 
 tC e = 
 
 tc e = 
 
 fC 
 
 (Aw/cos 0) 2 
 3,420 
 
 (kw/cos 0) 2 
 2,480 
 
 (kw/cos 0) 2 
 8,320 
 
 (kw/cos 0) 2 
 5,670 
 
 (kw/cos 0) 2 
 
 15,100 
 
 (kw/cos 0) 2 
 10,850 
 
 tC e = 
 tC e = 
 
 tc. = 
 
 (kw/cos 0) 2 
 23,500 
 
 (kw/cos 0) 2 
 16,380 
 
 (kw/cos 0) 2 
 32,100 
 
 (kw/cos 0) 2 
 21,800 
 
 (kw/cos 0) 2 
 22,200 
 
 (kw/cos 0) 2 - 
 15,470 
 
 For 2,300 volts the same formulas can be used if the load in 
 kw is multiplied by two. 
 
 The factor tC e still remains to be evaluated in order to plot 
 the curves. In the case of lighting circuits the load factor can 
 be taken as practically a constant and the shape of the load curve 
 can be also assumed to remain uniform. Hence a simple value 
 for tC e can be used instead of keeping it a variable in the equation, 
 as was done with power circuits. This factor, as used in Fig. 35, 
 was assumed as .04. The equivalent hours for lighting were 
 determined previously. The cost of energy combined with this 
 was found to give the above figure as an average. 
 
 The curve is plotted for two values of the cost of copper 
 (20 cts. and 30 cts.). The area in which the load ordinate inter- 
 sects the curve for copper cost indicates the wire size to be 
 used. The economical advantage of the size given is indicated 
 by the distance to the adjacent areas. This method of exhibiting 
 results has the advantage of showing graphically the limits 
 between which one can work and will show readily the effect of 
 an increase or decrease of load. 
 
 Use of Regulators. In connection with the determination of 
 the most economical wire size for a lighting circuit it is sometimes 
 necessary to include a consideration of the cost of a regulator. 
 As with power circuits, it may be desirable to determine whether 
 
LIGHTING CIRCUITS 
 
 135 
 
 it is more economical to use large conductors without a regulator 
 than smaller conductors with a regulator. The annual charges 
 on investment and losses for the regulator must be included with 
 those of the line in this case. With lighting circuits, however, it 
 is usually more necessary to have good regulation than with 
 power circuits to prevent fluctuation in the illumination. Hence 
 it is often preferable to use a regulator in any case, regardless of 
 the exact economy, in order to keep the voltage at or near the 
 center of the load as nearly constant as possible. 
 
 Other Problems on Lighting Circuits. With lighting circuits 
 
 200 
 
 1000 
 
 1200 
 
 400 600 800 
 
 Load in KW. 
 FIG. 35. Economical wire size for single-phase primaries. 
 
 there appear also some of the same problems met with in the 
 preceding chapters, -such as the economical load which can be 
 carried on a line already in place, when an additional circuit 
 should be installed or the wire size increased, etc. As the solution 
 is not essentially different from that for other types of lines, 
 these questions need not be discussed in detail here. 
 
 Increasing Capacity of an Overloaded System. It has been 
 the case in many, if not all, large cities, and also in a number of 
 small communities, that the systems of lighting circuits, which 
 a few years ago were apparently perfectly satisfactory for a long 
 time to come, have become inadequate on account of the large 
 increase in the utilization of electricity and in view of the prospec- 
 
136 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 tive further increases in the future. This is especially true where 
 comparatively low primary voltages are in use, such as 1,100 or 
 2,300 volts, and where single-phase circuits, running from the 
 usual type of main sub-stations, is the practice. The change to 
 be made in such a system is a problem warranting the most 
 careful consideration. It is quite probable that future develop- 
 ments may, in a few years, make any provision, which could now 
 be made in the light of present conditions, again inadequate. 
 This cannot be foreseen as yet however. The best that can be 
 done is to provide for an increase at the same rate as at present 
 for a reasonable number of years, leaving, if post, !e, a good 
 opportunity for further changes at the end of that time, to care 
 for the unforeseen developments. Several me ods of providing 
 for this increase in capacity have been tried in different cities 
 with good success. The principal ones will bt noted i :re with 
 brief notes on the advantages claimed and some factors which 
 affect the making of such changes. 
 
 Probably the simplest method is that of increasing the voltage 
 used. 'The advantage of such a change was mentioned in the 
 chapter on " Power Circuits," i.e., the increase in load carried, 
 with the same per cent voltage drop, will be equal to the square 
 of the increase in voltage. A change in station transformers, 
 station equipment and line transformers is necessary. The 
 economical advantage can be studied by a careful consideration 
 of the annual costs including, of course, the costs of making the 
 change. 
 
 Another method is to change from a single-phase to a three- 
 phase system for the main lighting circuits. The advantage 
 gained is the well-known advantage of three-phase over single- 
 phase. Twice the load can be carried for the same per cent power 
 loss and per cent voltage drop. Balanced against this advantage 
 is the cost of making station changes, of installing the third 
 conductor, etc. 
 
 A combined increase in voltage and a change from single- to 
 three-phase is sometimes used, such as a change, from a 2,300- 
 volt, single-phase to a 4,600-volt, three-phase system. This 
 method has the advantage of obtaining an increase in capacity, 
 due to both changes with the cost of making the change very 
 little greater than for either change alone. 
 
 A method which has been used recently in a number of places 
 with apparent satisfaction is the installation of a four-wire, three- 
 
LIGHTING CIRCUITS 137 
 
 phase system, with grounded neutral. For example, if 2,300- 
 volt, single-phase circuits have been in use, the change is made to 
 three-phase with a voltage of 2,300 volts from each phase to 
 neutral. Some of the advantages claimed are: 
 
 1. The reduction in voltage drop and power loss due to three- 
 phase transmission from substation to feeding points or branches. 
 
 2. The reduction in total number of conductors due to the 
 combination of three former single-phase circuits with six conduc- 
 tors into one three-phase circuit with four conductors. 
 
 3. The utilization of the same branch single-phase circuits and 
 distribution transformers without change. 
 
 4. The load need not be so well balanced as with three-phase, 
 three- wire circui"' 'on account of the use of a neutral conductor. 
 Also the ^hutting' down of one phase will not cut off service on the 
 other pM'ses. 
 
 5. The station changes are somewhat simpler than for a 
 straight increase in voltage or change to a three-phase, three- wire 
 system. 
 
 6. The ability to carry three-phase loads on the same circuit 
 if desired. 
 
 Some disadvantages noted are the increase in voltage from 
 phase wire to ground on single-phase branches, the concentration 
 of fairly large loads on one circuit, and the fact that the increase 
 in voltage is not as great as could be made by a straight doubling 
 of voltage. 
 
 Another method which has also been tried out in a few places 
 is that of extending the secondary transmission system. Instead 
 of attempting to carry the load all out of main substations, on 
 heavy feeder circuits, at primary voltage, small automatic or 
 semi-automatic substations are established at desirable feeding 
 points and these are supplied through high-voltage lines or 
 cables. The primary circuits running from these substations are 
 comparatively short and a corresponding advantage in regulation 
 is accomplished. 
 
 No one of the above methods of increasing the capacity of the 
 lighting circuit system can be recommended as most advan- 
 tageous for all cases. The one best applicable to any system 
 must be chosen by a careful study of present loads, conditions 
 in the substations and on the lines, probable future loads, the cost 
 and practical difficulties of making the change, and adaptability 
 of the new system to still further changes when necessity de- 
 
138 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 mands. This study should be based on as complete a determina- 
 tion as possible of annual costs before and after the change and 
 the comparative annual costs with various alternative changes. 
 
 Importance of Good Service. The importance of good service 
 in connection with lighting circuits should be emphasized. It is 
 here especially that a desire for economy should not lead the 
 engineer to practices which will endanger the quality or con- 
 tinuity of the service rendered. A fluctuation of voltage is easily 
 discernible in the effects on lighting and gives rise to many 
 complaints. An interruption of service, especially if for any 
 considerable length of time, discommodes a great number of 
 customers, sometimes with very serious consequences. Economy 
 is always desirable, but it is false economy to save a few dollars 
 on construction and thereby lose customers. The aim should 
 be to reduce cost to a minimum which is consistent with service, 
 at least as good as that to which the customers have been 
 accustomed. 
 
CHAPTER XIII 
 SECONDARY DISTRIBUTION SINGLE-PHASE 
 
 STUDY OF MOST ECONOMICAL DESIGN FOR SECONDARIES 
 VOLTAGE DROP CONDUCTOR SIZE TRANSFORMER 
 SIZE LENGTH OF SECONDARY 
 
 In working toward the efficient and economical design of the 
 central-station system as a whole no link in the chain connecting 
 the consumer with the coal pile may be overlooked. The ulti- 
 mate purpose of all study in this direction is to enable energy to be 
 delivered to the customer at the least possible cost per unit, 
 while at the same time good service is maintained. To this 
 purpose considerable attention has been paid to generating plant, 
 transmission lines and substations but on the final link before 
 reaching the customer the distribution lines the tendency 
 has been to apply ''rule of thumb" methods and " experience" 
 only to the layouts. When it is considered that, even in a well- 
 designed system, the investment in distribution lines will often 
 be from one-fifth to one-fourth of the total investment on the 
 system, and that the energy losses on these lines will be equal to 
 or somewhat more than one-half of the total loss between the 
 generator and the customer, it may be expected that a study of 
 the economical design of distribution lines will be found of great 
 profit. 
 
 There are several conditions pertaining to the secondary system 
 which make the careful layout of such a system especially impor- 
 tant. The number of transformer installations is so large and the 
 lines spread over so great an area that constant or very frequent 
 inspection is impossible. The load is subject to irregular in- 
 creases. In districts which are newly built up, new services are 
 constantly being added. In old districts, new appliances are 
 being purchased and the load on old services thereby increased. 
 On this account any design must be made to cover a period of 
 years and the increase in load for that period estimated from 
 past experience. On the other hand, care must be taken not to 
 install too much capacity and thereby increase the cost beyond 
 the limits of economy. The problem must be carefully studied 
 to obtain the balance between low cost and good service for any 
 particular case. 
 
 139 
 
140 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 The problems of secondary distribution economics deal chiefly 
 with the wire size and the size and arrangement of transformers. 
 The voltage is usually limited to a small range of values by past 
 practice, transformer standards, lamps, motor and other appli- 
 ance standards, etc. The layout of pole locations, while requiring 
 a considerable amount of engineering skill and experience, is 
 usually dependent on local conditions, the arrangement of lot 
 lines, convenience in reaching services, probable future business, 
 etc. rather than on purely economic considerations. Of course, 
 many local problems arise in the layout and construction of 
 secondary lines, in the solution of which economics should be 
 considered. Changes in type of construction should be looked 
 at from an economic view point as well as from that of the me- 
 chanical design only. 
 
 There are several general types of problems relating to second- 
 aries. They might be classified in general as those of : 
 
 (a) Single-phase secondaries in cities and large towns. 
 (6) Single-phase secondaries in small towns and country. 
 
 (c) Three-phase secondaries on large power installations. 
 
 (d) Three-phase secondaries on small power installations. 
 
 In this chapter the problem of single-phase secondaries will 
 be discussed, especially in reference to well built up districts 
 where the load may be considered to have practically a uniform 
 distribution. 
 
 Secondaries for Uniformly Distributed Load. In attacking 
 such a problem we can often determine from tests and from past 
 experience what the density of the loading is and how it will 
 increase for some years in advance. We are usually limited to 
 certain stock sizes of transformer and of wire, on any system, due 
 to practical considerations of manufacturing and stock keeping. 
 The problem then is to determine the proper combination of wire, 
 transformer and transformer spacing in order to give good condi- 
 tions of operation and also to show the least cost per year for the 
 load densities expected during the period of time under considera- 
 tion. It is clearly understood that no definite rules can be 
 established which will fit all conditions. The variations in the 
 problem are too many. The most that can be done is to furnish 
 means for readily discovering the limitations of any problem 
 and of proceeding within these limitations to the most economical 
 design. 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 141 
 
 The study has been carried forward from three different angles. 
 First, from the theoretical; second, from a semi-practical, that is, 
 by adopting certain standards and studying their behavior; 
 third, from a purely practical, giving the designer data on the 
 costs of various transformers and wire sizes under the conditions 
 ordinarily encountered in practice. 
 
 In all this discussion it has been assumed that the loading is 
 such that it may be considered as uniformly distributed along the 
 line. The unit used is called load density, given in kilowatts 
 per 1,000 ft. The line is assumed to be three- wire secondary 
 spaced 42 in. between outside wires. The cost of right-of-way, 
 poles, crossarms and insulators is not included in any of the com- 
 putations as it is assumed this would be the same under any given 
 condition. Also the difference in length of primary for different 
 transformer spacings is not considered. In actual design under 
 known conditions a correction should be made for this. The 
 loading conditions are taken as those of residence-lighting 
 districts although the same methods could be adapted to any 
 other conditions of loading if its characteristics were known. 
 Transformers are assumed to be in the center of the secondary 
 served, feeding both ways. 
 
 DISCUSSION OF METHODS USED IN DERIVING EQUATIONS 
 AND THEIR APPLICATION 
 
 Theoretical. In the theoretical discussion ideal conditions 
 are assumed which will rarely if ever be met with in practice, 
 but it can be shown by a study of the results how they may be 
 applied to practical conditions. These assumptions are that 
 the line is indefinite in length so that the transformers may be 
 placed at any exactly determined spacing and that the spacing 
 will change with the load; that the transformer is always of a 
 size just equal to the load to be carried, that is, equal to the 
 load density at peak load times the spacing; that the wire may 
 be of any cross sectional area and vary with the load. Such a 
 condition could only be obtained in a case where the load showed 
 only seasonal variations and no yearly increase. However, 
 in practice we usually design for a certain period at the end of 
 which it is assumed the load density will be a certain amount. 
 
 The general method has been to obtain an expression for the 
 annual cost per 1,000 ft. of line and to determine by finding the 
 
142 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 first derivative and setting it equal to zero, the condition under 
 which this annual cost is a minimum. This is the most 
 economical condition. 
 
 Annual Cost of Secondary Distribution. The general equation 
 for the annual cost is first obtained as follows : 
 
 Annual cost per 1,000 ft. of installation = Y 
 
 Y = (Total annual charges on transformers per 1,000 
 ft. of line) + (Total annual charges on 
 line per 1,000 ft. of secondary) = Y T 
 + Y L (55) 
 
 Where Y T = (-) (Purchase price + cost of handling + 
 
 cost of installation -f cost of lightning 
 arresters and equipment) + Cost of core 
 
 and copper loss + cost of inspection j ~ 
 
 g T = Per cent interest + depreciation + taxes on 
 
 transformer. 
 S = Spacing of transformers in feet. 
 
 The core loss is practically a constant quantity for 24 hr. per 
 day throughout the year. The copper loss on the other hand 
 depends on the load. If the characteristic variation of this load 
 from hour to hour, day to day and month to month is known, 
 the average loss per day can be determined in terms of the year's 
 peak load. In this case the peak load is assumed to be just 
 equal to the capacity of the transformer. The cost of energy 
 at the transformer must also be carefully determined. The cost 
 for copper loss will be considerably higher than that for core loss 
 on account of the lower load factor. The sum of all these items 
 makes up the annual cost on a transformer. 
 
 It was found that if the value of the transfoimer annual cost 
 is plotted against the transformer size that the curve for values 
 between and 25 kw. may be approximated by a straight line 
 of the formula YT' = K\ + K 2 T, T being the transformer size 
 and KI and K 2 constants to be determined for any particular 
 
 combination of transformer cost, energy cost, etc. (see Fig. 36). 
 
 i nnn 
 This becomes Y T = -~- (#1 + K 2 T) per 1,000 ft., where S is 
 
 the length of secondary belonging to any one transformer or the 
 distance between transformers where banked. 
 
SECOND AR Y DISTRIB UTION SINGLE-PHASE 
 
 143 
 
 Assuming a transformer size just sufficient to carry the load, 
 then 
 
 T-T S 
 LD 
 
 Where L D = load density in kw. per 1,000 ft. 
 
 mi. TT- 1>000/ T , T ,, -L/jr>O \ /!-r>\ 
 
 Then Y T = _y (*. + ^, j (56) 
 
 50 
 
 I 30 
 
 c 
 
 Transformers assumed to be fully loaded a f 
 yearly peak load. 
 
 Curve of total cost approximates equation 
 Y T '=fr, + K z T 
 
 ) s_ 10 15 ao 
 
 Ske of Transformers in KW=T 
 
 FIG. 36. Annual charges on transformers. 
 
 25 
 
 The annual cost on the line includes interest, depreciation and 
 taxes on the investment cost of the wire in place, including 
 purchase price and cost of installation, also the cost of annual 
 energy loss due to resistance. The copper loss is arrived at by 
 the same method as the copper loss on the transformer, that is, 
 by use of the equivalent average number of hours per day at 
 full load or equivalent hours. 
 
144 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Y L = Investment cost of material per 1,000 ft. of line + instal- 
 lation charges per 1,000 ft. of line H ^ X cost of 
 copper loss in secondary. 
 
 Y L = ^(3 X 1,000 X w X Ccu + C sr ) + ^^l 2 (^) * X j X 
 
 f X2X < X365X Io] ^ 
 Where g L = per cent interest + depreciation + taxes on line, 
 
 w = weight of insulated wire in pounds per foot, 
 C cu = cost of insulated wire per pound, 
 C sr = cost of stringing 1,000 ft. of line, 
 / = total current in secondary at transformer, 
 p = resistivity of wire per mil foot, 
 Cez = cost of copper loss in secondary per kilowatt-hour, 
 t = equivalent hours per day which yearly peak load 
 should continue in order to give an PR loss equal 
 to the total actual PR loss for the year, 
 A = cross-sectional area 'of wire in circular mils, 
 E = voltage between outside wires of secondary, 
 Cos = power factor of load. 
 j = L D S 
 
 E cos 
 and 
 v SL 
 
 
 (3,000 wC cu + S sr ) 
 
 (57) 
 
 The total annual cost per 1,000 ft. of installation is now 
 obtained by adding these two quantities, annual cost of trans- 
 formers and annual cost on line, and the equation obtained as 
 shown below. 
 
 Then Y - (* + ) + ^ (3,000 W C e , + 
 
 Equation 58 gives the total annual cost per 1,000 ft. of instal- 
 lation as a function of the spacing and load density. 
 
 Most Economical Voltage Drop. One of the most important 
 controlling factors in determining the length of a secondary or the 
 spacing of transformers is the maximum allowable voltage drop. 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 145 
 
 It has usually been considered that the most economical condition 
 of operation would be with a voltage drop higher than would be 
 allowable for good service. In our case, 3 per cent drop has been 
 considered the limiting value, as luminosity curves for Mazda 
 lamps show a reduction as high as 18 per cent with 5 per cent 
 voltage drop while 3 per cent shows over 10 per cent reduction. 
 Considering the voltage loss in the service drops, which cannot be 
 figured closely on account of variable conditions and the fact that 
 the load is not absolutely uniformly distributed, 3 per cent is 
 considered the highest value commensurate with good operation. 
 It must be determined, then, if under certain conditions, a smaller 
 voltage drop than this will be more economical. 
 
 In order to obtain the most economical per cent voltage drop it 
 is necessary to obtain F as a function of the per cent voltage drop. 
 
 This is done as follows: 
 
 W = total load on secondary in watts, 
 V = voltage drop on secondary in per cent of de- 
 livered voltage, 
 
 B = constant relation between per cent voltage drop 
 and the per cent power loss. 
 
 A F 9 * 7 ' 
 Then W = L D S = 
 
 Whence 8 
 
 300 BS 
 E 
 
 17.32 
 
 Substituting this value for S in Eq. 58, 
 
 Then F = 1? K l + K 2 L D + ^ (3,000 w C cu + C ar ) + 
 
 17.32 
 
 # 2 AV 
 
 r r T 2nf r 1 
 
 60.83 ," P -&\ 
 , D I AE 2 cos 2 0J 
 
 (17.32) 2 BL, 
 
 = 17,320 ^ JP^p V ~ y * + 0.2028 P * C * L V 
 E \ A B cos 2 6 
 
 + K 2 L D + ^ (3,000 X C cu + C sr ) (59) 
 
 The most economical per cent voltage drop is obtained when 
 the first derivative of Y with respect to V equals 
 dY 
 
 W ' 
 
 = -y 2 = 17,320 K,-^^ V ee -H+ 0.2028 J^j^ =0 
 10 
 
146 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Solving for V ec 
 
 V ec = 1,223 
 
 r cos 2 
 
 (60) 
 
 Equation 60 gives the most economical per cent voltage drop 
 as a function of the load density. 
 
 By assuming values for the constants to fit particular condi- 
 tions this expression for V can be plotted against load density 
 for various standard wire sizes. These curves show that, as 
 load density increases, the most economical voltage drop de- 
 creases and, under the conditions assumed in the curves here 
 plotted, the most economical voltage drop falls below 3 per cent 
 at load densities which are often encountered with such loads 
 (see Fig. 37). 
 
 | . 
 
 Load uniformly distributed- Transformer size equal fv load 
 
 10 15 20 25 
 
 Load Density in Kw. per 1000 feet = 
 
 FIG. 37. Most economical voltage drop in per cent of delivered voltage. 
 
 Most Economical Transformer Spacing. In order to obtain 
 the most economical length of secondary or spacing of trans- 
 formers it is necessary to have Y as a function of S. This is 
 obtained from Eq. 58. 
 
 (60.83 
 
 AE 2 cos 2 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 147 
 
 The most economical spacing is obtained when the first deriv- 
 ative of V with respect to S equals 
 
 Solving for S ec 
 
 A* . 
 
 Equation 61 gives the most economical spacing of transformers 
 as a function of the load density. 
 
 It is necessary to limit the range of application of Eq. 61 to 
 conditions where the voltage drop is less than 3 per cent. A 
 second equation must be developed for 3 per cent drop to apply 
 where the most economical drop would be greater than 3 per 
 cent. Practical considerations limit the drop to that value. 
 
 From above 
 
 S = 
 
 300 BL D 
 
 Using V = 3 per cent 
 
 = Transformer spacing for 3 per cent drop. 
 Then, summarizing, 
 
 H 
 
 which is general for all cases. If now the constants are evaluated 
 these curves may be plotted for various sizes of wire, using, for 
 any particular load density, the equation which shows the 
 shortest spacing. We obtain the set of curves, Fig. 38, giving 
 the transformer spacing which will give, with any wire size, the 
 greatest economy, providing good operating conditions are 
 maintained by having no voltage drop greater than 3 per cent. 
 Most Economical Transformer Size. It is a simple matter 
 with this data at hand to derive the curves showing the most 
 economical transformer size for any load density, providing the 
 transformers are spaced most economically. Since it was 
 assumed in the beginning that the transformer would be just 
 
 S 
 
 large enough to carry the load, T ec = L D r7 ^ (Eq. 64) where S ec 
 
 1,000 
 
148 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 is the value taken from the curves for most economical spacing. 
 This is the most economical size for any load since the annual 
 charges on the investment represented by the transformer is a 
 
 6000 
 
 5000 
 
 Load uniformly distributed- Trans former size jusf equal rofoad. 
 
 15 20 25 30 
 
 Load Density in Kw. per 1000 feet = L O 
 
 40 
 
 45 
 
 FIG. 38. Most economical spacing of transformers (limited by a maximum 
 allowable voltage drop of 3 per cent). 
 
 15 20 25 JO 
 
 Load Densiiy in Kw.per 1000 feel- = L 
 
 FIG. 39. Most economical transformer size (being just equal to the load at the 
 most economical spacing). 
 
 much greater proportion of the total annual charge than the cost 
 of energy losses. Therefore the use of a larger transformer, even 
 though under-loaded, would be more costly (see Fig. 39). 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 
 
 149 
 
 Most Economical Wire Size. It is now possible to attack the 
 problem of the most economical size of wire for any load density. 
 We will assume that it is feasible to use the most economical 
 transformer size at its most economical spacing for any load 
 density, modified by the limiting 3 per cent voltage drop require- 
 ment. Then if we substitute in our original equation (Eq. 58) 
 the expressions for S used in plotting the curves for most eco- 
 nomical spacing and for spacing limited by 3 per cent drop in 
 voltage, we obtain two expressions for the annual cost per 1,000 
 ft. in terms of load density and cross-sectional area of wire (A). 
 It is necessary to introduce two approximations. The weight 
 per foot of wire (w) enters the equation, also the quantity B which 
 
 1.12 
 LOB 
 CO 1.04 
 1.00 
 
 B= Re 
 
 a 
 
 tot 
 
 3; 
 
 Cur 
 
 atlonbet'w 
 nd percen 
 
 jes taken 
 y voltage 
 ve approjtit 
 
 sen percan 
 f power to 
 it 42 "spa 
 
 t- voltage c 
 ss in alim 
 'ing and 
 
 'rop. 
 
 / 
 
 / 
 
 
 
 50 
 
 nates equ 
 
 i+i on 0=A 
 
 ffs* 
 / 
 
 / 
 
 
 
 
 3t 
 
 
 
 
 / 
 
 Y 
 
 
 
 
 ^ 
 
 5 3 S i 
 
 ight of T. &.W.P. Wire in Ibs.per 10 
 
 
 
 / 
 
 / 
 
 
 ^ 
 
 ^ 
 
 -^ 
 
 
 
 A 
 
 
 ^ 
 
 ^ 
 
 Weight o 
 per 100 f1 
 Curve app 
 
 "insulated 
 T.B.W.P.w 
 roximates 
 KWfr 
 
 too (p 
 
 wire in po 
 re. 
 equation 
 
 unds 
 
 0* 
 
 ^ 
 
 -"" 
 
 
 
 
 W= 
 
 er foot) - 
 
 
 WJ 
 
 >0 
 
 ) 10,000 20,000 30,000 40,000 50,000 60,000 10,000 80,000 90,0 
 
 Site of Wire inCircwIar Mils- A 
 
 FIG. 40. 
 
 is the constant relation between per cent voltage drop and per 
 cent power loss for any size of wire. It is found by plotting values 
 of w for standard sizes of wire of the range of sizes which would be 
 used in secondaries that the expression W = K s + K 4 A is a very 
 close approximation, K 3 and K^ being constants (see Fig. 40). 
 Also it is found that the value of B for any size of wire may be 
 approximated very closely by the straight line function B = 
 K$ + K&A, where K 5 and K & are constants (see Fig. 40). These 
 must be derived from the particular values of B which apply 
 to the conditions being studied since these values vary for differ- 
 ent spacings between wires. Substituting these expressions in 
 the equations referred to above we obtain the two general 
 
150 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 expressions for annual cost per 1,000 ft. in terms of wire size for 
 maximum economy of transformer spacing and for 3 per cent 
 voltage drop, Eqs. 65 and 67. These are differentiated with 
 respect to A and the Eqs. 66 and 68 are obtained between 
 the most economical wire size and the load density for most 
 economical spacing and for 3 per cent voltage drop. 
 Substituting the value S ec (Eq. 61) for S in Eq. 58 
 
 + C 8r ) 
 
 P tC e2 
 + 60.83 
 
 If w = K * " (see curve 40). 
 
 The equation for annual costs per 1,000 ft. becomes 
 
 +*** 
 
 Simplifying 
 
 30(^3 + K*A)C CU + C ar (65) 
 
 Equation 65 gives the annual cost per 1,000 ft. of line, using the 
 most economical spacing of transformers. 
 
 The most economical cross-section of wire is obtained when the 
 first derivative of Y with respect to A equals or 
 dY 
 
 = _ V 744 f) !^ ^ . _^ , J/L_ y. on/?- /nr 
 
 3 X L# 2 cos 2 0J D ec h 100 X 
 Solving for A ec 
 
 154 TKJptC^ ( . 
 
 Aec = LD 
 
 Equation 66 gives the most economical cross-section of wire 
 using the most economical spacing of transformers. 
 
 It is necessary to limit the application of Eq. 66 to less than 
 3 per cent voltage drop and develop the equation for most 
 economical wire size with 3 per cent drop. This is done as 
 follows: 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 151 
 
 From Eq. 62, the spacing which will give a 3 per cent voltage 
 drop is, 
 
 E rr 
 
 = W\BT D 
 
 B may be expressed as a function of A as follows: 
 B = K$ + K&A (see curve 40). 
 
 S = f Q 
 
 Substituting the value of S in Eq. 58, the expression for annual 
 costs per 1,000 ft. of line (the spacing being limited for a 3 per 
 cent voltage drop) becomes 
 v 1,000X1 
 
 4 3 per cent 
 
 10 
 
 CU + Csr] 
 
 AE 2 cos 2 6 " 100 (K 6 + KtA)L D 
 Simplifying 
 
 3 per cent ~ 
 
 cu + C sr ] 
 
 + 0.6083 - 2 e r ^ (67) 
 cos 2 e(K$ + 
 
 . Equation 67 gives annual cost per 1,000 ft. of line using a 
 
 spacing which limits the voltage drop to 3 per cent at full load. 
 
 The most economical cross-sectional area is obtained when the 
 
 first derivative of 3 percent with respect to A is equal to 0. 
 
 dY 3 per cent. _ ^ 
 
 dA 
 
 ptC e zL D 
 
 - .6083 
 From which 
 
 T 14 
 
 ' I i-r^'" 
 
 cu (68) 
 
152 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Equation 68 gives the most economical cross-sectional area of 
 wire when the spacing is limited by a 3 per cent voltage drop 
 
 The constants were evaluated and these curves plotted, the 
 3 per cent curve for low load densities and the maximum 
 economy curve for high loading. They furnish a graphic repre- 
 sentation of the most economical size of wire to use under any 
 load density providing ideal conditions obtain in the way of 
 transformer size and spacing. See Fig. 41. 
 
 s,ww 
 
 II innnn 
 
 Tnmsfe 
 econor 
 
 rmer she assumed 
 jical spacing /imifea 
 
 iu&teeyual 
 
 +o load at 
 age drop 
 
 most 
 #4w 
 
 IE 
 
 
 ^f 
 
 
 
 
 
 
 
 
 j^jgSSJ 
 
 
 Cross- Sectional Area in Cl'rcolar Mils 
 
 _1_J_JLJ 
 
 
 
 
 
 ** 
 
 *** 
 
 
 
 
 
 
 
 ^^~ 
 
 
 * 
 
 6 WIRE 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 3 5 10 I 
 
 70 25 30 } 
 
 b 40 4 
 
 Load Density in Kw.per 1000ft = L O 
 
 FIG. 41. Most economical wire size. 
 
 Purpose of Theoretical Curves. At first glance it may appear 
 as if these curves, being obtained on the basis of such theoretical 
 assumptions, could have very little practical value. However, 
 when attacking a practical problem of this nature the data from 
 these curves may be used as the basis upon which to start the 
 calculations of annual costs under operating conditions. If, 
 for example, the present load density and the load density which 
 is to be expected at some certain future time are known, by going 
 to the theoretical curves we may determine (a) whether the volt- 
 age drop is to be limited by the 3 per cent maximum, (6) what 
 would be the most economical conditions of transformer size and 
 spacing for present operation and for operation at that future time, 
 and (c) what standard size of wire will be most economical over 
 the period. The curve for the most economical wire size covers, 
 for each standard size, such a range of load densities that we 
 should be able at once to select our wire size without further 
 computation. Having determined this and knowing what 
 stock sizes of transformers and practical spacings come the 
 nearest to fitting the ideal conditions over the period under 
 
SECOND AR Y DISTRIB UTION SINGLE-PHASE 
 
 153 
 
 consideration, we can then investigate, as will be shown later, 
 the comparative economy of such various methods of installa- 
 tion as could be used in this particular case. In other words, 
 
 100 
 
 SO 
 
 10 
 
 t 
 
 60 
 o 
 
 o 
 o 
 
 c 
 '\n 
 8,40 
 
 iO 
 
 Trans forme resizes and Spacings assumed to be 
 those most theoretically economical limited 
 by 3 % maximum voltage drop. 
 
 Annual cost includes line and transformers. 
 
 5 10 15 ZO 5 30 
 
 Load Density in Kilowatts per 1000 feet 
 
 FIG. 42. Curves showing comparative economy of various wire sizes in sec- 
 ondary installations. 
 
 these theoretical curves give certain limitations on which we may 
 proceed to further more practical investigation. 
 
 Semi-practical. In order to present our results in a little 
 more concrete and practical form and to show the exact com- 
 parative economy between various types of installation, espe- 
 
154 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 cially with respect to the size of wire to be used, a series of curves 
 were developed showing the exact annual cost under various 
 conditions. These are called the semi-practical curves (Figs. 
 42 and 43). 
 
 Annual Cost of Standard Wire Sizes Working under Ideal 
 Conditions. The first condition was assumed to be that in 
 which the most economical size of transformer could be used, 
 spaced the most economically or, where necessary, for 3 per cent 
 maximum voltage drop. A curve was plotted for each of the 
 three standard sizes of wire, No. 6, No. 4 and No. 2, showing the 
 annual cost at various load densities (see Fig. 42). This is, in 
 reality, simply plotting Eqs. 65 and 67 as developed above. 
 
 Annual Costs per 1,000 ft. of Installation for Any Combination 
 of Standard Sizes of Wire and Transformers. As the next step 
 in proceeding from the general problem to the concrete example 
 various combinations of standard sizes of transformers with 
 standard sizes of wire were assumed and curves developed 
 showing the annual cost of each of these combinations at various 
 load densities. The transformer spacing was still assumed to be 
 always the theoretically best spacing for each particular load. 
 This enables us to compare for example the economy of a 10-kw, 
 transformer and No. 4 wire with that of a 15-kw. and No. 6 
 wire at any load density. 
 
 The method of developing these curves has some points of 
 interest although the equations are merely variations of our 
 general equation for annual cost per 1,000 ft. It is seen that for 
 any size of transformer, as the load density increases a certain 
 point is reached where the spacing is no longer governed by the 
 allowable voltage drop but by the size of the transformer itself. 
 Hence each curve will consist of two parts, the lower where the 
 voltage drop governs the spacing, and excess transformer capacity 
 is provided, the upper where the transformer size governs the 
 spacing and the voltage drop is less than the allowable. The 
 total annual cost is made up of five items : 
 
 1. Transformer core loss. 
 
 2. Transformer copper loss. 
 
 3. Copper loss on the line itself. 
 
 4. Fixed charges on the transformer (interest, depreciation, 
 taxes, inspecting, tests, etc.). 
 
 5. Fixed charges on the line. (Interest and depreciation.) 
 Each of these five elements was analyzed as to constants and 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 155 
 
 variables, considering the load density L D as the chief variable, 
 and the transformer and wire sizes constant for any given condi- 
 tion. It was found that the equations took the following form : 
 
 Y = K S L D H + K 9 L D + (K 7 + K lo )L D * + K n 
 when the voltage drop and wire size governs, and 
 
 Y = (K 12 + #13 + K 15 )L D + (Ku + Xu) 
 when transformer size governs. 
 
 The first is an equation of a third degree curve in LD* breaking 
 into a straight line (the second equation) at the critical point 
 where the spacing for 3 per cent drop fully loads the transformer. 
 The equation for each constant was then developed and evaluated 
 for each combination of wire and transformer. The expressions 
 for costs here given differ from those given in the theoretical 
 discussion in that here actual stock sizes of transformers and 
 standard wire sizes are used. The results were then plotted as 
 shown in Fig. 43. The derivation of these curves is a good 
 example of the method of developing a general curve by the 
 use of symbols for all constants and then evaluating these sym- 
 bols to fit a given condition. 
 
 The derivation follows: 
 
 The load possible on a given wire with a 3 per cent drop is 
 given by the formula: 
 
 3AE* AE 2 
 
 W = 
 
 B X 300 X S WOBS 
 
 W = L D S and / 
 
 L D S 
 
 E cos 
 
 77! I ~l 
 
 S = Tft\/T>7- wnen limited by voltage drop (see Eq. 62). 
 
 JLU \ JjLjj) 
 
 L S 
 The total load on the transformer = D 
 
 1 j 
 
 1 000 T 
 S = ' T - when limited by transformer capacity. 
 
 Li D 
 
 The following items enter into the total annual charges per 
 1,000 ft. of installation. 
 
 (a) Transformer Core Loss. This is assumed constant for a 
 given transformer for all loads. 
 
 Core loss = constant. 
 
 1 000 
 Annual charge per 1,000 ft. = ^ X C eJ X 24 X 365 X 
 
 core loss 
 1,000 
 
156 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 C el X 24 X 365 X IPX core loss v 
 
 - X Lj>H (limited by voltage 
 drop) 
 
 or 
 
 ei X 24 X 365 X core loss T 
 
 1 000 T ~~ (l irm ted by transformer size) 
 
 Whence, 
 
 . d X 24 X 365 X 10 X core loss 
 
 _ ei 
 
 X 24 X 365 X core loss 
 
 i,ooor 
 
 C e i = cost of core loss per kilowatt-hour 
 Core loss in watts. 
 
 (b) Transformer Copper Loss. 
 
 1,000 / 365 X t 
 
 / CdX 
 
 _ 1,000 / L,,^ 8 365 X 
 
 s \^ COS 2^^ ^ooo 
 
 L D *R T C e 2 X 365&S 
 J5/ 2 cos 2 6. 
 
 R T C e2 X 365 X t^jj 
 
 cos 2 ! ---- Lz> ^ (limited by voltage drop) 
 
 X 365 X 1,000 X t X T , 
 
 ~E 2 cos 2 (9 ~ D ( limited b y trans- 
 
 former size) 
 
 Where 
 
 ez 3652 \/-^ 
 
 TV- _ \Z> 
 
 cos 2 ^ 
 365,000* X T 
 
 (c) Secondary Copper Loss. 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 157 
 
 Secondary copper loss = ^P 2 (^) XjXgX2X*X 
 
 365 X 
 
 1,000 
 365C e2 S 2 
 
 by volt- 
 
 e , , .. ^ . , . . , 
 
 or = ^-sj 2~Z1 (when limited by transformer size) 
 
 = K L , 
 
 Where 
 
 cos 2 0600 
 
 (d) Fixed Charges on Transformer. Constant for any size of 
 transformer. 
 
 l,000r g T (Transformer cost + lightning arrester +1 
 S LlOO cost of installation) -f inspection costJ 
 
 10 4 [ (Transformer cost + lightning ar- 
 rester + cost of installation) + in- 
 spection cost 
 (when limited by voltage drop) 
 = X 10 L D 
 
 1 1 g T (Transformer cost + lightning arrester +1 
 T 100 cost of installation) + inspection cost I L D 
 (when limited by transformer size) 
 
 = Ki$L D 
 Where 
 
 10 4 r g T (Transformer cost + lightning arrester! 
 
 /J L 1 00 + cost of installation) + inspection cost J 
 
 -^ If g T (Transformer cost + lightning arrester +~] 
 
 Llb rLlOO cost of installation) + inspection cost J 
 (e) Fixed Charges on Wire. 
 
 CU -f C tr ) 
 
158 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 (/) Total Annual Cost. The total yearly cost per 1,000 ft. of 
 line is the summation of these five items. 
 
 too 
 
 90 
 
 80 
 
 ! TO 
 
 Load uniformly distributed./ / 
 Voltage drop most / I / 
 economical maximum 3% ' / 
 
 'jm 
 
 GENERAL FORMULAE 
 
 # f As y- f/f 7 ^; ^ 
 (Where 3% dropgoverns transformer spacing) 
 
 (Where transformer she gove rns 
 trans former spacing) 
 
 5 10 15 20 ZS JO 
 
 Load Density m KW.per IOOOfeetL 
 
 FIG. 43. Curves showing comparative economy of various combinations of 
 secondary installations. 
 
 Hence 
 Y s per cent 
 
 / 
 
 ' -j- 
 
 (when limited by the voltage drop) 
 Y = KizLiD + Ki S L D + KU + Ki$L D 
 - (K 12 + X 13 + ^i 5 )L c + (X 14 + 
 
 limited by transformer size) 
 (see curve Fig. 43). 
 
 u (69) 
 
 (when 
 (70) 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 
 
 159 
 
 Purpose of Semi-practical Curves. These semi-practical curves, 
 although reducing the variable elements, still retain enough of 
 the ideal condition so that they cannot be used as an absolute 
 criterion but merely as a guide. They do show however con- 
 cretely the relative economy of the various standard sizes of wire 
 when used under the most favorable conditions and this may be 
 taken as a guide to their comparative behavior under all condi- 
 tions. The second set of curves also shows concretely the 
 relative economy of the various transformer sizes with any one 
 
 eooo 
 
 5000 
 
 
 I I I I 
 
 J Wire Secondaries = 244/122 Volts of cusfome 
 mast economical transformer spacing 
 limited bu 3% voltage drop \ 
 Power Factor 95% } ' 
 
 L oad uniformly distributed 42 inch spacing 
 between outside wires. 
 
 FIG. 44.- Load curves for secondaries. 
 
 size of wire as well as the relative economy of various sizes of wire 
 with any size of transformer. This comparison of economy is 
 valuable in showing the exact amount which the annual cost of one 
 installation is greater or less than another. It often occurs that 
 where the difference in cost is not great, other advantages are 
 sufficient to more than offset it and lead to the choice of the more 
 costly. The spacing of transformers is here considered to be the 
 maximum allowable throughout, with the transformer carrying 
 its maximum allowable load. This limits the general appli- 
 cation of these curves in practice and hence like the first series 
 they are chiefly useful in establishing limits and as a basis for the 
 design. 
 
160 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Practical. We now come to the development of the curves 
 which the designer may use in testing the economy of any design 
 and thereby choose the most economical from several alter- 
 natives. Here no "most economical" conditions need be 
 assumed. The curves simply represent annual costs as they 
 occur under any condition which may be encountered. 
 
 Load Curves for Secondaries. The first curve is a development 
 from the two theoretical curves, the most economical transformer 
 spacing and most economical transformer size. 
 
 15 20 15 30 
 
 Load Densi-ty in KW.per 1000 ft-. = L, 
 
 
 FIG. 45a. Line cost curves. Annual cost per 1,000 ft. of 3 No. 6 secondaries in- 
 cluding fixed charges and cost of lost energy. 
 
 From Eq. 63 
 
 S ec = 2.02 
 
 By plotting the transformer size against the spacing we obtain 
 for each size of wire a curve showing the most economical spacing 
 or the spacing limited by 3 per cent voltage drop for any total 
 load on the transformer (see Fig. 44). By drawing diagonal 
 lines, one for each load density desired, we can now show for any 
 particular load density, the maximum economical spacing, and 
 the minimum transformer size with that density and spacing. 
 This curve merely simplifies the former two and serves the same 
 purpose, not introducing any new principles. It is evident that 
 any point below the curve will indicate a drop less than the value 
 used on the curve. This curve is of use in determining what 
 
SECOND AR Y DISTRIB UTION SINGLE-PHASE 
 
 161 
 
 alternative designs may be feasible with any load and standard 
 equipment and what changes may be made to care for an increase. 
 Line Cost Curves. The equation for annual charges on the 
 line (Eq. 57) is next developed numerically. 
 
 Y L - (3,000,0, + <?)+ 60. 
 
 35 
 
 40 
 
 5 10 15 20 25 30 
 
 Load Density in KW.per 1000ft = L D 
 
 FIG. 456. Line cost curves. Annual cost per 1,000 ft. of 3 No. 4 secondaries in- 
 cluding fixed charges and cost of lost energy. 
 
 10 
 
 35 
 
 40 
 
 45 
 
 15 20 25 30 
 
 Load Density in KW.per 1000 ft. = L 
 
 PIG. 45c. Line cost curves. Annual cost per 1,000 ft. of 3 No. 2 secondaries in- 
 cluding fixed charges and cost of lost energy. 
 
 All constants were evaluated and a curve plotted for each 
 desired spacing 100-ft. intervals were used showing the 
 annual charges per 1,000 ft. in terms of the load density for each 
 
 standard size of wire (see Fig. 45, a, b, c). 
 11 
 
162 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Transformer Cost Curves. The third set of curves shows the 
 annual cost on the transformer for any loading (see Fig. 46). 
 
 
 \ 
 
 c 
 
 i 
 
 c 
 
 frto/ 
 7/"^r^ 
 
 /7C/W^ 
 
 ?fe/7e 
 
 innut 
 isforn 
 PS //xe 
 'SK/^ 
 
 Jftfel 
 
 ?/C05 
 
 7er3 ^ 
 d cha 
 vses 
 
 
 
 t on\ 
 nder 
 rges 
 
 'anoL 
 varioi. 
 znd c 
 
 s siz 
 jsloa 
 ?sf 
 
 ?s 
 
 *y 
 
 +_ 3500 
 
 1 
 g 3250 
 
 
 
 fc_ 3000 
 o 2150 
 
 \ 
 
 \ 
 
 ^ 
 
 ' 
 
 / 
 
 
 i 
 
 
 ' 
 
 ___Ji 
 
 
 -^ 
 
 / 
 
 
 
 o i |;?U 
 
 I 2500 
 1 
 
 . CO 
 
 \ * 
 
 \ " "0 
 
 \ J) J V 
 
 \ 1- 
 
 \ 
 
 
 ' 
 
 > 
 
 f 
 
 r 
 
 
 
 
 Scale for Reducing "Mai Annual Cost 
 ooi-J o ?3 G; J g ? 
 
 5 S g s S g * 
 
 \ 
 
 U. c 
 
 w '.Z An 
 
 1 S 
 
 * 
 
 ^ 
 
 #^ 
 
 / 
 
 
 
 
 
 |1 
 
 \K C - n 
 
 -U^30 
 
 
 // 
 
 
 
 
 
 
 
 
 ^J 
 
 \ ^ 
 \ 
 
 jr 
 
 
 
 
 
 
 
 
 V 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 J 
 
 10 15 tQ 25 30 35 40 
 
 FIG. 46. Transformer cost curves. 
 
 Annual cost of the transformer = 
 
 QT (cost of transformer + cost of lightning arrester + 
 
 y 
 
 100 
 
 cos ^ f installation) + inspection 
 + cost of core loss 
 4- cost of copper loss 
 
 = ~~; (cost of transformer + cost of lightning arresters. 
 
 1UU 
 
 + cost of installation) 
 + cost of inspection 
 
 + C., X 365 X 24 X 
 
 E 2 cos 2 
 
 -R T X 365 X t X 
 
 1,000 
 
 (71) 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 163 
 
 The equation for each of the standard sizes of transformers 
 2, 5, 10, 15 and 25 was developed and plotted. Since this curve 
 shows total annual cost on a transformer and not cost per 1,000 
 ft. of installation, a scale was added on the diagonal at the left 
 by use of which, with a pair of triangles, the cost per 1,000 ft. 
 may be obtained by the principle of similar triangles. 
 
 8 Yr- 
 
 1,000 Y T 
 
 1,000 
 
 . .YT- iv x -g- 
 
 (Y T > = total annual cost on a transformer) 
 
 (Y T = annual cost of transformers per 1,000 ft. of installation) 
 (see Fig. 47). 
 
 Hence by adding the diagonal scale at the left, Y T may be 
 obtained from Y T > as follows by the method of similar triangles. 
 Draw a line from the value of Y T , obtained 
 on the vertical scale to the value of S used, 
 on the diagonal scale. Draw a parallel line 
 through 1,000 ft. on the diagonal scale and 
 where it intersects the vertical scale will be 
 found the desired value of Y T . 
 
 Cost of Replacing Transformers. Two ^ 
 
 more items of cost are of interest to the Q Q> 
 
 designer and those are arbitrarily fixed by ^ 
 
 local conditions, the cost of changing the 
 size of transformers in the same location 
 and the cost of changing the location of a 
 transformer. These will be practically con- 
 stant for all sizes and may be determined in any case from 
 local cost records. 
 
 Application of Practical Curves. We are now ready to furnish 
 the designer with the information necessary to test the relative 
 economy of any two alternative designs. He first determines 
 his wire size from a study of the theoretical and semi-practical 
 curves. Then, going to the load curves he may determine his 
 alternatives in transformer size and spacing. Assume that 
 conditions point to the alternative of installing 10-kw. trans- 
 formers at a long spacing, changing to 15-kw. at a shorter spacing 
 after a certain period of years, or of installing the 10-kw. at the 
 shorter spacing now and merely changing sizes at that time. 
 
164 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 From our curves the exact cost per 1,000 ft. for each year under 
 consideration may be obtained by using the correct loading and 
 spacing and, at the proper time, adding the cost of either 
 changing location or changing size. The total of the annual 
 costs for each design gives the total cost over the period under 
 consideration and a comparison of the totals shows exactly the 
 relative economy of the designs over the whole period. These 
 curves may be applied to any such problem since they are based 
 not on the assumption of ideal conditions but cover any actual 
 condition which might occur in practice. They can be used 
 in cases where the transformer spacing cannot be uniform on 
 account of local conditions of pole spacing, secondary length, and 
 street and alley arrangement, a very usual case. When there is 
 doubt about the wire size a study of the various possible combina- 
 tions making use of these curves will soon determine the size for 
 greatest economy. Similar curves can also be developed to suit 
 other classes of problems such as concentrated loads, loads with 
 characteristic variations different from those of the residence load 
 used here, as in business districts, power loads, etc. 
 
 Example of Application of Practical Curves. A concrete ex- 
 ample of the use of the above curves may be helpful. Assume 
 that tests on a district show a load density of 8 kw. per 1,000 ft., 
 with No. 4 secondary wire already in place. Our load curves 
 show for that loading and size of wire, 12.8-kw. load at 1,800 ft. 
 spacing to keep within 3 per cent drop in voltage. We wish to 
 provide for an increase in load which we will estimate may go to 
 15 kw. per 1,000 ft. in 6 years. For the present a 10-kw. 
 transformer spaced at 1,400 ft. would care for the load while at 
 15 kw. per 1,000 ft. there would be required a 15-kw. transformer 
 at 1,000 ft. spacing or a 25-kw. at 1,200 ft. In order to avoid 
 too many changes we may space 10-kw. transformers at 1,000 ft., 
 changing in 3 years to 15-kw. or we may put in 15-kw. trans- 
 formers now at 1,500 ft., changing the location in 2 years to 
 1,000 ft. Other alternatives might be considered but these two 
 will serve as an example. . 
 
 For the first alternative, assuming uniform increase in load 
 density of 1% kw. per year. 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 165 
 
 First year. 
 
 Line cost - LD = 8 kw., S = 1,000 $16.00 per 1,000 ft. installation 
 
 Transformer cost 10 kw. at 8-kw. load. ... 32 . 00 per 1 , 000 ft. installation 
 
 For year $ 48. 00 
 
 Second year. 
 
 Line cost - L D = 9% kw., S = 1,000 16. 30 per 1,000ft. 
 
 Transformer cost - 10 kw. at 9% 32 . 70 per 1 ,000 ft 
 
 For year $49. 00 
 
 Third year. 
 
 Line cost - L D = \H kw., S = 1,000 16.80 per 1,000 ft. 
 
 Transformer cost - 10 kw. at 10^ 33 . 70 per 1,000 ft. 
 
 For year $50. 50 
 
 Feurth year. 
 
 Line cost - L D = 12K kw., S = 1,000 17.30 per 1,000 ft. 
 
 Transformer cost - 15 kw. at 12^ 42. 90 per 1,000 ft. 
 
 Cost of changing size (10 kw. to 15 kw. on 
 
 same pole) 7 . 00 per 1,000 ft. 
 
 For year $67 . 20 
 
 Fifth year. 
 
 Line cost - L D = 13% kw., S = 1,000 17.80 per 1,000 ft. 
 
 Transformer cost 15 kw. at 13H 43 . 60 per 1,000 ft. 
 
 For year $61 . 40 
 
 Sixth year. 
 
 Line cost - L D = 15 kw., S = 1,000 18.40 per 1,000 ft. 
 
 Transformer cost - 15 kw. at 15 44 . 40 per 1,000 ft. 
 
 For year $ 62.80 
 
 Total for 6 years $338. 90 per 1000 ft. 
 
 Second alternative 
 First year. 
 
 Line cost - LD = 8 kw., S = 1,500 17.20 per 1,000 ft. 
 
 Transformer cost 15 kw. at 12-kw. load. . 35.50 per 1,000 ft. 
 
 For year $ 52.70 
 
 Second year. 
 
 Line costLz? = 9% kw., S = 1,000 $18.00 per 1,000 ft. 
 
 Transformer cost 15 kw. at 15^ 37 . 20 per 1,000 ft. 
 
 For year $ 55.20 
 
 Third year. 
 
 Line costLo = WH kw., S = 1,000 16.80 per 1,000 ft. 
 
 Transformer cost 15 kw. at 10% 42.30 per 1,000 ft. 
 
 Cost of changing location 20. 50 per 1,000 ft. 
 
 For year $79. 60 
 
 Fourth year same as first alternative (less charge for changing size) $60.20 per 1.000 ft. 
 
 Fifth year same as first alternative $61 .40 per 1,000 ft. 
 
 Sixth year same as first alternative $62 . 80 per 1,000 ft. 
 
 Total for 6 years $371 . 90 per 1,000 ft. 
 
166 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 A saving of $33 per 1,000 ft. of installation, or approximately 
 10 per cent of the total cost over a period of 6 years by the first 
 method thus demonstrating its economy. It is well to note that 
 a large part of the difference in cost is due to the fact that in the 
 first case the size of transformer is changed while in the other 
 the location but not the size is changed. If a further refinement 
 of the comparison is desired, interest may be considered on the 
 yearly items up to the end of the period under consideration. 
 Usually such refinement is not necessary however. 
 
 Conclusions. A study of all these curves gives considerable 
 aid in determining certain standards of design as well as the final 
 particulars for any special problem. There also may be obtained 
 a definite knowledge of the behavior of secondaries under various 
 conditions of loading and operation. It is purposed here to take 
 up each curve in detail, to bring out its characteristics and its 
 possible use. 
 
 Most Economical Voltage Drop. The curves on voltage drop 
 show that the most economical condition varies inversely as the 
 cube root of the wire size also inversely as the cube root of the 
 load density. 
 
 For low load densities the economical drop is high but decreases 
 rapidly, while at high loading the decrease is comparatively slow. 
 It is clearly shown that the most economical voltage drop may be 
 well under that allowable for good service for loads which may be 
 often encountered in practice. Under the conditions and prices 
 assumed in the present case the 3 per cent limit seems to have 
 some justification by economy for ordinary loads. 
 
 Two conditions must be considered which might affect these 
 curves, i.e., the price of materials and cost of energy and the fact 
 that here the transformer was considered just sufficient to carry 
 the load while ordinarily, when designing for an increasing load, 
 the transformers are underloaded. It is seen from the equation 
 of the curves for economical voltage drop that the cost of copper 
 does not affect this discussion. This is due to the fact that the 
 annual cost is based on a unit of 1,000 ft. hence for any given price 
 of copper the annual cost per 1,000 ft. of three-wire line is a 
 constant no matter what the load. The cost of energy enters as 
 an inverse factor to the % power. Also it is a small element in 
 the factor KI, which is also to the % power but in the direct ratio. 
 Hence an increase in the cost of energy would increase both the 
 numerator and the denominator but the latter slightly more than 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 167 
 
 the numerator, hence all the curves would be raised slightly. 
 This effect would be small, however, for ordinary fluctuations. 
 In the case of an increase in the transformer price there would be 
 no change in the curves providing the increase were proportional 
 to the size since the factor K\ would not be affected by such an 
 increase. 
 
 In ordinary design for an increasing load the transformer would 
 be made larger than sufficient to carry the present load to allow 
 for the anticipated increase. A study of the curves for the vari- 
 ous components of the annual charge on a transformer and the 
 equation resulting therefrom, Y T ' = KI + K Z T, will show that 
 if they are developed with the transformer working below its 
 rated loading, and if the percentage of underloading is kept the 
 same for all sizes, the factor K\ will be very little affected, the 
 effect being similar to an increase in price proportional to size. 
 Since this is the only part of this equation that enters into the 
 equation for most economical voltage drop it follows that if a 
 design could be limited to any given percentage of underloading 
 throughout, the curves would still show the most economical 
 condition of voltage drop. 
 
 Most Economical Transformer Spacing. These curves for the 
 most economical transformer spacing (Fig. 38) are derived from 
 the same general expression for annual cost per 1,000 ft. as those 
 for economical voltage drop. Hence, the same results might be 
 expected from the use of either of these sets of curves with the 
 exception that where the most economical spacing would give a 
 maximum voltage drop of more than 3 per cent we have corrected 
 it for that value making it such as to give 3 per cent. 
 
 These curves show for very low load densities, extremely high 
 spacings which are probably much greater than it would be 
 practicable to use since for such a distance and such light loads 
 the effect of the non-uniform loading would be considerable. As 
 is shown by the equation, the spacing for 3 per cent drop varies as 
 the square root of the wire size while for greatest economy it 
 varies as the cube root. It also varies inversely with the load 
 density, to the square root in the first case, the % power in the 
 second. For ordinary loadings encountered in practice and the 
 usual range of wire sizes it is seen that spacing of from 1,000 
 to 2,000 ft. is the most economical and practicable. For the 
 higher loadings the most economical spacing decreases very 
 slowly, remaining over 500 ft. up to high values of L D . 
 
168 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Changed conditions would have a similar effect on these curves 
 as on those for economical voltage drop, in the range of values for 
 which the most economical voltage drop governs the spacing. 
 That is, a rise in the price of energy would lower the curves 
 slightly, the prices of wire and transformers would not have a 
 noticeable effect. For the condition of underloaded transformers, 
 if the proportion of underloading were fixed there would be slight 
 change. In practical designing, however, when considering the 
 amount of this margin in transformer capacity to be used, it 
 might be found relatively more economical to use a transformer 
 size somewhere near the theoretically most economical and obtain 
 the margin in capacity by using a spacing less than the most 
 economical spacing. This may have some advantage over using 
 the most economical spacing, as shown by the curves, and a larger 
 size of transformer than the most economical, when the design 
 is to cover several years and the cost of changing sizes and loca- 
 tions is taken into account. Hence care must be used in placing 
 too much dependence on the strictly theoretical values in 
 practical design. The choice must be tested by the actual year 
 to year costs as shown by the cost curves. 
 
 Most Economical Transformer Size. The curves for the most 
 economical transformer size simply show the size of transformer 
 which will carry the load when the spacing is the most economical 
 or just enough to give 3 per cent voltage drop. They have 
 relatively less practical value excepting that it is from these and 
 the spacing curves combined that the practical load curves are 
 obtained. 
 
 Most Economical Wire Size. The wire size is the first thing to 
 determine in a design and must be chosen to cover long periods of 
 increase in load as replacement of secondary wire is very costly. 
 Hence for secondaries a standard must be chosen for installation 
 in new work which will show good economy through the greatest 
 range of conditions to be encountered. The curves seem to 
 indicate clearly that under the conditions and prices assumed 
 No. 6 wire should be used as a standard in all new work, in dis- 
 tricts where ordinary residence lighting load is expected. The 
 economy curve rises very rapidly at low densities up to about 
 20,000 cir. mil. or nearly No. 7 at about 7 kw. per 1,000 ft. 
 From here the rise is less rapid but still considerable until it 
 crosses the value of No. 6 wire at 15-kw. load density. The 
 load density of 15 is a normal loading. It would not be ad vis- 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 169 
 
 able to use any size less than a No. 7 since the loadings at the 
 smaller values are subject to such rapid increase. Even at No. 7 
 the economical load is fairly small (7 kw. per 1,000 ft.). On 
 the other hand, the curve rises slowly after passing No. 6 and 
 only reaches No. 5 at a loading of about 31 kw. and No. 4 at 
 40 kw. which are high densities and to be encountered only in 
 special cases. It is interesting to note that for all values below a 
 No. 6 wire the economical size is governed by 3 per cent voltage 
 drop while above that the most economical drop governs, the 
 curves crossing at 19-kw. load density. 
 
 Since the curves were figured at a low copper price, in case of an 
 increase in price, the curves would be lowered, i.e., a smaller wire 
 size would be indicated for any particular loading. An increase in 
 energy cost would slightly raise the curve, an increase in trans- 
 former price if proportional to size would not affect the discussion. 
 Since the curves were figured on the assumption that the trans- 
 former spacing was the most economical and the size just equal to 
 the load, a change in these conditions might affect the most 
 economical wire size somewhat. A fixed proportion of under- 
 loading as above shown would have little effect but if different 
 conditions of spacing were assumed, the design should be tested 
 by use of the cost curves for various sizes of wire. 
 
 Semi-practical Curves. The curves which we call semi- 
 practical show a little more concretely the relative economy of 
 installations with the various sizes of wire, in dollars per year 
 per 1,000 ft. They show the actual annual cost for different 
 types. The excessive cost of No. 2 wire for ordinary loads is 
 clearly demonstrated being from $3.50 to $6.00 per year more 
 than No. 4 for loadings up to 15 kw. per 1,000 ft. 
 
 When we go from the ideal size of transformer to practical stock 
 sizes, still assuming the best spacings to be used, there are some 
 conditions in which the relative wire economy is somewhat 
 different. These curves also give an indication of transformer 
 economy. It seems to be quite clearly shown that, under the 
 assumed conditions, the use of large transformers such as 25 kw. 
 is not justified except with very heavy loading, even considering 
 the possible reduction in the number of transformers and hence 
 in the core loss. The increase in the investment cost more than 
 equalizes such saving. 
 
 Practical Curves. The use of the cost curves in designing has 
 already been explained. It may now be readily seen how a study 
 
170 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 of the theoretical and semi-practical curves applied to any 
 problem will give a basis upon which to formulate a design which 
 can then be tested for actual economy by application of the exact 
 costs to be expected. We can determine from this, in case of a 
 new line, the size of wire and then the spacing and size of trans- 
 formers which will care for several years of increase. The exact 
 number of years will be determined by the rate of increase 
 together with the economy of the design, including cost of 
 changing sizes and locations. Or, in case of remodeling an old 
 district, we start with a given size of wire which although perhaps 
 not the most economical, will not justify the cost of change. We 
 can then choose and space our transformers most economically 
 with regard to that size of wire. In a special case where no 
 increase in load is expected the theoretical curves will give 
 exactly the design to use. In other cases where the loading, 
 voltage, etc., are somewhat different, by proper substitution in 
 the theoretical formulae, curves could be plotted which would 
 apply to that particular condition. 
 
 General. The curves given here should not be accepted for 
 general application to design problems. The costs and condi- 
 tions of loading used were of local derivation and apply only to 
 the organization and the time for which they were obtained. 
 Similar curves should be developed for the study of conditions in 
 any other locality and they should be revised from time to time 
 to meet changing conditions. These examples are given here 
 merely to indicate the characteristics of such curves. 
 
 It is evident that no very simple means of correctly designing 
 a distribution system in regard to transformers and secondary wire 
 can be made available due to the many varying conditions encoun- 
 tered and the large number of factors to be taken into account. 
 
 The elements of good judgment and experience are as necessary 
 in the solution of these problems as in any other problem of 
 engineering. The object of this study has been to analyze and 
 evaluate the factors of the design of a single-phase secondary 
 system of the type considered, that lend themselves to such 
 definite analysis and to present the results as aids in the appli- 
 cation of good judgment and experience to the best possible 
 solution of the problem. This problem has been dwelt on in some 
 detail since it is thought that the methods used and the principles 
 brought out are typical for a large number of such types of 
 problems. 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 171 
 
 Secondaries for Scattered Load. A very common question 
 arising in rural districts where the load is scattered, is that of 
 how far it is economical to extend secondary from a present 
 transformer location to reach a new customer rather than to hang 
 a new transformer. It is thought that this problem is of suffi- 
 cient interest to warrant a brief mention here. 
 
 The comparison should be made, of course, on the basis of 
 annual costs. 
 
 The annual cost on the installation in case the secondary is 
 extended is 
 
 Y 8 = annual charges on investment on line of length 
 
 D - D s 
 
 -f- cost of PR loss on extended secondary 
 -f- cost of increase in I 2 R loss in transformer 
 -+- cost of increased copper loss in transformer. (72) 
 Where D = distance from present transformer to new load, 
 
 Ds = distance from present transformer end of present 
 
 secondary, 
 
 (It is assumed that the present transformer is large enough to 
 carry the increased load.) 
 
 The annual cost in case the primary is extended and a new 
 transformer used, is 
 
 Y p = annual charge on investment on line of length 
 
 D - D p 
 
 + I 2 R loss in total length of primary D 
 -f- total annual charges on new transformer including 
 fixed charges and energy losses. (73) 
 
 Where D p = distance from present transformer to end of primary. 
 In the problem considered, it is assumed that the cost of right- 
 of-way, poles, crossarms, etc. would be the same in either case. 
 If the expressions for F 8 and Y p are put in the form of equa- 
 tions in terms of the load, voltage, wire size, cost of energy, etc. 
 and F s equated to Y p , a solution may be obtained for D, the 
 distance at which economy changes from a secondary to a 
 primary extension. 
 
 The following expression was obtained using constants apply- 
 ing to a particular system. 
 
 (TF 2 \ / 1 1 \ W 
 
 457 -) ( --J-) = -26(Z>. - D p ) (.175w> + .0125) - .021l|- 2 
 
 (21.6 ^^ + (W+ 2TFi) Rn - WRn) + K 
 
172 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Where W = new load in watts, 
 
 E = the secondary voltage, 
 EI = the primary voltage, 
 
 w = weight per foot of conductor (primary and sec- 
 ondary assumed same size for small load), 
 Wi = present load on present transformer, 
 R t i = equivalent resistance of present transformer, 
 Rtz = equivalent resistance of new transformer, 
 K = fixed charges and annual cost loss cost on trans- 
 former, 
 
 If both primary and secondary were two No. 6 wire and EI = 
 4,600, E = 112 the equation becomes 
 
 L3821) 
 
 W 2W,\ W 
 
 n ~ " 
 
 oo 
 
 In the limiting case, where D p = D and D s = the cost of 
 primary extension would be a minimum and of secondary exten- 
 sion a maximum. If TI T 2 = 2 kva. and TI is fully loaded 
 
 18,550- 4,180 
 
 (74) 
 
 This indicates the distance to which it is economical to carry 
 any load, TF, on secondary. 
 
 Since, for 3 per cent voltage drop with the above conditions 
 
 D= 417 
 
 TF/1,000 
 Solving simultaneously 
 
 1,000 
 
 This indicates that, for any load over 200 watts, economy 
 need not be considered if the maximum allowable voltage drop 
 is fixed at 3 per cent, since it is economical to carry such a load on 
 secondary to any distance at which the voltage drop is 3 per cent 
 or less. 
 
 For loads less than 200 watts, Eq. 74 may be plotted in a curve 
 if desired. 
 
 The above solution has been given very briefly and all but the 
 chief steps omitted. It indicates the method which can be 
 
SECONDARY DISTRIBUTION SINGLE-PHASE 173 
 
 followed in studying a number of similar problems. The rule 
 established in the particular case shown has been found very 
 useful in laying out rural extensions. 
 
 Space will not permit the elaboration of more problems of 
 single phase secondaries. Among the others often encountered 
 are the following : economy of replacing a small secondary with a 
 larger size instead of hanging an additional transformer (see 
 Chap. X on " Reconstruction Problems"); replacing wire larger 
 than the economical size with a smaller size; the economical size 
 of secondaries and arrangement of transformers for electric 
 range loads; economy of leaving dead wire in place if it is to be 
 utilized later; and many others. 
 
CHAPTER XIV 
 POWER SECONDARIES 
 
 POWER SECONDARY vs. SEPARATE TRANSFORMERS ECONOMICAL 
 SIZE OF 30 SECONDARIES 
 
 The power secondaries on any system are, as a rule, by no 
 means as extensive as the lighting secondaries. In districts where 
 the power load is very heavy, it is very often advisable to supply 
 each customer from separate transformers. In districts where 
 the power load is scattered, the distances are usually too great to 
 carry more than a very few customers on one secondary. There 
 are, of course, many cases where several small or medium sized 
 loads are grouped in a relatively small area, such as a number of 
 small factories or a block of stores. For such conditions, it is 
 usually practicable to use a few large transformer installations with 
 power secondaries. While the proportion of the total system 
 investment represented in power secondaries in relatively small, 
 nevertheless the amount of load handled on any one installa- 
 tion is usually large compared with that on a lighting secondary. 
 Hence, although a consideration of their economy, as a general 
 proposition may not seem so important, in individual cases it may 
 be quite profitable. 
 
 It very rarely happens that the load on a power secondary is 
 so arranged that it may be considered as a distributed load, 
 either uniformly or in accordance with any other definite law. 
 Neither does it often occur that power secondaries can be made 
 continuous and the transformers spaced as desired. Hence, the 
 problem is usually one of concentrated loads of a given size to be 
 transmitted a definite distance. There must be a study of each 
 particular case rather than of the type of installation in general. 
 
 Two kinds of problems are quite commonly met with in connec- 
 tion with power secondaries. It must be decided, for any case of 
 a small or medium sized load, whether it is preferable to carry it 
 on a separate transformer or tap it to a power secondary, provid- 
 ing one is available or can be installed. If it is to be thus handled, 
 the most economical size of conductor should then be determined. 
 
 Regulation and Continuity of Service Important. The deciding 
 factor is quite likely to be some other consideration than that of 
 economy only. It must first be determined whether the load can 
 
 174 
 
POWER SECONDARIES 175 
 
 be carried from the present installation, with a reasonable sized 
 secondary, without exceeding the allowable voltage drop. Often- 
 times, a present transformer installation can be moved to a new 
 location and the secondaries rearranged to accommodate such 
 additional loads. Even though it is found easily possible to carry 
 the load in this way there are some cases where the importance of 
 continuity of service may indicate that a separate installation is 
 advisable. Any trouble on one customer's service or in the 
 transformer would disable the services of all other customers. 
 Hence, any customer whose service is especially subject to inter- 
 ruptions, or any one whom an interruption would seriously 
 discommode should be given separate transformer installations. 
 For the same reason, it is well not to concentrate too many power 
 services on one secondary, especially services to manufacturing 
 plants. 
 
 Economy of Secondary Instead of Separate Transformers. 
 If it has been decided that the load in question can and should 
 be carried on secondary, it still remains to determine whether 
 such an installation is economical. If the annual charges 
 including cost of energy loss on the secondary, necessary to reach 
 the customer, is greater than the annual charges on a separate 
 transformer installation, the latter should be used. If the whole 
 installation is to be rearranged, the annual charges on the whole 
 cost of making the change must be included. For small loads or 
 loads near a present transformer, a secondary installation is, in 
 most cases, more economical. The cases to be questioned are 
 those where heavy secondaries of considerable length are 
 necessary. 
 
 For example, suppose a new load of 25 hp. is to be served, which 
 is 800 ft. from a present installation. In order to get proper 
 regulation, it will be necessary to string No. secondaries the 
 whole distance. The transformer which at present is a 75 kva., 
 well loaded, must be changed to a 100 kva. It will be assumed 
 that the necessary poles will be the same for either a secondary 
 or a primary extension to the customer. To determine the most 
 economical installation, we must compare the annual costs of 
 the two alternatives. These are made up as follows: 
 Secondary Installation. 
 
 1. The annual charges on 800 ft. of three No. secondary, 
 including cost of conductor, crossarms, pins, insulators, and labor 
 cost for installing. 
 
176 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 2. Annual cost of energy losses over the new secondary. 
 
 3. Annual charges on cost to change transformers. 
 
 4. Annual charges on increase in transformer investment from 
 a 75 to a 100 kva. 
 
 5. Annual cost of increase in transformer energy losses. 
 Primary Installation. 
 
 1. Annual cost on 800 ft. of primary, unless the present 
 primary extends past the new load. 
 
 2. Energy losses on the primary. 
 
 3. Annual cost on a new transformer installation of 25 kva., 
 including losses. 
 
 Some such figures as the following may be obtained: 
 
 Secondary installation Primary installation 
 
 No. 1 $ 39.00 No. 1 $ 15.00 
 
 No. 2 32.00 No. 2 2.50 
 
 No. 3 2.00 No. 3 118.00 
 
 No. 4 35.00 
 
 No. 5.. 14.00 $135.50 
 
 Total $122.00 
 
 The secondary installation will have advantage of $13.50 per 
 year which at 15 per cent represents a capitalization of $90.00 
 
 If the primary is already in place, the advantage would be 
 slightly with the separate installation, although the difference 
 in cost is small. 
 
 Economical Size of Secondaries. When it has been decided 
 upon that a load is to be carried on secondary, either after some 
 such consideration as the above, or when, with a separate installa- 
 tion, in order to locate the transformer conveniently, it is necessary 
 to string a few spans of secondary from the transformer to the 
 service, there still remains the question of the most economical 
 size of conductor to be used. Let us assume for this example 
 small 30 power installations for which the load is considered 
 continuous during the time of operation. 
 Equation of Annual Cost 
 
 Annual cost on 30 secondary = r gr(cost of wire + cost of stringing) ~| 
 
 per 100ft. +g(costof poles, fixtures and guys) (75) 
 L-f- cost of energy loss. 
 
 Where g per cent interest, taxes and depreciation 
 = 13 per cent for wire. 
 
 Assume poles, fixtures and guys the same for all cases and 
 neglect cost in making comparison. No doubt with the heavier 
 
POWER SECONDARIES 177 
 
 sizes of wire, additional cost will be found necessary for heavier 
 cross arms, insulators, etc. and additional guying. The study 
 could be made so as to include these, as was done in the case of 
 primary lines in Chap. XI, but in this case this factor will be left 
 out of the computations. In using the resulting curves it can be 
 kept in mind and will have the effect of increasing slightly the 
 economical load on those sizes. 
 
 Annual charge for energy loss (30) = 
 
 PR X 365 X t X ~~ = .365/ 2 /ttC. 
 
 kw 2 
 Annual charge (energy) = 365,000 Dr - ^ tC e 
 
 In the above t = equivalent hours per day, 
 
 If t w = equivalent hours per week which is assumed in this case 
 to be hours per week which motor runs and power is given in 
 horsepower. 
 
 E = 220 volts; cos = .80 
 
 fjp* x (.746) 2 
 Annual charge = 52,000 Dr '-^ t w C e 
 
 EJ COS (/ 
 
 = mZDrt w C e HP* 
 
 D = distance one way in hundreds of feet. 
 r = resistance per wire per 100 ft. 
 Annual charge per 100 ft. (energy) = .933r^C e #P 2 
 The cost of the conductor per 100 ft. was determined from the 
 number of pounds per 100 ft. for different sizes, the cost of ties, 
 and the proportional charge for freight and injury to returned 
 reels. This was put in the form K m C cu + K n , so that the effect 
 of a change in price of copper could be studied. The labor cost 
 was determined for average conditions and reduced to a cost per 
 100 ft. for each size. The proper overhead loading percentage was 
 added to both material and labor. Combining these two charges, 
 the expressions for annual charges on the conductor in place for 
 each size were determined as shown in Table 17, p. 179, column 2. 
 Column 3 of the same table gives the resistance per 100 ft. of 
 the various sizes of conductors and column 4 the corresponding 
 expression for annual charge for energy loss, found by inserting 
 the proper value of r in equation above. The sum of columns 2 
 and 4 would give the total annual charges per 100 ft. for each 
 size of conductor. 
 
 12 
 
178 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 Equations of Equal Cost. If the expression for the total 
 charges on one size of wire is equated to that for any other size, 
 the- resulting equation would represent the conditions for which 
 there is no economical advantage of one size over the other. 
 This is done for each adjacent size in the table and the resulting 
 expressions are given in column 5. 
 
 If we consider two prices of copper, the results for any inter- 
 mediate price can be interpolated. In the table below the 
 expressions for equal annual cost are given with the copper price 
 substituted, using prices of 20 cts. and 30 cts. per pound. 
 
 TABLE 18 
 
 Size of 
 Wire 
 
 Size of 
 Wire 
 
 If C cl 
 
 , = .30 
 
 If C c 
 
 t = -20 
 
 6 to 4 
 
 t w C e HP* 
 
 = 55.99 
 
 t w C e HP* 
 
 = 39.04 
 
 4 to 2 
 
 t w C e HP 2 
 
 = 165.77 
 
 t w C e HP 2 
 
 = 113.47 
 
 2 to 
 
 t w C e HP* 
 
 = 465.00 
 
 t w C e HP* 
 
 = 317.20 
 
 to 00 
 
 t w C e HP* 
 
 = 788.50 
 
 t w C e HP 2 
 
 = 547.50 
 
 00 to 000 
 
 t w C e HP* 
 
 = 1276.80 
 
 t w C e HP* 
 
 = 889.80 
 
 000 to 0000 
 
 t w C e HP* 
 
 = 1784.40 
 
 t w C e HP* 
 
 = 1214.40 
 
 Value of t,Ce. In previous chapters the variation of the cost 
 of energy with the load factor and hence with the equivalent hours 
 has been explained. In "Appendix A" the method of approxi- 
 mating the value of t w C e is carried out. In this case, the loads are 
 considered small, consisting of not more than two or three motors, 
 which would give a flat load curve. The value of equivalent 
 hours per week, t w , would then be practically equal to the aver- 
 age number of hours per week which the motors are run. In. case 
 of loads of a different character, a more accurate determination of 
 t w would be necessary. The values of t w for any load factor, and 
 the corresponding values of t w C e used are given below. 
 
 TABLE 19 
 
 t w LOAD FACTOR t^Ce 
 
 .0 .0 .0 
 
 16.8 .10 .467 
 
 33.6 .20 .619 
 
 50.4 .30 .756 
 
 67.2 .40 .894 
 
 84.0 .50 1.015 
 
 100.8 .60 1.139 
 
 117.6 .70 1.258 
 
 134.4 .80 1.358 
 
 151.2 .90 1.450 
 
 168.0 1.00 1.543 
 
POWER SECONDARIES 
 
 179 
 
 
 . "5 ? 00 ' CJI "~! CC 
 
 !!!; 
 
 1 
 
 "3 
 
 s 
 
 co *o 10 ^ O O O ^ 
 t o 3 "* 
 
 ^ 8 ^ g 
 
 
 "3 
 
 c<i 52 * t^ ,-H Tt< CN 
 
 C0-gofe 
 
 g 
 
 II II II II II II II 
 
 II II II II 
 
 o 
 
 1 
 
 10, iH, 10, N IBs S, iHn IQ. 
 l&5l&5l&3iaHl&3 tqiSb 
 
 III ill 
 
 a 1 
 H 
 
 
 C* O 
 
 
 
 10 (N 
 
 
 I i 8 
 
 8 
 
 tjC 02 
 c8 -^ 
 
 H- a. l^ 1 
 
 1 1 1 
 
 'o > ' 
 
 ^ '^ '^ o 
 
 O O O 
 
 'S a> 
 
 S S S of 
 
 i! ?! o 
 
 
 CD CJD CO ^O 
 
 ^O C^ I s - 
 
 1 I 
 
 O O O O 
 
 i i 
 
 o *- 
 
 
 
 O II 
 
 
 
 
 
 CO CO O5 IM 
 
 o o >o o 
 
 00 O *O 
 
 11 -S ^J 
 
 S 
 
 888 
 
 S Ji 
 
 
 
 9 *~ 
 
 
 
 EH 
 
 
 
 
 05 IH 05 M 
 
 Cq iO ^O 
 
 
 
 1-1 05 00 
 
 
 u, t. 
 
 03 i-J 
 
 l-H 1-H 
 
 a 
 
 t "t + t 
 
 3 
 
 _o 
 
 o 
 d 
 
 O O 2 o 
 
 C5 "5 O 
 .-H 05 O O 
 CO O5 CO -^ 
 
 ^ e 
 
 rH <N l-H 
 
 co 
 
 ic i t^ d id 
 
 >0 i-I CO 
 
 d 
 
 ^H M 
 
 <N CO CO 
 
 8 
 
 II II II II 
 
 II II II 
 
 d 
 
 ^-x ^-^ 
 
 ^ v ^-s 
 
 
 
 ^ IN r^ > 
 
 ^^ ^ -^ 
 
 
 CO i-l i-H ^H 
 
 CO >0 t- O 
 
 eS o 
 
 03 
 
 ^H iH i-I (N 
 
 (N (N CO 
 
 43 
 
 1 1 1 1 
 
 + i | 
 
 O 
 
 ~l 1 ~1 "i 
 
 i \ 
 
 "3 
 
 3333 
 
 222 
 
 a 
 
 O O O O 
 
 rt< iO CO CO 
 
 O O 
 
 T)H CN 05 
 
 d 
 
 s s s? s 
 
 
 
 CO CO CO CO 
 
 CO CO CO 
 
 
 X X X X 
 
 XXX 
 
 
 CO CO CO CO 
 
 CO 00 CO 
 
 Q1 
 
 CO * iM O 
 
 8 i 1 
 
 J 
 
 d d d d 
 
 odd 
 
 CQ 
 
 fc 55 ^ ^ 
 
 K K K 
 
 
 CO CO CO CO 
 
 CO CO CO 
 
180 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 The curve, Fig. 48 is plotted from the above figures, giving 
 the value of t w C e for any value of t w . 
 
 Curves for Economical Conductor Size. The two sets of 
 curves given in Fig. 49 and Fig. 50, can now be plotted from the 
 equations in Table 18, using as coordinates the average number of 
 hours of operation per week and the load in horsepower (the load 
 is given in horsepower for convenience in use with loads consist- 
 ing of one or two motors which are so rated). The curves are 
 similar to those given for " Power Lines" in Chap. XI. The 
 conductor sizes given pertain to the areas between the curves, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^--* 
 
 . ' 
 
 ~^"^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 ^ 
 
 ^"^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 Value 
 value 
 
 i Of 
 
 soff 
 
 ^Q. 
 (ho 
 
 for & 
 urs p 
 
 ifftn 
 rwe 
 
 
 
 
 
 
 
 
 
 
 / 
 
 ^ 
 
 
 
 *) 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Hours per Week 
 
 FIG. 48. Values of t w C e for different values of t w (hours per week). 
 
 since the curves themselves are the loci of points of equal cost. 
 For example (using 20-ct. copper) three No. 2 secondary, is more 
 economical than three No. 4 for a 20-hp. load, if it operates more 
 than 6J^ hr. per week. It is less economical than three No. 
 if the 20-hp. motor operates more than 54 hr. per week. Simi- 
 larly for a load operating 4 hr. per week, three No. is the most 
 economical secondary for loads between 21J^ hp. and 28 J^ hp. 
 If higher-priced copper is being used, the corresponding loads and 
 hours of operation will be larger, as shown by Fig. 50. The 
 distance which the point representing a given load lies from either 
 boundary curve is an indication of the amount of economical 
 advantage of the size of wire shown, over the next adjacent size. 
 Naturally, these curves apply only to concentrated loads of 
 the character assumed. Similar curves may be developed for 
 any type of load desired by the inclusion of the proper factors in 
 making up the equation. For loads which are not concentrated, 
 the secondary may be studied by dividing into sections. Curves 
 for large power loads can be similarly developed. In that case, 
 however, it is probably better to designate the load in kilowatts 
 
POWER SECONDARIES 
 
 181 
 
 120 
 
 110 
 
 S'zes of wire shown belong to AREAS BETWEEN CURVES. 
 Points on curves represent conditions for which the annual 
 cost Is the same for either adjacent wim&ize. 
 
 ASSUMPTIONS: -Load 'consists of one motor ore 
 ~no variation from full load while 
 
 alent 
 
 yrequ/yalenf 
 'leitison. "T 
 
 Cost of construction other than wire stringing 
 -same, for all sizes. 
 
 '0 10 20 30 40 50 GO 10 80 90. 100 110 120 130 140 150 160 110 
 
 Hours per Week 
 
 FIG. 49. Curves showing most economical size of wire for three-phase 
 
 secondaries. 
 
 I I ! I I 1 I 
 
 Sizespfwire shown belong fo AREAS BETWEEN CURVES. 
 
 Points ori curves represent conditions for which the annual cost 
 s the same for either adjacent wire size. 
 
 ASSUMPTIONS'. -Load consists of one motor or equivalent no. variatior 
 from full load while it is on. 
 Cos tof construction other than wire stringing 
 same for all sizes. 
 Copper at 3 cents per Ib 
 
 10 20 30 40 50 60 10 80 90 100 HO 120 130 140 150 ISO HO 180 
 
 Hours per Week 
 
 FIG. 50. Curves showing most economical size of wire for three-phase 
 
 secondaries. 
 
182 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 and use values of t = equivalent hours per day, as was done in the 
 case of power primaries, Chap. XI. 
 
 Problems in Power Secondaries Comparatively Simple. The 
 problems relating to power secondaries are comparatively simple. 
 With good cost data on the installation of such secondary and 
 transformers, load curves for power secondaries as shown in 
 Chap. VII, and curves for economical conductor size, nearly all 
 such problems as are discussed in this chapter may be readily 
 solved. Power secondaries may be, of course, a relatively less 
 important part of the system, than transmission lines, primaries 
 or lighting secondaries. Notwithstanding this fact, however, 
 the study of their economy should not be overlooked. 
 
CHAPTER XV 
 UNDERGROUND LINES 
 
 VOLTAGE CABLE SIZE ROUTE NUMBER OF DUCTS IN A 
 DUCT LINE ARRANGEMENT OF DUCTS AND CABLES 
 
 The major part of this book has been devoted to the problems 
 connected with overhead lines. This was done not because the 
 economic study of underground lines is any less important than 
 that of overhead. The opportunities for effecting economies are 
 just as great, or perhaps greater, on underground lines on account 
 of the greater cost of construction. It is a fact, however, that on 
 most of the central-station systems in this country, the under- 
 ground lines are of small extent as compared with the overhead. 
 Overhead work is preferred where possible, underground being 
 used chiefly in congested down-town areas and for transmission 
 lines where high voltage is not permitted by the municipality, or 
 is considered unsafe. In some other countries, the case is quite 
 different, underground work predominating. In any case, the 
 principles employed in solving the problems of underground lines 
 are the same as those used for overhead which have been ex- 
 plained and exemplified. Only the conditions of the individual 
 problems are different. Some of the affecting factors and special 
 problems relating to underground lines will be taken up in this 
 chapter. 
 
 The necessity rarely arises to make a choice, on the basis of 
 economy, between an underground installation and an overhead. 
 Overhead, as a rule, costs only a fraction of the amount necessary 
 to install underground for the same load. An overhead trans- 
 mission line, for example, can be built for between $3,000 and 
 $4,000 per mile to carry the load which would require an under- 
 ground installation costing $15,000 to $18,000 per mile. Over- 
 head also has the advantage of greater flexibility when changes 
 are necessary to accommodate increases in load. The choice of 
 underground is usually based on considerations of necessity (in 
 congested areas), safety, sightliness, etc. 
 
 The nature of the construction on underground lines makes 
 the necessity for standardization and for the best possible work- 
 manship very evident. The comparative inaccessibility and 
 
 183 
 
184 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 the high cost of installation make repair work very expensive. 
 True economy lies in using all means possible to reduce such 
 repair work to a minimum. 
 
 Problems Similar to those of Overhead Lines. A great many 
 of the problems of underground lines are very similar to those 
 already discussed for overhead. There is, for example, the same 
 general classification into transmission lines, primaries and 
 secondaries, with the special questions arising with each. The 
 economy of any installation should be similarly studied with 
 respect to voltage, voltage drop, conductor size, transformer size 
 and location, most economical route, etc. The solution of these 
 problems with respect to underground lines is limited by their 
 character. Inflexibility and high cost of construction make it 
 necessary that original installations be designed with sufficient 
 thought toward probable future conditions, even at the expense 
 of apparent present economy in some cases. There are some 
 other problems concerning the construction itself, which apply 
 only to underground lines, such as the number and arrangement 
 of ducts in a run, and of cable in the ducts, etc. These will be 
 taken up after some of the questions of voltage, conductor size, 
 route, etc. have been briefly considered. 
 
 Voltage. The determination of the most economical voltage 
 is rarely dependent on the economics of the underground system 
 alone. In transmission, the voltage is limited by the type of 
 cables available. At present, about 33,000 volts is the practi- 
 cable limit, although higher voltages are being considered and will 
 probably be used in the near future. For power lines, one or two 
 standard voltages are usually chosen for the whole system. 
 While the economics of the underground lines should be con- 
 sidered in this choice, it is by no means the only factor. A 
 study of the system as a whole is necessary. 
 
 Conductor Size. A new element is introduced in the study of 
 the most economical cable to carry any load or the most econo- 
 mical load for any cable. The limiting factor in this case is 
 usually the current carrying capacity of the cable. This is 
 governed by the heating of the cable. It depends not only on the 
 size of conductor and on the insulation but on the location of 
 the cable in the duct run, the number of ducts, the condition 
 of the surrounding soil and the shape of the curve of the load 
 on the cable itself and on the other cables in adjacent ducts. 
 
 The question of the heating of cables has been attracting a 
 
UNDERGROUND LINES 
 
 185 
 
 great deal of attention recently and a number of articles on this 
 subject have appeared in the Journal of the A. I. E. E. and other 
 publications. The study is still in the making and, while much 
 valuable data has been collected, there is still much to be done. 
 At present there is no generally accepted standard on such 
 matters. It is safe to say that in practically all cases of ordinary 
 loads, the apparent economical load for a cable, basing the figures 
 on the normal life of that cable, will be considerably more than 
 the safe current carrying capacity, if the cable is to fulfill its 
 
 8.00 
 
 TOO 
 
 6.00 
 
 500 
 
 ~Z 4.00 
 
 3.00 
 
 150 200 
 
 load in Amperes 
 
 FIG. 51. Annual cost per 1,000 feet per ampere transmitted No. 00 underground 
 cable 23,000 volts. 
 
 normal life. For example, Fig. 51 shows the annual cost of 
 transmitting loads over No. 00 underground cable at 23,000 
 volts for various values of equivalent hours. In making the 
 computations, depreciation was figured on the basis of a 20-year 
 life for the cable. It is seen that the most economical loadings 
 on that basis would be between 200 and 300 amp., depending on 
 the load factor (or the equivalent hours). Figure 52 gives the 
 average allowable load for such a cable in various combinations 
 with similar cables, based on the heating of the cable. It shows 
 that the safe loading in duct runs of the usual size is in the 
 neighborhood of only 100 amp., or about 50 per cent of the most 
 economical load, as shown by Fig. 51. It might be thought 
 
186 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 possible to run the cable at a higher average load than that given 
 as the safe load, letting the increased depreciation due to the 
 shortening of the life of the cable be balanced against the in- 
 creased economy of operation. Not enough data is available on 
 the extent to which a cable is damaged by an overload to warrant 
 any figures on this subject. It is quite probable that the damage 
 done would be greater than the economy effected in most cases. 
 In the example represented by Fig. 51, if the load were such as 
 to reduce the life of the cable to 10 years, the apparent most 
 economical load at tC e = .10 would be about 225 amp. How- 
 ever, if a load of 140 amp. or greater could be carried without 
 reducing the life of the cable to less than 10 years, the annual 
 
 5 6 7 8 9 IO II 12 
 No. of Ducts and Cables per Duct Line 
 
 FIG. 52. Curves showing current carrying capacity per cable vs. number of 
 cables per duct line for three conductor paper and lead cables. 
 
 cost per ampere would be less than that at 100 amp. with a 
 20-year life for the cable. If a cable is to be run at a load greater 
 than that considered as perfectly safe from a standpoint of heat- 
 ing, there is the further consideration of increased dielectric 
 losses and increased resistance of conductor. The problem has 
 great possibilities when more data becomes available. 
 
 Figure 51 also shows how the most economical load increases 
 as the load factor, and hence the value of equivalent hours, 
 decreases. Since cable heating has somewhat of a cumulative 
 effect, it is generally possible to carry higher loads at low than at 
 high load factors. The safe load will, therefore, bear somewhat 
 the same relation to the most economical load, in any case. 
 Hence it appears that, in most conditions met with in practice, 
 
UNDERGROUND LINES 187 
 
 the load carried will be governed by the capacity of the cable 
 rather than by economy as usually considered, although, of 
 course, economy is realized by not overloading a cable to the 
 point of injury. 
 
 Economical Route. The most economical route for an under- 
 ground line may be an important consideration in its layout. 
 On account of the high construction cost, the use of the shortest 
 possible route is even more important in this case than with 
 overhead lines. Other things being equal, this might often point 
 to the use of private right-of-way. The relative inaccessibility 
 of such lines on private property, the possibility of interference by 
 future buildings, etc., unless the property is bought outright, 
 and the difficulty of draining manholes in many cases, generally 
 makes it preferable to keep such lines on the public highway. 
 The number and location of manholes necessary may have 
 considerable bearing on the choice of a route. In runs of only 
 a few ducts, the cost of manholes is a large proportion of the total 
 and hence is relatively very important. In runs with a large 
 number of ducts the manhole cost is less important compared 
 with the total cost of the line, but in any case, the route requiring 
 the fewest manholes, other things being equal, has considerable 
 advantage. 
 
 Secondary Distribution. The problem of underground sec- 
 ondary distribution is quite different from that on overhead 
 lines. Transformer locations are limited to the manholes which 
 in turn must be spaced for convenience in installing and main- 
 taining the cable. The secondaries and services must be designed 
 to transmit the required load with ample provision for future 
 contingencies. The changing of secondaries of services or of 
 transformer locations is a considerably more serious matter than 
 on overhead lines. In general the design of an underground 
 secondary system is a problem requiring a great deal of care and 
 good judgment in predicting future loads and providing for 
 them in an economical manner. 
 
 Number of Ducts in a Duct Line. There are a number of 
 problems relating to the construction and arrangement of ducts 
 and cables which are worthy of attention. An example will be 
 given here of a method of determining the most economical 
 number of ducts which should be placed together in one duct 
 run. While it would not always be practicable to limit the 
 number of ducts to the figures which might result from such a 
 
188 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 study, the knowledge thus gained would at least be an aid in 
 making the choice. For simplicity, in this example it will be 
 assumed that the ducts are all filled, the cables are all of the same 
 size, and the loads carried on all the cables are of equal amounts 
 and identical characteristics. Naturally such a condition would 
 rarely be found in practice. However, if the method of attacking 
 the problem is once established, it can be extended to cover other 
 cases of dissimilar cables and loads. 
 
 The basis for determining economical conditions is the annual 
 cost per ampere transmitted over the line as a whole. The 
 equation for annual cost of the line is made up as follows : 
 
 Annual cost per 1,000 ft. = g (cost of 1,000 ft. of duct line, of n 
 
 conduits, in place) 
 + g (average cost per 1,000 ft. for 
 
 manholes complete with sewer 
 
 connections) 
 + g (cost per 1,000 ft. of n cables 
 
 installed) 
 -1- cost of energy losses per year over 
 
 1,000 ft. of n cables. (76) 
 
 Cost of Duct Line. The cost of installing a duct line is made up 
 of: 
 
 1. Cost of excavating, backfilling, paving, etc. which is nearly 
 proportional to the width of the duct line. If the cross-section of 
 the duct is square, or nearly so, the width is practically propor- 
 tional to the square root of the number of conduits, \/n. 
 
 2. Cost of materials used and labor of installing which is 
 practically proportional to the number of conduits, n. 
 
 3. Cost of transportation, tools, water connections, etc. which 
 is practically the same for all sizes. 
 
 The total cost of the duct in place is then represented approxi- 
 mately by the expression KI + K 2 \/n + K 3 n where K\, K z and 
 K 3 are constants to be determined from actual field costs on 
 several jobs. 
 
 Cost of Manholes. The cost of building a standard manhole 
 is usually easy to determine. The cost of drainage and sewer 
 connections will vary, of course, with different conditions en- 
 countered. An average figure must be assumed. By deter- 
 mining the average number of manholes per 1,000 ft. of duct, the 
 
UNDERGROUND LINES 
 
 189 
 
 annual charges per 1,000 ft. for manholes complete may be 
 easily computed. 
 
 Cost of Cables. The cost per foot of cable is readily deter- 
 mined from current prices and average labor costs of the system 
 under consideration. 
 
 Energy Losses. The energy loss depends on the current 
 
 4000 
 
 2 4 6 8 10 fc 14 16 
 Ducts and Cables per Duct Line 
 
 FIG. 53. Curves of total current carrying capacity of duct lines vs. number of 
 ducts for various 3 conductor paper and lead cables. 
 
 carried. If the number of conduits in the duct is large, the 
 allowable current per cable will be smaller than in a duct run of 
 few conduits on account of the increased heating effect. A 
 study of the current carrying capacities of lead covered cables 
 under various conditions is given by Ralph W. Atkinson, in 
 the Journal of the A. I. E. E. for September, 1920. By use of 
 the figures and charts given in that paper, the variation of the 
 allowable current on a cable with the number of conduits in the 
 duct line was derived, for three types of cables, and plotted in 
 Fig. 52. Figure 53 is derived from Fig. 52 and gives the total 
 allowable number of amperes carried on the duct line as a whole, 
 for any number of conduits. 
 
 If I c = the allowable current per cable, 
 
 and r = the resistance per 1,000 ft. of one conductor of one 
 cable. The annual cost of energy losses per 1,000 ft. 
 
 = (3J.V X i X 365 X j^) 
 
190 
 
 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 g. The value of g, i.e., the per cent of interest, taxes, depre- 
 ciation, etc. will vary for the different classes of material. Such 
 figures as the following can be assumed for any problem. 
 
 Interest 7 per cent 
 
 Taxes 2 per cent 
 
 Depreciation, etc., on ducts 2 per cent 
 
 on manholes 5 per cent 
 
 on cables 5 per cent 
 
 Total Annual Cost. After the various individual costs, as 
 indicated above, have been evaluated, they may be combined to 
 give the total annual cost. Figures 54, 55, and 56 show, for 
 
 6 8 10 12 14- 16 
 Ducts and Cables 
 
 FIG. 54. Curves showing annual costs per 1,000 feet vs. number of ducts and 
 cables for No. 00 3-conductor cables (23,000 volts). 
 
 the three different types of cable, the variation of the cost of 
 the different items given above as a function of the number of 
 ducts in the line. The numerical values, of course, apply only to 
 the local conditions for which they were derived. The curve for 
 total annual cost is a summation of the separate charges. If the 
 total annual cost for any given number of ducts and cables is 
 divided by the total allowable current carried, as given by Fig. 
 53, the cost per ampere is obtained. This is given in the upper 
 curve on Figs. 54, 55 and 56. 
 
 Results Shown. The curves indicate clearly the high cost of 
 less than four ducts in a run. For larger lines, the point of econ- 
 
UNDERGROUND LINES 
 
 191 
 
 omy is not so plainly shown. There appears to be comparatively 
 little difference in economy between duct runs of from 4 to 16 
 
 10,000 
 
 4- 6 8 10 \1 
 Ducts and Cables 
 
 FIG. 55. Curves showing annual costs per 1,000 feet vs. number of ducts and 
 cables for 450 cm. 3-conductor cables (4,600 volts). 
 
 ^ 4 G 8 10 12. 14 16 
 Ducts and Cables 
 
 FIG. 56. Curves showing annual costs per 1,000 feet vs. number of ducts and 
 cables for 200 M cm. three conductor cables (2,300 volts). 
 
 ducts. The curves for 2,300- volt cable, Fig. 56, indicates a 
 minimum point at about eight ducts, but the curve is fairly flat. 
 The reason for the small difference in cost shown is due to the fact 
 
o 
 
 o 
 
 o 
 
 192 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 that as the number of ducts increases the allowable current 
 per cable decreases. It would seem to be indicated, in this case, 
 that there would be considerable advantage in not building duct 
 runs to provide too far into the future. If the total cost per 
 ampere of a six- or eight-duct run is no more than that of a 16, 
 it would be better to build the smaller size and when that is filled, 
 build another of the same size, thus saving the investment on 
 empty ducts for a considerable period. Of course, if different 
 cost figures were used or different assumptions as to cable sizes 
 and loadings were made, the points of greatest economy might be 
 more pronounced. Hence these curves must be considered as 
 examples of the method only and not for general application. 
 
 Arrangement of Ducts. Another detail of construction which 
 might offer a profitable field for investigation is the arrangement 
 of ducts in a duct run. It is realized that the practical difficulties 
 in construction may be 'the deciding factor in 
 this matter. However, it must be kept in 
 mind that the carrying capacity of a cable 
 depends a great deal upon its location with 
 respect to other cables. The center cable in 
 a nine duct run arranged in a square will have 
 considerably less capacity than the outside 
 FIG. 57. Arrange- cables, while the corner cables will have more 
 ment of ducts in duct capacity than those between (Fig. 57) . This 
 is due to the relative capability of heat disper- 
 sion of the various positions. It would be an interesting problem 
 to determine how much extra expense would be justifiable in 
 order to use some arrangement which would accomplish better 
 heat dispersion. The problems of increasing the current capacity 
 of a duct line by using the center ducts for some form of a cool- 
 ing system or of flooding the runs are similar. No definite data 
 is at present available on these questions. 
 
 Arrangement of Cables in a Duct Line. The question of the 
 relative heating of cables leads to the problem of the proper 
 arrangement of cables in a given duct run. It rarely happens 
 that the cables are all of the same size or kind or carry similar 
 loads. The diversity between loads may have considerable 
 bearing on the most economical arrangements of the cables. 
 Obviously a cable carrying a heavy load with a high-load factor 
 can operate more efficiently if placed in an outside duct where the 
 heat can be more rapidly dissipated. Similarly, if a cable carry- 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
UNDERGROUND LINES 193 
 
 ing night lighting load only is placed in an interior duct, sur- 
 rounded by ducts containing cables with day power loads only, 
 and the peak loads do not overlap, the lighting cable can be 
 operated at a higher load than would be considered safe with all 
 cables similarly loaded. 
 
 Possible Savings Large. The possibilities for economic 
 investigations on underground lines are large. The loads carried 
 are fairly heavy as a rule, since underground work is usually done 
 in. congested districts. The construction cost is high. Savings 
 of a few per cent mean a relatively large amount of money in the 
 long run. The examples presented above and the other problems 
 mentioned are intended merely as suggestions for study along 
 this line. The available data on heating of cables, allowable 
 current carrying capacity, ageing, etc. is as yet so unreliable that 
 conclusive solutions of any problem are difficult. 
 
CHAPTER XVI 
 THE SYSTEM AS A WHOLE 
 
 There still remain a great many problems, both large and small, 
 which are continually confronting engineers in charge of distribu- 
 tion systems. Some of these are special cases but can be solved 
 by an adaptation of the principles indicated herein. Others 
 apply to other parts of the system than the distribution lines, 
 such as to details in the design of the generating station or the 
 substations. No attempt has been made to cover such questions 
 specifically, although the general methods could be applied. 
 There are still other problems which deal with the system as a 
 whole. While it is not within the province of this work to discuss 
 such questions at length, a few of the most important will be 
 mentioned here. 
 
 A considerable amount has been published at different times 
 and at various places about the proper location of a generating 
 station. Theoretically, it should be located in such a way, with 
 respect to the loads to be carried, that the total annual cost on 
 the completed system will be a minimum. This does not 
 necessarily mean that the best location is at the center of gravity 
 of the loads as is sometimes stated. If we consider two equal 
 loads, one operating 24 hr. per day and the other 1 hr. per day, it 
 is obvious that the generating station should not be half way 
 between but, rather, nearer the load with the high load factor. 
 From a practical standpoint, the location theoretically best is 
 rarely attainable. There are many other important considera- 
 tions such as transportation facilities for fuel, available supply of 
 cooling water, practicable building sites, etc., which govern the 
 choice. It might be said that, with the present-day design of 
 stations and the large capacities being attained, the matter of 
 available water supply is becoming all important. The question 
 of generating energy at the mouth of coal mines and transmitting 
 to distant points has been much talked about recently. Such a 
 practice seems at present, to be limited, however, to exceptional 
 cases where an ample water supply may be had. In such cases 
 
 194 
 
THE SYSTEM AS A WHOLE 195 
 
 economy is realized if the energy can be so generated and trans- 
 mitted to the point of utilization at a less cost per year, including 
 investment charges and cost of losses on transmission lines, than 
 the cost of the same energy generated at the feeding point, includ- 
 ing the cost of transporting the coal by freight. In any case the 
 economy of any location for a generating station is determined 
 only by a complete study of the annual costs of all alternatives. 
 
 The location of a substation is a somewhat similar problem. 
 In this case, however, the limiting factors are usually not so 
 many, and the choice may be made on a more theoretical basis. 
 A careful consideration of the loads to be carried, the lengths of 
 the necessary lines, both underground and overhead, and the 
 energy losses involved, will usually be very profitable. It often 
 occurs that the expenditure of a little more money in the purchase 
 of the best possible site may be repaid several times over in the 
 economies effected on the distribution system. 
 
 The previous chapters in this book have discussed the problems 
 pertaining to each part of the system with very little regard for 
 the relation of that part to the system as a whole. If the study 
 of the economics of the distribution system is to be made com- 
 plete, this inter-relation must be considered. For example, 
 we may determine the most economical drop in voltage on a 
 transmission line and also on a power line. However, in order to 
 properly serve the customer, the voltage regulation must be 
 maintained to a certain standard. The problem is, then, to 
 determine the most economical design and arrangement of trans- 
 mission lines, substation, power lines and regulators to accomplish 
 the desired result. This may possibly be somewhat different 
 than the solutions for the individual parts. Such a study 
 involves a careful determination of annual charges on all types of 
 construction and a complete computation of the cost of energy 
 loss. 
 
 Immediate results should not necessarily be expected from an 
 application of engineering economics to an established dis- 
 tribution system. In some cases the money saved on one or two 
 lines may prove the work well worth while. In other cases, the 
 results will be evident only after some length of time. It is 
 rare that a system can be brought up to an economical standard 
 in a short time. Usually the engineer must be content to 
 improve conditions gradually, by designing new extensions and 
 rebuilding old work, here and there, in accordance with economi- 
 
196 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 cal principles. The final results will be shown in the improved 
 efficiency on the system as a whole. Often, this will take the 
 form of a reduction in overall losses on the system, although this 
 is not necessarily the case. The economical percentage of loss 
 will depend on the relation between construction costs and the 
 cost of energy, which may be very different for different systems. 
 Hence the percentage of energy lost is not a true measure of 
 economy. The ideal condition of maximum efficiency to which 
 it is the purpose of all economic study to contribute, is that 
 conditions in which every customer is provided with a reasonably 
 good quality of service at the least possible cost over the whole 
 system. 
 
CHAPTER XVII 
 INDUSTRIAL PLANT PROBLEMS 
 
 APPLICATION TO INDUSTRIAL PLANT PROBLEMS OF THE 
 
 PRINCIPLES OF ECONOMICS OUTLINED FOR ELECTRICAL 
 
 DISTRIBUTION 
 
 This book has dealt primarily with electrical distribution prob- 
 lems from the point of view of the central station. The purpose, 
 in general, has been to indicate means of studying a distribution 
 system with the view to transmitting electrical energy from the 
 generating plant to the customer at the least possible over-all 
 cost. The second part of the book so far has dealt entirely with 
 the problems encountered in the various parts of such a distribu- 
 tion system. The consumer, however, is interested but indirectly 
 in such problems, in that a reduction in central station costs may 
 lead to lower rates or better service. The consumer of any 
 considerable amount of energy, however, such as a large indus- 
 trial plant, has numerous problems of his own which may be 
 classed as problems of electrical distribution. These may be 
 viewed from the standpoint of the producer or the buyer of 
 electrical energy according to whether he produces his own energy 
 or buys it from the central station. In some cases these problems 
 are very similar to those of the central station. In other cases 
 they may be quite different. In any case, however, where the 
 question of economy enters, the principles explained in Part I 
 will be found fundamental. Annual cost is the basis of compari- 
 son for any alternative propositions. In general, the solution 
 of many of the problems may be simplified by the use of a general 
 equation such as that described -in Chapter VI. The most 
 economical condition will be discovered, only if all items of 
 expense including the cost of energy losses as well as the fixed 
 charges on investment are considered. 
 
 Voltage Regulation. The problem confronting the electrical 
 engineer in an industrial plant is, essentially, the same as that of 
 the distribution engineer as stated in Chaper I, i.e., to realize 
 the greatest economy possible consistent with good service. In 
 this case good service depends upon two factors. In the first 
 
 197 
 
198 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 place, the central station must furnish reasonably good regulation 
 at the consumer's service. This is usually more or less regulated 
 by contract but may depend somewhat on the character of load 
 imposed by the consumer as to power factor, fluctuation, etc. 
 A customers load may be of such a nature that it is practically 
 impossible to give good regulation at his service and other services 
 on the same line may be similarly disturbed. In such a case it is 
 usually necessary for the customer to change his equipment or 
 method of operation in some way so as to remedy the difficulty. 
 The second factor is the interior distribution of the plant itself. 
 Assuming that good regulation is furnished by the central station, 
 it is essential that the arrangement of the plant electrical dis- 
 tribution be such that good voltage conditions are maintained at 
 the points of utilization. By plant distribution is meant all 
 parts of the electrical circuit including that of apparatus of 
 utilization. All such problems may at first be more electrical 
 than economic. However, good service being accomplished, it is 
 then essential to investigate the matter of economy. 
 
 Problems of Power Distribution. The problems pertaining 
 to the actual wiring in an industrial plant will be very similar to 
 those of the larger distribution system which has been heretofore 
 described. For small plants it will be simply a case of secondary 
 distribution. The sizes of conductor can be readily determined 
 by a consideration of the annual charges on the cost of the con- 
 ductor in place versus the cost of losses. In the case of the 
 consumer, the unit cost of lost energy is more easily determined 
 than for the central station as it appears very decidedly in his 
 monthly bills. For larger plants, with several separate buildings, 
 it may be advisable to distribute partly at primary voltage. 
 For still larger ones, generating their own power, the problem 
 of high voltage transmission may enter. In any such case the 
 problems of the central station are more nearly approached and 
 the examples cited in the preceding chapters will apply. 
 
 In many plants the question arises as to the comparative 
 advantages of electrical distribution of power as compared with 
 mechanical, i.e., individual motors on every machine instead of 
 large motors with mechanical transmission to a group of machines. 
 Of course, there are many factors in such a problem, including 
 the type of machines used and method of operation. In general 
 a study of the total annual cost of operation including fixed 
 charges, electrical and mechanical losses, and a consideration 
 
INDUSTRIAL PLANT PROBLEMS 199 
 
 of any possible difference in production, labor costs, maintenance, 
 etc. will be of great value in determining the proper choice. Such 
 a study will be based on the principles explained in Part I. 
 
 Problems of Equipment. The choice of proper equipment may 
 have considerable effect on the economy of plant operation. For 
 customers buying energy at primary voltage there is first the 
 selection of proper primary switch house equipment. Trans- 
 formers must be selected with the view of probable increase in 
 load. However, if too large units are installed, not only is there 
 a waste of investment but the additional core losses and the effect 
 on the power factor may be important. It will sometimes be 
 found economical in such a case to use an open delta installation 
 until the load justifies the installation of the third unit. Where 
 reserve transformers are required in case of trouble where a 
 shutdown would be serious, the choice of the size of units will 
 be influenced by that fact. In any such case as well as in the 
 selection of other equipment such as switches, etc., a study of the 
 total annual cost is essential. 
 
 A very common fault in the selection of apparatus of utilization 
 is that of overmotoring. Where careful engineering has not been 
 done, the tendency often is to install larger motors than are neces- 
 sary for the use required with the idea that the reserve power ob- 
 tained is advantageous. The investment charges in such a case 
 are larger than necessary. Also, any considerable underloading 
 of induction motors brings down the power factor and this may 
 have considerable effect on the regulation. In case poor power 
 factor is penalized in the rates it is important to maintain as 
 good a value as possible. A careful study of the diversity factor 
 between machines on the same motor is important in its 
 proper size. In some cases synchronous motors operated to im- 
 prove power factor or static condensers will be found advisable. 
 Similar considerations will be encountered with other types of 
 equipment. Old inefficient equipment may often be replaced 
 to advantage, although it must be kept in mind that beyond a 
 certain point efficiency is not always economy. In all cases, the 
 annual cost must always be considered as an important factor 
 along with probably load increase, regulation, reserve power, etc. 
 
 Problems of Operation. In this country the establishment of 
 rates modified by power factor considerations is becoming more 
 and more important and is at present receiving considerable 
 attention from the central stations. Poor power factor not only 
 
200 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 occasions additional power losses all the way back to the genera- 
 tors but it increases the difficulties of maintaining good regula- 
 tion. It would therefore seem just that a customer with a low 
 power factor should pay more per kilowatt hour than one with a 
 high power factor. With this in mind it is essential that every 
 user obtain as good a power factor as possible from his plant. 
 This may be done by a selection of equipment of proper size and 
 type for the load as mentioned above and also by the use of 
 condensers, either static or synchronous. The expenditure 
 justified in order to increase power factor can be determined by 
 a study of the reduction in the power bills so effected. In this 
 way it will be discovered whether it will be a paying proposition 
 to establish a power factor of say 95 per cent or of 80 per cent, etc. 
 
 The plant load factor will usually be an important considera- 
 tion in a study to increase economy. At present most power 
 rates are based on some form of demand charge and some form 
 of kilowatt hour charge. It is obviously advantageous to reduce 
 the demand charge if possible, spreading the energy used at the 
 peak through the rest of the day. This is especially true where 
 the rates are of such a form as, the demand load for "a" hours at 
 "b" cents per unit and the reminder at "c" cents. The demand 
 may often be reduced by a careful study of the time of starting 
 up motors and of overlapping operation of various equipment. 
 In some cases the reduction in the annual bills will warrant the 
 purchase of new equipment to reduce the demand. 
 
 Other Problems. It is not within the province of this book to 
 attempt to discuss in any detail all problems in electrical distribu- 
 tion which occur in an industrial plant. It has been attempted 
 above to outline some of the questions encountered and give a 
 general idea of the field of application of economic principles to 
 such problems. There are many others which are less strictly 
 electrical or not at all electrical which will bear a similar study. 
 The questions of proper illumination as affecting production, 
 safety devices, labor saving devices, etc. are all more or less 
 economic problems. The main point must be always kept in 
 mind that, once good service is accomplished, the greatest advan- 
 tage lies with any installation for which the total annual cost, 
 everything considered, is less than any other. 
 
APPENDIX A 
 METHOD OF APPROXIMATING ENERGY COST 
 
 It very often happens that it is desirable to study the economics 
 of different types of circuits before it has been possible to under- 
 take any detailed determination of energy cost as outlined in 
 Chap. IV. It has been customary, with most engineers doing 
 such work, to assume in such a case, an approximate average 
 figure such as .01 per kilowatt-hour or some other figure which is 
 known to be somewhere near the average for the whole output 
 of the system. A little closer approximation may be made, 
 keeping the principles set forth in Chap. IV in mind, by determin- 
 ing roughly the variation in cost with the load factor for the class 
 of load being considered. While not very accurate, such figures 
 are at least better than a rough guess of a cost to cover all cases. 
 
 An example of such a determination will be worked out here to 
 show how it is possible to handle such a problem. The general 
 methods used may be adopted to other similar cases. 
 
 The loads considered are power loads in the suburban dis- 
 tricts, i.e., such that considerable transmission-line cost is 
 involved. As was explained in Chap. IV, the demand cost for 
 energy losses is very nearly proportional to the amount of load 
 at time of station peak. It was assumed that the average 
 demand cost at the generating-station switchboard equals $15.76 
 per kilowatt and the average kilowatt-hour cost = $ .0036 per 
 kilowatt-hour. (These costs are usually available and fairly 
 correct in most companies.) The demand cost per kilowatt-hour 
 will be inversely proportional to the load factor, considering load 
 at generating station only. Reducing the demand cost to a cost 
 per kilowatt-hour: 
 
 At 100 per cent load factor, demand charge = 
 
 .0018 per kilowatt-hour. 
 For any other load factor, demand charge = .0018 
 
 Load factor (ex- 
 
 pressed as a fraction not as per cent.) 
 
 In order to determine the energy cost at any point on the 
 
 201 
 
202 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 system beyond the generating-station, the costs given above 
 must be increased by the cost of transmission and distribution. 
 Also diversity factors must be taken into account. 
 
 A study of the property classification of the system indicated 
 a division approximately as follows: 
 
 PER CENT 
 
 Generating plant 40 . 
 
 Underground conduit, etc 6.5 
 
 Poles and fixtures 7.0 
 
 Transmission lines 24 . 5 
 
 Distribution 19.3 
 
 Transformers 2.7 
 
 We may then assume, roughly, that the demand charge exclusive 
 
 of the generating plant is 60 per cent and of the generating plant 
 
 40 per cent of the total. For a load of 100 per cent load factor 
 
 the total demand charge would then be simply = 
 
 generating-station demand charge 
 
 .40 
 
 = 2.5 generating-station demand for 100 per cent load factor. 
 
 For any other load factor, the diversity between loads will 
 affect the cost. A kilowatt load at the customer will not mean 
 a kilowatt at the generator unless the peaks happen to coincide. 
 The diversity factors were assumed in this case to be those given 
 in the " Standard Handbook" for general power loads as follows: 
 
 PER CENT 
 
 Between transformers 74 
 
 Between power lines 87 
 
 Between substations 91 
 
 It is assumed that these factors hold good for a load factor 
 of 25 per cent. Then, 1 kw. at the customer becomes 
 
 1 X 1 =1 kw. at the transformer. 
 . 74 X 1 = . 74 kw. on the power line. 
 .87 X .74 = .61 kw. at the substation. 
 .91 X .61 = . 586 kw. at the generating-station. 
 
 These ratios combined with the percentages of the total prop- 
 erty values as given above, give the approximate share of each 
 kilowatt at the customer in the total demand charge for the 
 system. For example, if the transformer investment is 2.7 per 
 cent of the total, and 1 kw. at the customer represents 1 kw. at 
 
APPENDIX 
 
 203 
 
 the transformer, that kilowatt should take 2.7 per cent of the 
 total demand charge for its share in the transformer investment. 
 If it becomes only .586 kw. at the generating station and the 
 generating station represents 40 per cent of the total investment, 
 the kilowatt in question should be charged with 23.5 per cent of 
 the total demand charge for its share in the station. In order to 
 make property and diversity factor classifications coincide and 
 as an approximation, it was assumed that the investment in 
 distribution and poles and fixtures could be taken to represent 
 power lines, and that the investment in transmission and 
 underground could be applied to substations. 
 
 Then for 1 kw. at the customer 
 
 
 Per cent total inve 
 
 stment 
 
 Share in total 
 demand cost 
 
 On transformer 
 
 1 kw. X 2 7 
 
 
 2 7 
 
 On power line 
 
 .74kw. X (19.3 + 
 
 7.0) 
 
 19.5 
 
 At substation 
 At generating station 
 
 .64 kw. X ( 6 . 5 + 
 . 586 kw. X 40 
 
 24.5) 
 
 19.8 
 23.5 
 
 Total 
 
 
 
 65 5 
 
 
 
 
 
 Then if total demand for any load factor 
 
 _ gen.-sta. demand for that load factor 
 
 .40 
 The demand charge for 1 kw. at the customer with the load factor assumed 
 
 (25 per cent) 
 
 gen. -station demand at 25 per cent load factor X 0.655 
 
 .40 
 = 1.63 X gen. -station demand at 25 per cent load factor 
 
 From the two values thus obtained for 100 per cent and 25 per 
 cent load factors and the point for load factor equals zero, the 
 curve shown on Fig. 58 was plotted showing by what amount the 
 generating station demand cost at any load factor should be 
 multiplied to give the demand charge at the customers in question. 
 Similarly the kilowatt-hour charge must be increased to 
 account for losses in transmission and distribution. It was 
 assumed that the total loss between generating station and 
 customer would be not far from 30 per cent and, for the loads 
 considered, it was assumed that it would be accurate enough to 
 increase the generating-station, kilowatt-hour charge by 30 per 
 
204 ECONOMICS OF ELECTRICAL DISTRIBUTION 
 
 cent to give the charge at the customer. The average kilowatt- 
 hour charge would then = .036 X 1.36 = .0047. 
 
 The following table may then be derived. The first column 
 gives the average number of hours of operation per week (t w ) at 
 
 ~ent Generating Station Demand Charge /io 
 o t- - r^ * 
 
 jt O W O u 
 
 
 
 
 
 
 
 
 ^ 
 
 -^ 
 
 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 * 
 
 
 
 
 / 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 '0 O.I 02 0.5 OA 0.5 0.6 0.7 0.8 0-9 1.0 I.I 12 
 Load Factor 
 
 FIG. 58. Three-phase secondary. Factor by which demand charge at gene- 
 rating station for any load factor must be multiplied to obtain demand charge 
 at load. 
 
 full load corresponding to any load factor. The last column 
 gives the corresponding values of t w C e . These are used as shown 
 in Chaps. XI and XIV. The values of t w C e are plotted in Fig. 59 
 
 10 ZO 30 40 50 60 70 80 90 100 110 120 130 140 ISO 160 170 180 
 Hours per Week 
 
 FIG. 59. Values of t w C e for different values of t w (hours per week) . 
 
 The cost of energy obtained in this way is very approximate 
 but may serve the purpose until some more accurate determina- 
 tion can be made. The figures here given must be considered 
 as examples only and not as representative of present-day costs 
 on any system. 
 
APPENDIX 
 TABLE 20 
 
 205 
 
 *. 
 
 Load 
 factor 
 
 Station 
 demand 
 
 Multi- 
 plier 
 
 Demand 
 at load 
 
 Total at 
 load 
 
 M7. 
 
 .0 
 
 .00 
 
 .0 
 
 1.00 
 
 .00 
 
 .0 
 
 .0 
 
 16.8 
 
 .10 
 
 .018 
 
 1.28 
 
 .0231 
 
 .0278 
 
 .467 
 
 33.6 
 
 .20 
 
 .009 
 
 1.52 
 
 .0137 
 
 .0184 
 
 .619 
 
 50.4 
 
 .30 
 
 .006 
 
 1.72 
 
 .0103 
 
 .0150 
 
 .756 
 
 67.2 
 
 .40 
 
 .0045 
 
 1.90 
 
 .0086 
 
 .0133 
 
 .894 
 
 84.0 
 
 .50 
 
 .0036 
 
 2.05 
 
 .0074 
 
 .0121 
 
 1.015 
 
 100.8 
 
 .60 
 
 .003 
 
 2.20 
 
 .0066 
 
 .0113 
 
 1.139 
 
 117.6 
 
 .70 
 
 .0026 
 
 2.32 
 
 .0060 
 
 .0107 
 
 1.258 
 
 134.4 
 
 .80 
 
 .00225 
 
 2.41 
 
 .0054 
 
 .0101 
 
 1.358 
 
 151.2 
 
 .90 
 
 .002 
 
 2.47 
 
 .0049 
 
 .0096 
 
 1.450 
 
 168.0 
 
 1.00 
 
 .018 
 
 2.50 
 
 .0045 
 
 .0092 
 
 1.543 
 
INDEX 
 
 Atkinson, Ralph W., 189 
 B 
 
 "B", 59 
 
 Backbone transmission lines, 71 
 
 Balance factor, 38 
 
 Cost, records, 11, 14 
 
 unit, 12 
 
 variation, effect of, 11 
 Costs, 11 
 
 classification of, 23 
 
 method of making classification, 
 25 
 
 I) 
 
 Cables, arrangement of in duct line, 
 
 192 
 
 cost of, 189 
 Conductor economy, transmission 
 
 line, 83 
 Conductors, cost of, 75 
 
 economical size for power 
 
 secondaries, 159 
 size, lighting circuits, 132 
 power circuits, 111, 121 
 transmission line, 83 
 underground, 184 
 Copper loss, secondary, 156 
 
 transformer, 156 
 Core loss, transformer, 155 
 Cost, annual, 12, 18 
 
 annual on physical property, 18 
 for secondary distribution, 
 
 142 
 
 total for secondary line, 158 
 consumers, 24 
 demand, 24 
 
 apportioning of, 27 
 at any point, 29 
 data necessary for, 31 
 of energy, 22 
 
 variations in, 32 
 first, 12 
 
 kilowatt hour, 27 
 labor, formulas for, 14 
 of lost energy, 32, 75, 118 
 output, 24 
 
 apportioning of, 27 
 
 Data, empirical, 7 
 
 exact, 7 
 
 Demand charge, 204 
 costs, 24 
 
 apportioning of, 27 
 at any point, 29 
 data necessary for, 31 
 factor, 38 
 
 Density, load, 140, 141 
 Depreciation, 19, 20 
 Distribution of load over several 
 
 lines, 124 
 of power in industrial plants, 
 
 198 
 
 secondary lines single phase, 139 
 lines underground, 
 
 187 
 
 Diversity factor, 38 
 Doherty, H. L., 23 
 Duct line, cost of, 188 
 Ducts, arrangement of, 192 
 
 economical number in duct 
 run, 187 
 
 E 
 
 Economics, application to distribu- 
 tion problems, 5 
 Eisenmenger, H. E., 26, 29 
 Energy charge, 204 
 cost, 22 
 
 method of approximating, 201 
 variations in, 32 
 loss, cost of underground, 189 
 Equation, general, 48, 50 
 
 207 
 
208 
 
 INDEX 
 
 Equipment, industrial plant, 199 
 
 Equivalent hours, 39 
 
 corrections for, 44 
 for lighting circuits, 40, 45 
 for power circuits, 120 
 for various loads, 45 
 relation between load factor 
 and, 45 
 
 Financial conditions, 9 
 
 G 
 
 "g", 49 
 
 Generating station, location of, 194 
 
 H 
 
 Hopkinson, Dr. John, 23 
 
 I 
 
 Increase in capacity, 135 
 Industrial plant problems, 197 
 Insulators, cost of, 75 
 Insurance, 19 
 Interest, 18 
 
 Kelvin's law, 48 
 
 Kilowatthour, total charge per, 32 
 cost, 27 
 
 Lighting circuits, 128 
 
 Lightning arresters, cost of, 75, 79 
 
 Line layouts, points to consider in, 9 
 
 secondary, annual cost, 143 
 Lines, underground, 183 
 Load, characteristics, 37 
 
 curves, for secondaries, 160 
 
 density, 140, 141 
 
 Load, most economical for transmis- 
 sion lines, 84 
 prediction on lighting circuits, 
 
 129 
 
 Loading, 13 
 Lost energy, cost of, 32 
 
 M 
 
 Maintenance, 19 
 Manholes, cost of, 188 
 
 O 
 
 Obsolescence, 20 
 
 Operation problems, industrial 
 
 plant, 199 
 Output cost, 24 
 
 apportioning of, 27 
 Overhead expense, 13 
 
 Parts of system, relation to whole, 9 
 Patrolling, cost of, 76 
 Power circuits, 109 
 
 two in place of one, 121 
 voltage, 110 
 distribution, industrial plant, 
 
 198 
 factor, 37 
 
 improvement, 111 
 loss and voltage drop, 55 
 secondaries, 174 
 
 Predicting load on residence lighting 
 circuits, 129 
 
 R 
 
 Rate making, cost of energy for, 22 
 Reconstructed line, annual cost, 98 
 total investment represented, 
 
 96 
 utilization of old materials, 
 
 99, 104 
 division between parallel lines, Reconstruction problems, 96 
 
 126 Regulation, improvement by two 
 
 factor, 39 circuits, 124 
 
 increase in, 8 industrial plant voltage, 197 
 
INDEX 
 
 209 
 
 Regulators with lighting circuits, 
 
 use of, 134 
 
 Repair, maintenance and, 19 
 Replacement of serviceable line, 21 
 Results, presentation of, 51 
 Right-of-way, cost of, 74 
 Route, economical underground line, 
 
 187 
 most economical transmission 
 
 line, 89, 107 
 
 Secondaries, economical size of 
 
 power, 176 
 line cost curves, 161 
 load curves for, 160 
 power, 174 
 
 economy of instead of sepa- 
 
 rate transformers, 175 
 S0 for scattered load, 171 
 Secondary copper loss, 156 
 
 distribution, annual cost, 142 
 single phase, 139 
 underground, 187 
 line, annual cost, 143, 154 
 transmission lines, 81 
 Service, good, 7, 138 
 Shortening a line, economy of, 100. 
 Silver, A. E., 73 
 
 Span, standard transmission line, 82 
 Special apparatus, cost of, 75 
 Substation structures, cost of, 75 
 Substations, location of, 195 
 Switches, cost of, 75, 79 
 Symbols, 50 
 System as a whole, the, 194 
 
 Taxes, 18 
 
 "t C.", 117 
 
 Towers, cost of, 74 
 
 Transformers, annual cost of, 142 
 
 Transformers, copper loss, 156 
 core loss, 155 
 cost of, 75, 77, 79 
 curves, 162 
 replacing, 163 
 fixed charges on, 157 
 size, 147, 168 
 spacing, most economical, 146, 
 
 167 
 
 Transmission line, backbone, 71 
 problems, 71 
 secondary, 81 
 
 Underground lines, 183 
 
 Voltage, drop, 58 
 
 approximate method for 
 
 secondaries, 64 
 most economical for second- 
 
 aries > 144 > 166 
 
 Pwer circuits, 111 
 
 and Pwer loss, 55 
 P ower circuits, 110 
 regulation, in industrial plants, 
 
 197 
 standard for transmission lines 
 
 ^2 
 underground lines, 184 
 
 W 
 
 Waste energy, cost of, 32 
 
 Wire, secondary, fixed charges on, 
 
 157 
 
 size, power circuits, 111, 121 
 S< lighting circuits econom- 
 
 ical, 132, 149, 168 
 transmission lines, 83 
 underground lines, 184 
 

 LD 2l-ioo m . 8> > 34 
 
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