192 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002

Optimal Selection of Conductors for DistributionFeeders

Sujit Mandal, Member, IEEE and Anil Pahwa, Senior Member, IEEE

AbstractDesigning a distribution system requires many stagesof planning and rigorous calculations. Selection of conductors fordesign and upgrade of distribution systems is an important partof the planning process. An ideal conductor set should have themost economic cost characteristics, sufficient thermal capacity inthe largest conductor to take care of situations with very high loadand it should provide proper voltage at the farthest end under peakload conditions. In this paper, a method for selection of optimal setof conductors is presented. Several financial and engineering fac-tors are considered in the solution. The intent is to arrive at a so-lution, which will be the most economical when both capital andoperating costs are considered. Simulations have been performedto obtain results based on different criteria and the results are com-pared.

Index TermsCosts, economics, planning, power distribution,power distribution lines, power system planning.

I. INTRODUCTION

SELECTION of conductors for design and upgrade of distri-bution systems is an important part of the planning process.After taking all the factors into consideration, utilities select fouror five conductors to meet their requirement [1]. This selectionis done mainly based on engineering judgment. Historical fac-tors also play a role in the selection process, i.e., if a companyhas been using a particular size of conductor, they would wantto continue to use that size unless there are compelling reasonsnot to do so.

The available literature consists of work of only a fewresearchers on finding the best set of conductors in designinga distribution system. Funkhouser and Huber worked on amethod for determining economical aluminum conductor steelreinforced (ACSR) conductor sizes for distribution systems [2]in 1955. They showed that three conductors (2/0, 266 MCM,397 MCM) could be standardized and used in combinationfor the most economical circuit design for the loads to becarried by a 13-kV distribution system. They also studied theeffect of voltage regulation on the conductor selection process.The work done by Wall et al. [3] was published in 1979 inwhich the authors considered a few small systems to determinethe best conductors for different feeder segments of thesesystems. The study done by Ponnavaikko and Rao in 1982 [4]

Manuscript received November 15, 1999; revised August 6, 2001. This workwas supported by the National Science Foundation under Award EEC-9527345.

S. Mandal was with the Department of Electrical and Computer Engineering,Kansas State University, Manhattan, KS, 66506 USA. He is now with TechnicalSystem Planning, Entergy Services, Inc., New Orleans, LA 70113 USA (e-mail:smandal@entergy.com).

A. Pahwa is with the Department of Electrical and Computer Engineering,Kansas State University, Manhattan, KS 66506 USA (e-mail: pahwa@ksu.edu).

Publisher Item Identifier S 0885-8950(02)01068-4.

suggested a model to represent feeder cost, energy loss costand voltage regulation as a function of conductor cross-section.The researchers proposed an objective function for optimizingthe conductor cross-section. Tram and Wall worked on similargrounds in 1988 where again the authors took different exam-ples of feeder systems and calculated the best conductor foreach feeder segment based on specific requirements of voltageand losses [5]. Anders et al. published their work in 1993 [6]where they analyzed the parameters that affect the economicselection of cable sizes. The authors also did a sensitivityanalysis of the different parameters as to how they affect theoverall economics of the system. In 1995, Leppert and Allen[7] suggested that conductor selection is not only based onsimple engineering considerations such as current capacityand voltage drop but also on various other considerations,e.g., load growth and wholesale power cost escalation. Willis[1] gives a very broad idea about the line economics and thevarious factors that affect the selection of conductors. In thispaper, we have presented a systematic approach for selectionof an optimal conductor set. Several financial and engineeringfactors are considered in the solution. The intent is to arrive at asolution, which will be the most economical when both capitaland operating costs are considered.

II. BASIC LINE ECONOMICS

Every conductor has a unique cost versus load characteristic.So for proper choice and analysis of the conductors, it isnecessary to obtain these characteristics for all of them. Initialinstallation, annual operation and maintenance and losses arethe three components of the total cost. Initial installation isa one-time cost and it is incurred whenever the line is built.This cost is different for different conductor sizes since heavierhardware is needed for larger conductors and also handlingcost is higher for larger conductors. Annual operation andmaintenance cost is also higher for lines with larger conductors.This is mainly due to the fact that utilities would spend moremoney on maintenance of those lines that carry higher loadsince failure of these lines would impact a larger number ofcustomers. This cost could have an annual escalation, but inmost planning studies such escalation is neglected [8], [9].Thus, the present worth of fixed annual expense of $ /yearfor operation and maintenance over a period of years at adiscount rate of can be determined by multiplying it by thepresent worth factor , where

(1)

08858950/02$17.00 2002 IEEE

MANDAL AND PAHWA: OPTIMAL SELECTION OF CONDUCTORS FOR DISTRIBUTION FEEDERS 193

The losses in the lines, which are a function of the peak load,contribute to the variable part of the cost. If there were any loadgrowth, this cost would increase every year. Thus, if the peakload in the first year of operation of the line is MW, losses dueto this load can be computed by first determining the current,which is

(2)

where is the line-to-line voltage in kV and is the powerfactor. Hence, for an annual loss factor (ratio of average loss toloss at peak load) , the total energy losses in the first year forone mile of a three-phase line with a resistance of ohms permile are

Losses kWh/mile (3)

Multiplying (3) by (cost of energy in $/kWh) gives the costof losses in $/mile for the first year. Now, if we consider thatpeak load grows at a rate per year, the losses will grow at therate , which is equal to 2 since losses are proportional tothe square of the peak load. We have assumed that all the otherterms in (3) remain fixed. The present worth of a quantity thatescalates with a rate is obtained by multiplying the value inthe first year by a present worth factor [9], where

(4)

To obtain the total present worth cost of a line with a specificsize of conductor, we add the installation cost, the present worthof operation and maintenance and the present worth of lossesand obtain the following expression of total present worth cost.

mile (5)where

Cost Cost

and

Cost initial installation cost in $/mi. Cost opera-tion and maintenance cost in $/yr/mi.

Thus, the total present worth as a function of peak load turnsout to be a quadratic function [1], [10]. We can similarly findpresent worth cost of lines of different conductor sizes and whenthe total present worth cost versus peak load characteristics ofdifferent conductors are plotted on the same graph, we get agraph of the type shown in Fig. 1. An important thing to noteis that the plots of different conductors intersect with each otherand thus for every value of peak load there is a conductor withthe least cost.

III. LOADING AND LOAD REACH

Reach of a conductor is defined as the distance up to which acertain load can be delivered over the line without violating thevoltage drop limits [1]. Thus thermal reach of conductor wouldbe the distance up to which the conductor can move power at its

Fig. 1. Economic characteristics of a set of four conductors.

thermal loading limit (maximum allowed current) while main-taining the voltage within the specified limits. Similarly, the eco-nomic load reach of a conductor is defined as the distance up towhich the conductor is capable of carrying power equal to thatdetermined by the upper limit in the economical loading rangewithout violating the voltage drop limits.

Reach of a line with a specific conductor size can be deter-mined by first finding the voltage drop in that line. The percentvoltage drop per mile for a distribution feeder is approximatedby [11]

mile(6)

where is the current carried by the conductor, is thepower factor and is the inductive reactance of the line in /mi.If the allowed voltage drop is , the reach of the conductoris given by

Reach miles (7)

When , this reach is termed as thermal reach andwhen , this reach is called economic reach where

is the current carried by the conductor at the point of in-tersection of its curve with that of the next higher conductor.This point gives the economic loading limit of the conductor,i.e., that particular conductor is not economical to use beyondits economic loading limit. Thus, every conductor in a set has aloading range within which it can transmit power most econom-ically compared to other conductors. This range is called theeconomic loading range for that particular conductor. If a reachhigher than the economic reach is desired from a conductor thenthat conductor is derated or the loading limit is reduced to keepvoltage drop within the limits. In distribution system design itis a common practice to keep the reach of all the conductors ina selected set to be the same. Therefore, some or all of the con-ductors might require derating based on the desired reach. Fig. 2shows the same set of conductors as shown in Fig. 1 when allthe conductors have a common reach of 4.7 mi. Adjustments inscale are made in the axes of Fig. 2 compared to Fig. 1 to ac-commodate the important features of the curves. Note that the

194 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002

Fig. 2. Economic cost characteristics of a set of four conductors with acommon reach of 4.7 mi.

economic loading range of 477 MCM conductor is 3.8 MW to7.8 MW in Fig. 1, but after derating to obtain a common reachof 4.7 mi its economic loading range became 3.7 MW to 4.8MW. In other words for a peak load between 4.8 MW and 7.8MW, 477 MCM has the least cost but we have to use the highersize (795 MCM) since the voltage drop for the desired reach ishigher.

IV. CRITERIA TO SELECT CONDUCTOR SETS

Selection of a good conductor set is very important for properplanning of distribution systems. An ideal conductor set shouldhave roughly equal economic loading range for each conductor,sufficient thermal capacity in the largest conductor to take careof situations with very high load and should be the most op-timal in terms of cost. Generally, it is very difficult to selecta conductor set which will meet all the criteria of an ideal setand a systematic procedure is not available for selection of agood conductor set. Most utilities choose conductors for theirsystem based on experience and historical applications. Usually,the number of conductors in a set is limited to four or five forproper management of inventory.

In our approach to solve the problem, we have fixed thesmallest and the largest sizes of the conductors initially. Gener-ally the smallest conductor is not selected based only on loadingconsiderations, but also reliability is taken into consideration.Very small conductors have more tendencies to break underwindy and stormy conditions. The biggest conductor, on theother hand, is selected based on the maximum loading anddesired reach. It may turn out that even the biggest conductormay not provide the desired reach at the maximum loading. Inthat case either the loading or reach must be decreased. Oncewe know the smallest and the largest conductor sizes, the curvescorresponding to these conductors are drawn. Now the task isto select two or three more conductors in between (dependingupon whether we are looking for four or five conductors) suchthat the total cost of the conductor set is optimal.

To obtain optimal cost we could minimize the aggregate areaunder the plots of different conductors between the minimum

and the maximum load. For example, in Fig. 1 we would inte-grate from 0 to 18 MW with the lowest cost characteristics con-sidered for different load ranges. The conductor set with min-imum value for this area would have an average minimum costover the whole range of loading. Another approach is to draw astraight line that is tangent to the first and the last conductor asshown in Fig. 2 and then compute the area enclosed by the con-ductor curves and this line between the two points where the tan-gent touches the curves (or between the point where the tangenttouches the first conductors characteristics and the maximumload for the selected reach as in Fig. 2). During our study wefound that none of the conductors had plots that would go belowthis straight line. Hence, minimizing the total area under thecurves is equivalent to minimizing the area between the curvesand the straight line. However, instead of using this area for se-lection of optimal conductor set, a better approach is to givedifferent weights to areas in different loading ranges since inmost distribution systems a large percentage of feeders carryvery small load and a few feeders carry large load. For example,in a system discussed in [1], out of 24 800 miles of primaryvoltage line close to 56% of the total length of feeder sectionscarry power less than 0.5 MW. Around 11% of the total lengthof feeders carries power between 0.5 MW and 1 MW. Similarinformation is given for every increment of 0.5 MW. These num-bers are typical numbers and would be somewhat different fordifferent systems. A utility could very easily obtain such in-formation from their existing system. Data obtained from [1]was modified slightly to get Table I which shows the fraction oftotal feeder length with the given peak load in a typical distri-bution system. The modification included adjusting factors foreach range to make the sum of the fractions equal to 1. Thus, thenumber associated with each loading range gives the weight forthat range. Hence, we performed integration in steps of 0.5 MWand then multiplied the resulting area by the respective weightto yield the weighted area for selection of conductors.

Once the conductors are selected, they must be derated sothat all of them have the same specified reach. Derating theconductors in this manner could result in a sub-optimal set ofconductors. Another approach is to derate all the conductorsfor a specified reach before computing the weighted area fortheir selection. Any small amount of potential savings that mighthave been missed by fixing reaches after selecting the conduc-tors would come to focus by reversing the process, i.e., fixingreaches first and then selecting the conductors. The first ap-proach is called Method A and the second approach is calledMethod B in the rest of the paper.

V. EXAMPLES AND RESULTS

In the examples considered in this paper, 17 conductors span-ning from conductor # 1 to 795 MCM are included. Numeroussimulations were tried to test the concept [12] and results of twocases are presented in this paper. In one of the cases, four con-ductors were selected while in the other five conductors were se-lected based on enumeration of all feasible combination of con-ductors. For each case results are obtained using both Method Aas well as Method B. The distribution system consists of 12.47kV line-to-line voltage. For the base case, the discount rate is

MANDAL AND PAHWA: OPTIMAL SELECTION OF CONDUCTORS FOR DISTRIBUTION FEEDERS 195

TABLE IWEIGHTS ASSOCIATED WITH DIFFERENT LOADING RANGES

chosen to be 8% and the load growth rate is chosen to be 0.5%.Also, the loss factor is 0.46, power factor is 0.9, the cost of en-ergy is 3 cents/kWh and the planning horizon is 30 years.

For each of the cases, ten simulations were done with dif-ferent values of cost of energy, discount rate, planning duration,loss factor, load growth, power factor, installation cost and op-eration and maintenance cost to study sensitivity of the resultsto these parameters. Only one parameter was changed in eightsimulations while all the parameters were changed simultane-ously in two simulations. Changing these parameters did nothave a very significant effect on the best set of conductors. Inthis paper, results associated with the base case only are pre-sented. A summary of these results is given in Tables II and III.It can be seen that as expected different results are obtained fromMethod A and Method B.

Now, we are left with a crucial questionis Method A betteror Method B better? Therefore, further analysis was done toresolve this question. Firstly, a scenario with four conductorsin the set was studied and two common reaches of 4.7 mi and3.6 mi were used in the study.

Since the optimal conductor set is selected first and thencommon reach is fixed in Method A, the conductor set isindependent of the reach used. However, in Method B sinceconductor loadings are adjusted before selecting the optimalconductors, different sets of conductors were selected based onthe chosen reach. For a common reach of 4.7 mi conductors 2/0and 300 MCM and for a common reach of 3.6 mi conductors266 MCM and 477 MCM turned out to be the best conductorsin addition to # 1 and 795 MCM. The details of the conductorsets for a reach of 4.7 mi are provided in Tables IV and V. Tocompare these two sets, the weighted average deviation of thecost of a conductor set from the tangent line is determined,which is given by

Total Weighted AreaLoading range Adjusted Weight (8)

Since the total weighted area was calculated between thepoint where the tangent touches the first conductors charac-teristics and the maximum load for the selected reach, the sum

TABLE IIRESULTS FOR COMMON REACH OF 4.7 MILES BASED ON METHOD A

TABLE IIIRESULTS FOR COMMON REACH OF 4.7 MILES BASED ON METHOD B

TABLE IVBEST FOUR CONDUCTORS OBTAINED FOR A COMMON REACH OF 4.7

MILES USING METHOD A

TABLE VBEST FOUR CONDUCTORS OBTAINED FOR A COMMON REACH OF 4.7

MILES USING METHOD B

of the weights for the load ranges included in this calculationwould not be equal to 1. Hence, the values of these weightswere adjusted proportionally such that they add to 1 and thus theresulting weighted average deviation gave the real dollar valuein terms of $/mi. The adjusted average weighted deviationsof the two conductor sets from the linear cost characteristicsare $17 049/mi and $15 160/mi, respectively. Thus the latterconductor set offers an average saving of $1 889/mi for a reachof 4.7 mi. Although the cost is lower for the second conductorset, we found that the loading of conductors is more uniformfor the first set. It is particularly true for conductor 2/0, whichhas a small loading range that goes from 1.38 MW to 1.90 MWin the second set.

When the common reach was changed to 3.6 mi, the sameconductor set # 1, 266 MCM, 477 MCM and 795 MCM wasfound to be the best whether Method A was used or Method Bwas used. Details of the results are shown in Table VI. This is avery interesting result, which can be explained as follows. We al-ready know that to increase the reach of a conductor higher thanits economic reach, the conductors need to be derated to carrymaximum load lower than its economic loading limit. Thus,when the common reach is fixed at 4.7 mi after selecting the con-ductors, the conductors get derated significantly. This is becauseof the fact that the economic reaches of these conductors vary

196 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002

TABLE VIBEST FOUR CONDUCTORS OBTAINED FOR A COMMON REACH OF 3.6 MILES

USING BOTH METHOD A AND METHOD B

from 3.1 mi to 4.7 mi. So when the reaches of all the conductorsare adjusted to be 4.7 mi, all except one of them (266 MCM,which has the economic reach of 4.7 mi) get derated by a hugemargin. But when the reach is fixed at 3.6 mi the conductors arenot derated by a big margin, which implies a smaller deviationfrom already selected characteristics. Hence, for smaller reachthe results are the same whether Method A is used or Method Bis used. Note that average weighted deviation from the straightline is lower for a reach of 3.6 mi in comparison to a reach of4.7 mi. This can again be explained by the fact that for a lowerreach the derating of the conductors is lower than that for higherreach.

These results show that as the common reach of the conduc-tors is lowered, the total load range served by these conduc-tors increases. This is a very obvious result since the losses andvoltage drop in the line increase with increase in load on the line.Hence, when the conductors carry higher magnitude of power,they can transmit power only over smaller distance before vio-lating the voltage drop criterion.

Results of a similar analysis based on Method A and MethodB for a set of five conductors and common reach of 4.7 mi areshown in Tables VII and VIII. Again, we found that the resultsare better with Method B for a common reach of 4.7 mi. How-ever, only one conductor is different in the two sets. The con-ductor set obtained with Methods B yields a saving of $780/miover that found with Method A. Note that the savings are higherfor the similar cases with four conductors as shown earlier. Also,unlike the previous scenario where we were selecting four con-ductors, here the loading ranges of the conductors are more uni-form in the conductor set obtained with Method B. On the otherhand, in the set obtained with Method A, conductor 1/0 has avery small loading range of 1.38 MW to 1.61 MW.

When the reach was changed to 3.6 mi, the optimal conductorset was found to be the same using both Method A and MethodB. The results are given in Table IX. Since a similar thing hap-pened while selecting four conductors, explanation providedearlier is true for this case too.

The above-mentioned results show that Method B is betterthan Method A. However, another question that needs to be an-swered is whether it is better to select five conductors or fourconductors. Comparing the results of Tables V and VIII wherethe conductor sets have a common reach of 4.7 mi, it can beseen that the set of five conductors offers a saving of $2,468/mi.Similarly comparing the results of Tables VI and IX where thecommon reach is 3.6 mi, it can be seen that five conductors offera saving of $1,316/mi compared to the set of four conductors.These savings are very significant since distribution systemsare very extensive and cover large areas. Moreover, the loading

TABLE VIIBEST FIVE CONDUCTORS OBTAINED FOR A COMMON REACH OF 4.7

MILES USING METHOD A

TABLE VIIIBEST FIVE CONDUCTORS OBTAINED FOR A COMMON REACH OF 4.7

MILES USING METHOD B

TABLE IXBEST FIVE CONDUCTORS OBTAINED FOR A COMMON REACH OF 3.6 MILES

USING BOTH METHOD A AND METHOD B

ranges of all the conductors are significant and none of themcan be considered redundant. It is interesting to note that sav-ings for five conductors versus four conductors are higher thanthose obtained by using Method B versus using Method A.

VI. CONCLUSIONIt is very challenging to select an optimal set of conductors

for designing a distribution system. In this paper, a systematicprocedure has been suggested to achieve this goal. Minimizingthe weighted area between the cost characteristics of the con-ductors and a linear load versus cost representation provides asuitable approach to solve this problem. Analyses showed thatfixing a common reach for the conductors before selecting them(Method B) leads to potential savings that would not be avail-able if common reaches are fixed after selecting the conductors(Method A). It was also found that a set of five conductors pro-vides better economy than a four conductor set. The techniquespresented in this paper are very practical and can be very easilyimplemented by any utility.

REFERENCES[1] H. L. Willis, Power Distribution Planning Reference Book. New York:

Marcel Dekker, 1997, pp. 239283.[2] A. W. Funkhouser and R. P. Huber, A method for determining econom-

ical ACSR conductor sizes for distribution systems, AIEE Trans. PowerApparat. Syst., vol. PAS-74, pp. 479484, June 1955.

MANDAL AND PAHWA: OPTIMAL SELECTION OF CONDUCTORS FOR DISTRIBUTION FEEDERS 197

[3] D. L. Wall, G. L. Thompson, and J. E. D. Northcote-Green, An opti-mization model for planning radial distribution networks, IEEE Trans.Power Apparat. Syst., vol. PAS-98, pp. 10611065, May/June 1979.

[4] M. Ponnavaikko and K. S. P. Rao, An approach to optimal distributionsystem planning through conductor gradation, IEEE Trans. Power Ap-parat. Syst., vol. PAS-101, pp. 17351741, June 1982.

[5] H. N. Tram and D. L. Wall, Optimal conductor selection in planning ra-dial distribution systems, IEEE Trans. Power Syst., vol. 3, pp. 200206,Feb. 1988.

[6] G. J. Anders et al., Parameters affecting economic selection of cablesizes, IEEE Trans. Power Delivery, vol. 8, pp. 16611667, Oct. 1993.

[7] S. M. Leppert and A. D. Allen, Conductor life cycle cost analysis, inProc. Rural Electric Power Conf., 1995, pp. C2-1C2-8.

[8] H. Khatib, Financial and Economic Evaluation of Projects in the Elec-tricity Supply Industry. London, U.K.: IEE, 1997.

[9] R. M. Sigley Jr., Engineering economic analysis overview, in Tuto-rial on Engineering Economic Analysis: Overview and Current Appli-cations. Piscataway, NJ: IEEE Press, 1991.

[10] M. V. Engel, E. R. Green, and H. L. Willis, Tutorial on PowerDistribution Planning, M. V. Engel, E. R. Green, and H. L. Willis,Eds. Piscataway, NJ: IEEE Press, 1992.

[11] T. Gonen, Electric Power Distribution Engineering. New York: Mc-Graw-Hill, 1986.

[12] S. Mandal, Optimal selection of conductors for designing distributionsystem, M.S. thesis, Kansas State Univ., Dept. Elect. Comput. Eng.,Manhattan, KS, 1999.

Sujit Mandal (S97M99) received the B.Tech. degree in electrical engi-neering from the Indian Institute of Technology (IIT), Kanpur, India and theM.S. degree in electrical engineering from Kansas State University, Manhattan,KS in 1997 and 1999, respectively.

He worked as a Consultant at Power Technologies, Inc., Schenectady, NY,from 1999 to 2000. Presently, he is with Technical System Planning, EntergyServices, Inc., New Orleans, LA.

Anil Pahwa (S82M83SM91) received the B.E. (honors) degree in elec-trical engineering from Birla Institute of Technology and Science, Pilani, India,the M.S. degree in electrical engineering from the University of Maine, Orono,and the Ph.D. degree in electrical engineering from Texas A&M University, Col-lege Station, in 1975, 1979, and 1983, respectively.

Since 1983, he has been with Kansas State University (KSU), Manhattan,where he is presently Professor and Graduate Program Coordinator in the Elec-trical and Computer Engineering Department. From August 1999 to August2000, he worked at ABB-ETI, Raleigh, NC, while on sabbatical from KSU. Hisresearch interests include distribution automation, distribution system planningand analysis and intelligent computational methods for power systems analysis.

Dr. Pahwa is a member of Eta Kappa Nu and Tau Beta Pi.