Optimising the design of low-voltagecable networks

F.W. Walkden. B.Sc.(Eng). M.Sc.

Indexing terms: Cables and overhead lines, Networks

Abstract: A procedure is described for selecting cables for radially-connected low-voltage distribution networksof given layout, so that the volume of conductor in the network is a minimum. This corresponds to producing aminimum-capital-cost design when the cost of cables varies linearly with their conductor cross-sectional area.The procedure, based on analytical methods for the design of distributors, includes voltage regulation and thecurrent ratings of the cables as limiting factors in the design. It also permits other factors, such as the unbal-anced loading of phases and the presence of existing branches, to be allowed for. Results are given for twonetworks, with given costs for cables, and compared with results from a tried procedure which seeks to minimisecosts directly. The comparison indicates that the method can produce good results rapidly. It is concluded thatthis, coupled with the fact that programming and computation are sufficiently limited for sizeable networks tobe handled on a small microcomputer, suggests that it could be a useful design procedure.

List of symbols

p = conductor resistivityV = voltage regulation drop, per conductor, in a

network or subnetworkV'b = voltage regulation drop, per conductor, along

a branchV'j = voltage regulation drop, per conductor, at

nodejv = total volume of each conductorb = length of a branchIL = current loading per unit length/ = through current of a brancha = constant relating conductor cross-sectional

area and distance from the end in a contin-uously tapered distributor designed to have aminimum volume of conductor

X, \i = Lagrangian multipliers used in the designanalysis of stepped distributors

bj, Ij, Xj = b, IL, and X, respectively, of the branch termi-nating at terminal j

ak, lk = conductor cross-sectional area and length,respectively, of the /cth section of a steppedb r a n c h (k = 1,2, ...,n,ak+1 > ak)

ksk = Z h

A - Z aj+iaj(aj+i. - aj)

A> = Z aj+iaj(aj+i - aj)

j=k

1 Introduction

Methods to optimise cable networks for low-voltage dis-tribution have been the subject of previous papers(References 2 and 3). They describe procedures to obtain aminimum-cost design for a given layout, cost being thesum of the cable costs and the capitalised cost of the RI2

loss in the network. In each case they commence with adesign for the network based on the current ratings ofcables and modify this by an iterative procedure until therequired design is arrived at. Obviously this approach pro-duces a result more rapidly the closer the initial design isto the final one, as will be the case when the design of the

Paper 4215C (P8. P9), first received 27th February and in revised form 9th July1985

The author is with the Faculty of Engineering, Coventry (Lanchester) Polytechnic,Priory Street, Coventry CV1 5FB, United Kingdom

network is largely constrained by cable current ratings.But if the limiting condition for a significant part of thenetwork turns out to be voltage regulation then the com-putation to arrive at the required design can be very con-siderable. In a program based on the procedure describedby Snelson and Carson [2] the number of iterationsrequired is commonly in the 100s and convergence to asolution is found to be especially slow when the networkloading is relatively light.

It is sometimes the practice to discount the cost of thenetwork loss, owing to its uncertainty. This paperdescribes an alternative approach that can be used in suchinstances. It involves iteration only in the final stage of thecalculation when the design has been brought close to itsfinal state. The procedure is developed from the analyses inReferences 1 and 4, and it determines the minimumvolume of conductor required to satisfy the operating con-ditions of the network. This corresponds closely tominimum cable cost when (as is often the case with dis-tribution cables) there is a nearly linear relationshipbetween cable cost and cross sectional area of conductor.

2 Outline of the problem and method of solution

The problem discussed here is how to design, from stan-dard cables, a radially-connected network of a givenlayout and loading, generally consisting of both distribu-tors and feeders, so that the volume of conductor in thenetwork is a minimum. (The term 'distributor' is used todescribe a branch which has loads connected at intervalsalong it; the term 'feeder' to describe a branch along whichthere are no loads connected). The boundary conditions ofthe problem, imposed by the operating conditions of thenetwork are

(a) the current ratings of the cables(b) the limit of voltage regulation.

For a minimum volume design the current ratings shouldbe those corresponding to the thermal ratings of thecables. If less, economic current ratings could be usedinstead. This would introduce, though, the cost of losses asa factor in the design of those parts of the network wherecurrent rating was the limiting condition, which would notbe taken into account elsewhere in the network. Thedesign would not be entirely consistent and would depart,to some extent, from a minimum-volume design.

It is only where voltage regulation is the governing con-dition that the design problem becomes at all difficult.Where current ratings are exceeded before the limit ofvoltage regulation is reached, the design exercise is quite

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986 49

straightforward. If a network is designed with voltageregulation as the only limiting condition it is a simplematter to modify it if current ratings are exceeded at anypoint, even allowing for the fact that such modification willalter the regulation and induce changes elsewhere. On theother hand, if the network is designed on the basis of thecurrent ratings of the cables, it is much more difficult todetermine the most appropriate changes if the limit ofvoltage regulation is exceeded.

The proposed method of solution is divided into 3stages:

(a) An estimate is obtained of the distribution of voltageregulation in a minimum-volume design for the network,when voltage regulation is the only limiting condition.

(b) Working from this estimate, a design is producedtaking both boundary conditions into account.

(c) Incremental changes are made to the design to bringit to the condition for minimum volume.

As in previous methods [1-4] the loads connectedalong a distributor are represented as a uniform rate ofcurrent loading calculated from the ADMD of the individ-ual loads, and voltage regulation is calculated as RI drop,unity-power-factor loading being assumed. Lack of balancebetween phases and lack of diversity between loads can beallowed for in the way described by Snelson and Carson[2]-

3 Stage 1 : to determine an estimate of thedistribution of voltage regulation

The calculation begins on the assumption that the networkis designed with continuously-tapered conductors chosento make the conductor volume a minimum for a givenvalue of voltage regulation. This is taken as the initialdesign because it will match, more or less closely, the cor-responding design with standard cables, and because thedistribution of voltage regulation in such a design is easyto determine.

For each node of the network, a constant is determinedwhich is related to the regulation at that node. Both unbal-ance between phases and lack of diversity can be allowedfor in the calculation. These constants are then modified totake into account any branches whose volume cannot beminimised for one reason or another. The modified valuesare used to determine branch voltage drops in the secondstage of the design procedure. Fig. 1 represents the connec-tions at a node at the end of the network. The regulationdrop from node r to terminals t and u is V, and V's is thedrop from r to s. All 3 branches in the diagram areassumed to be distributors. For the sake of simplicity only2 branches have been shown radiating from s, the analysisnot being materially affected when there are more.

V supply\

Is ,b;

Fig. 1

50

A subnetwork at a node at the end of a radial network

The volume of conductor in a continuously-tapered dis-tributor designed for a voltage regulation V' has aminimum value (4/9)pILb3/V; if the distributor carries athrough current / in addition to its distributed load thisvolume becomes (4/9)pc2ILb'2/V, where c = 1- [(I/Ii)/b'Y12 and V = b + I/JL. (The derivation of these

expressions is given in Appendix 10).Thus, for the network in Fig. 1, the conductor volume,

when minimised in each branch is

v = $ - V's)} (1)

where XILb3 = Itb? + lub\Although the branch volumes are a minimum, v is only aminimum when dv/dV's = 0, which obtains when

V's = V'/k.

where

(2)

(3)

With these values of V's and ks the volume of the sub-network becomes

If the branch s is a feeder, these formulas are slightlyaltered. The volume of branch s, in terms of its throughcurrent /(= ZILb) is pIb2/V's, making the volume of thesubnetwork

v = p{lb2sIV's + %ULb3/(V - V's)} (5)

which, as before (eqn. 2), has a minimum value when

V's=V'/ks

but now

(6)

and the minimum volume of the subnetwork is

v = (7)

It is convenient to express this in terms of an equivalentdistributor to correspond to eqn. 4; i.e.

(8)

where be = (3/2)/cs bs,

and Ie = I/be

So, when the constant ks has been determined (in accord-ance with eqn. 2 or 4, as appropriate) the subnetwork atnode s can be replaced by an equivalent distributor of thesame volume, given by eqn. 4 or 8. This enables the sub-network at node r to be treated subsequently, in the sameway as that at node s, to determine kr corresponding to ks.

These constants correspond to nodal voltage drops and,hence, to branch voltage drops. At the first node from thesupply point, in accordance with eqn. 1,

V\ = V'k, (9)

where V is the regulation limit for the network.For the next node,

V2 = (V - V\)/k2 (10)

and so on.Adjustments are now made to take account of any

branches whose conductor volume cannot be minimised.The network is examined to identify branches where, on

1EE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986

the basis of the voltage drops just determined, the smallestcable will satisfy the voltage regulation condition. Thesebranches are removed as part of the minimum-volumenetwork, and the node k factors adjusted accordingly. Thebranches removed are treated as constant voltage drops inthe subsequent stages of the design, though they are liableto be brought back into the process of minimising conduc-tor volume, depending on how the voltage drops in thenetwork are amended in the final stage. Any existingcabling incorporated into the network is treated similarly.

4 Stage 2: design of branches from standard cables

Subject to modification if current ratings are exceeded, aminimum-volume design corresponding to the branchvoltage can be produced for all the branches in thereduced minimum-volume network.

If the branch is a feeder, a minimum-volume design isobtained with cables ak and afc_l5 where

Piv h

(11)

V'b being the branch voltage drop.If current ratings are exceeded, the branch design will

consist of one cable only, ak or larger.For each distributor a design constant X (which enables

cable lengths to be calculated) is determined. For a dis-tributor forming a terminal branch of the network, X =y/A/(2V'b/pIL — b2/an), where an is the largest cable to beused in the branch, ax being the smallest. For a distributorcarrying a through current / in addition to its distributedload,

X = jA'/ilV'JpI + (I/IL)2/ak - b'2/an),

where ak is the smallest cable in the branch design. (Thederivation of these expressions, the procedure for findingan and ak, and the relationship between X and cablelengths, are given in Appendix 10).

To check a distributor against current ratings it is onlynecessary to examine the minimum-volume design at twosections. This is because in the minimum-volume designthe maximum current density in each cable increases withcable size, except possibly in the largest, whereas currentdensity corresponding to the thermal rating decreases asconductor size increases.

In any cable, ak, of a minimum-volume design themaximum current density is

j k _ (iLsk + I)/ak (12)

and as

sk = ak+1aJX — I/IL for k up to n — 1,

(13)

which increases with k and is therefore a maximum whenk = n- 1.

In the remaining section (the largest cable, an) themaximum current density is

Jn = (lLb + I)K (14)

If b > all?., as it may sometimes be, Jn > Jn-i, in whichcase if Jn does not exceed the value corresponding to thecurrent rating it will not be exceeded elsewhere. In general,though, it will be necessary to check both Jn and Jn-V

If the design breaches current ratings, these replacevoltage regulation as a design constraint and, as a result,the voltage drop along the branch is reduced. Consequent-ly, the limiting value of voltage drop over the remainder of

the network is altered and the design of the other branchesis affected. This stage of the design commences, therefore,with the branch connected to the supply point, and workssequentially through the network, branch by branch, tothe ends. At each step the succeeding branch voltage iscalculated using the appropriate nodal k factor obtained atthe end of stage 1. In this way the effect of reductions inthe voltage drops of previous branches where currentratings have limited the design is allowed for.

Thus at the end of this stage a design has virtually beenproduced for each branch. Where voltage regulation hasbeen the limiting factor this amounts to values for ak and%_! in the case of feeders and, in the case of distributors,X, an, and ak (or ax). Where current ratings apply thedesign will have been completed.

5 Stage 3: final adjustments to minimise conductorvolume

At the end of stage 2 the design is more or less close to theminimum-volume state required. But, unless currentratings have determined the design of the entire network,there will be some scope for reducing volume, as thedesign constants for branches have been determined fromvoltage drops calculated for a continuously taperednetwork.

The design constant X obtained for distributors corre-sponds to a conductor volume

= anb'-ak(I/IL)~A'/X (15)

This varies with the branch voltage drop (V'b) as X is afunction of this quantity, and differentiating with respect toV'b yields

(16)

A feeder consisting of cables ak and ak-l has a conductorvolume

v = b(ak - afc_x) - akak^xV'J

and

dvdV'u

(17)

(18)

Referring to Fig. 1, if a small change is made to V's withoutchanging the regulation between node r and terminals tand u, then the change in the voltage drop along branch sis balanced by an equal but opposite change alongbranches t and u. If the 3 branches are distributors, theircombined volume is a minimum when V's is adjusted to avalue such that

When this condition holds, the corresponding conditionfor node r (the next node towards the supply) is

-Xr/Ir + X(X/IL) = 0 (20)

where the term ZCA//J includes all the branches connectedto node r other than the branch on the supply side.

Where there is a feeder, a term akak_x/I will appear ineqns. 19 and 20.

At the end of stage 2 the state of the design is such thatthe left-hand side of eqns. 19 and 20, and the correspond-ing quantity for other nodes, is not equal to zero. The aimin stage 3 is to bring the design to the state where all theequations corresponding to eqns. 19 and 20 are satisfiedsimultaneously. The conductor volume in the network willthen be a minimum.

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986 51

Starting at the ends of the network, the voltage drops atthe nodes are adjusted in sequence until the supply point isreached. The process is repeated until the required condi-tion is obtained.

At any stage in this process there may be brancheswhose design cannot be adjusted. These will includebranches whose design has reached a limit set by currentratings, or cable sizes, as well as branches consisting ofexisting cabling. If, for example, branch s in Fig. 1 cannotbe changed, the terms ljlt and XJlu in eqn. 19 are trans-ferred to the equation for node r. The equation for node sis eliminated, at least temporarily, from the calculationsand branch s is treated as a constant voltage drop. Thusthere are likely to be fewer equations to condition thanthere are nodes, and there is a tendency for their numberto decrease as the calculation proceeds.

6 Examples of results

Fig. 2 shows the connections of a 14-branch network, thedata for which are given in Table 1. Table 2 gives details

Table 4: Alternative designs for 8-branch network (branches1-8 in Fig. 2)

substa.1

> c

4

> 7 c

5 i

)

J 10

1

11

I

13

Branch

1234567

8

Cablesizemm2

7035357035703570

35

Design

lengthm

7538324548302525

46

A

% regulationalongbranch

2.721.171.021.331.510.700.78

1.75

atterminal

3.893.73

5.56

6.50

Design

lengthm

7538324548302535

36

B

% regulationalongbranch

2.721.171.021.341.510.700.78

1.74

atterminal

3.93.7

5.6

6.5

Fig. 2 Distribution network

Cable costs: design A-£1585; design B-E1600

of the cables available for the design. Table 3 gives twodesigns for this network, design A obtained by the methoddescribed in this paper and design B obtained by the pro-cedure described by Snelson and Carson in Reference 2.Table 4 gives corresponding results for a smaller networkconsisting of branches 1 to 8 in Fig. 2. The limit of regula-tion in each design was 6.5% of 240 V (phase voltage), andallowance was included for lack of diversity and for unbal-anced loading between phases.

The Snelson and Carson method, as applied to thedesigns here, works directly to minimise cable costs. The Adesigns, though showing some differences from the Bdesigns, produce practically the same cable costs, despite

Table 1 : Data for network in

BranchLength, mConsumersADMD/con.

175

1, kW 2

238

52

332

32

Fig.

445

12

2

CJI

4882

630

0—

725

72

87172

920

42

1018

42

1125

62

1255122

1385

82

1415

32

Table 2: Cables available for network

Conductor size (ak), mm2

Thermal current rating, ACost, £/km

135

1253604

270

1855162

3120250

7207

4185320

9130

5300420

12 466

Table 3: Alternative designs for 14-branch network (Fig. 2)

1234567

8

9

10

11

12

13

14

Design A Design B

Cablesize.

Branch mm2lengthm

% regulationalong atbranch terminal

lengthm

% regulationalong atbranch terminal

1853535

18535

18535

185

7538324548302544

120 2735 20

185 —

1201207035120

7035

18253916295615

1.731.171.020.931.510.560.78

1.34

0.62

0.37

0.46

1.11

1.11

0.48

2.902.74

4.17

3.99

5.18

6.50

6.50

5.86

7538324548302571

206

12253025

8515

1.731.171.020.931.510.560.78

1.12

0.62

0.32

0.46

1.36

1.37

0.48

2.92.7

4.2

4.0

5.0

6.5

6.5

5.6

Cable costs: design A-E3657; design B-E3658

the fact that the cable cost/size relationship (as given bythe figures in Table 2) is not quite linear. Both A designswere obtained after only 3 cycles through stage 3 of thedesign procedure, compared with 1215 iterations requiredby the Snelson and Carson method for the 14-branchnetwork and 150 for the 8-branch network.

The procedure was written as a program in Basic. Afterbeing loaded with this, a microcomputer with a 32k RAMwas left with sufficient capacity for networks of up to 120branches to be examined.

7 Conclusions

The Snelson and Carson method is a tried and provenprocedure, but is sometimes slow to converge to a solu-tion. That the two designs produced by the alternativeprocedure described here show almost the same cost(actually less in both examples) testifies to the accuracy ofthe method, and indicates that good results can beobtained after a very small number of iterations, evenwhen the relationship between cable cost and size departsto a degree from the linear. Moreover, the program of theprocedure is sufficiently compact, and the requirements ofthe computation for memory space sufficiently limited, forsizeable networks to be handled with a small micro-computer.

These features, plus the fact that allowance for the pre-sence of existing cabling in a given network can be accom-modated in the design procedure, suggest that it holdspromise as an effective and practical method for selectingappropriate lengths of cable to form the network.

52 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986

8 Acknowledgments

The author is indebted to the East Midlands ElectricityBoard, and particularly to Mr. George Mather of thatorganisation, for much information on distribution net-works and their design, without which the work describedin this paper could not have been done.

9 References

1 DAVIES, M.: 'Design of l.v. distributors from standard cable sizes',Proc. IEE, 1965, 112, (5), pp. 949-956

2 SNELSON, J.K, and CARSON, M.J.: 'Logical design of branched l.v.distributors', ibid., 1970,117, (2), pp. 415-420

3 HINDI, K.S., BRAMELLER, A, HAMAM, Y.M.: 'Optimal cableprofile of l.v. radial distributors: two mathematical programmingmethods', ibid., 1976,123, (4), pp. 331-334

4 WALKDEN, F.W.: 'Design of low-voltage distributors', IEE Proc. C,Gener., Trans. & Distrib., 1982, 129, (3), pp. 101-103

10 Appendix

10.1 Continuously tapered distributorsIt is shown in Reference 1 that the conductor volume in anunbranched distributor is a minimum, for a given voltageregulation, when the cross-sectional area varies in propor-tion to the square root of the distance (x) from the end. i.e.

a(x) = a (21)

At x = I/IL the current in the distributor is /, and thesection between this point and the supply can be regardedas a distributor carrying a through current / in addition toits distributed load. For such a distributor, therefore,minimum volume is obtained when

a(x) = ay

the volume, per conductor, then being

v = a{x) dxJo

where V = (b + I/IL), and c = 1 -The regulation drop along the distributor,

(22)

(23)

V'h = ILx)/a(x)'] dx

2c/a (24)

from which a can be obtained in terms of V'b to substitutein the expression for conductor volume, giving

v = $pILb'3c2/V'b (25)

When the distributor carries a through current / theanalysis is modified as follows. The expression for thevoltage regulation at the end of the distributor becomes

i

= Ph (30)

and the n equation involving the Lagrangian multipliers Xand ft are each modified by the addition of a term in I/IL

as typified by the feth:

lk/ak

from which set, by subtraction, is obtained

= ak(ak + 1 - ak_l)X, for k = 2 to (n - 1)

and

ln = b-(anan.JX-I/IL)

= 0 (31)

(32)

(33)

(34)

Substituting for /l5 lk and /„ in the expression for voltageregulation, and rearranging, yields

(I/IL)2/ai-b'2/an

(35)

where b' = b + I/IL and an is determined as described inReference 4.

The expression for lx shows that it is possible for it tohave a negative value. As this is impractical, it is necessaryto remove the cable ax, i.e. the smallest cable, from thedesign. In general this will be done repeatedly, as indicatedin Fig. 3, until the length of the smallest section is positive.

determine°n

set k=1\\

I

r

A 1

2Vb/pIL+(l/lL)2

k Q k H ?

/ak-b>2/an

increase k by 1

- r

10.2 Stepped distributorsThe design of an unbranched distributor using cables ax,a2, . . , an, is described in References 1 and 4. For aminimum-volume design the section lengths are

= ak(ak+1 - flk- J/A, for k = 1 to (n - 1)

and

where

/> =2V'b/pIL-b2/an

Per conductor, the volume is then

v = anb- All

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 1, JANUARY 1986

(26)

(27)

(28)

(29)

Fig. 3 Routine to determine X and akfor a distributor carrying throughcurrent

If this is obtained when ak is the smallest cable in thedesign, then

A''b/pIL + (I/IL)2/ak-b'2/an

The volume per conductor is

(36)

(37)j=h

and when substitution in terms of ). is made for \i thereresults

= anb'-ak(I/IL)-A'/X (38)

53