review of distribution system planning models: a model for optimal multistage planning

Review of distribution system planningmodels: a model for optimal multistage

planningProf. T. Gonen, M.Sc, Ph.D., Sen.Mem.I.E.E.E.,

and Prof. I.J. Ramirez-Rosado, Ing.

Indexing terms: Long-term planning, Modelling, Optimisation, Distribution networks, Mathematical techniques

Abstract: A representative set of electric power distribution system planning models published in the literaturehas been reviewed. The models have been classified according to their characteristics, from the point of view ofstages of the plan and overall time span; the methods of treating distribution feeders and/or substations in termsof cost representation, location and sizing problems; radiality and voltage drop considerations; and the mathe-matical programming techniques used to solve them. Some of the particular features of models have beendiscussed in detail. The paper also presents a model to solve the optimal sizing, location and timing of thedistribution substations and feeder expansion simultaneously. The model is based on mixed-integer program-ming and its objective function represents the present value of costs of investment, energy and demand losses ofthe system which take place throughout the duration of the plan. The objective function is minimised subject toKirchhoff's current law, power capacity limits, and logical constraints. The model developed allows explicitconstraints of radiality and voltage drops to be included in its formulation.

List of symbols

NSE

NFF

N

= set of nodes associated with proposed loca-tions for building substations in the plan

= set of nodes associated with existing sub-stations

= set of proposed substation sizes to be built inthe plan

= set of proposed feeder routes (between nodes)to be built in the plan

= set of paths (between nodes) associated withexisting feeders in the initial network to beexpanded

= set of proposed feeder sizes to be built in theplan

= set of nodes that are connected to node j orthat are proposed to be connected to node j

= sum of the existing and future node sets= set of nodes that are proposed to be connected

to node i at, or before, time t= set of nodes that are proposed to be connected

to node k at, or before, time t= number of nodes thai are proposed to be

present in the network at time t= power flow from node j associated with a pro-

posed substation of size X', at time t, MVA= power flow from node j referred to an existing

substation= power flow through route (i, k) associated with

a proposed feeder size X, at time t, MVA= power flow referred to an existing feeder= line-to-line voltage at node i at time t, kV= line-to-line voltage at node k at time t, kV= continuous and non-negative variable= continuous and non-negative deviation or

slack variables= 1 if a substation with size X', associated with

node j , is to be built at time t, 0 otherwise

Paper 4964C (P9), received 6th December 1985

Prof. Gonen was formerly with the Department of Electrical and Computer Engin-eering, University of Missouri-Columbia, Columbia, MO 65211, USA. He is nowwith the Department of Electrical Engineering, California State University, 6000JStreet, Sacramento, CA 95819-2694, USA. Prof. Ramirez-Rosado is with the EscuelaTecnica Superior de Ingenieros Industrials de la Universidad de Zaragoza, Zara-goza, Spain

(Xjt)t)E

nSF

nFF

(Uik)

( cos <D)fMow

TikXt

(Uik)nH

1 if a feeder with size X, associated with route(i, k), is to be built at time t, 0 otherwise1 if the power flow over the future route (i, k) isgreater than zero at time t, 0 otherwisesame as above but referred to an existingfeeder over path (i, k)number of elements in the set NSF

number of elements in the set NFF

period of the plan yearspeak power demand at node; at time tMVA capacity limit of existing substationassociated with node jMVA capacity limit of an existing feederassociated with route (i, k)MVA capacity limit of a proposed substationof size X', associated with node;MVA capacity limit of a proposed feeder ofsize X, associated with route (i, k)maximum acceptable voltage drop from eachsubstation to each node, %length of feeder connecting node i to. node k,kmresistance of feeder of size X, O/kmreactance of feeder of size X, Q/kmpower factor of network at time tminimum line-to-line voltage, kV

= vj^i -

= constant

V loo

a(cos O),

maximum {(Uik)x where X elarge positive constant

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 7, NOVEMBER 1986 397

1 Introduction

Expansion studies of a distribution system have been donein practice by planning engineers. The studies were basedon the existing system, forecasts of power demands, exten-sive economic and electrical calculations, and the planner'spast experience and engineering judgement. However, thedevelopment of more involved studies with a large numberof alternative projects using mathematical models andcomputational optimisation techniques can improve thetraditional solutions that were achieved by the planners.As expansion costs are usually very large, such improve-ments of solutions represent valuable savings. For a givendistribution system, the present level of electric powerdemand is known and the future levels can be forecastedfor one stage, e.g., one year, or several stages. Therefore,the problem is to plan the expansion of the distributionsystem (in one or several stages, depending on data avail-ability and company policy) to meet the demand atminimum expansion cost.

The overall distribution system planning problem hasbeen dealt with by dividing it into the following two sub-problems that are solved successively:

(a) The subproblem of the optimal sizing and/or loca-tion of distribution substations. In some approaches, thecorresponding mathematical formulation has taken intoaccount the present feeder network either in terms of loadtransfer capability between service areas, or in terms ofload times distance. However, the full representation ofindividual feeder segments (i.e. the network itself) has notbeen considered.

(b) The subproblem of the optimal sizing and/or locat-ing feeders. Such models take into account the full repre-sentation of the feeder network but without taking intoaccount the former subproblem.

However, there are more complex mathematical modelsthat take into account the distribution planning problemas a global problem and solving it by considering mini-misation of feeder and substation costs simultaneously.Such models may provide the optimal solutions for asingle planning stage. The complexity of the mathematicalformulation substantially increases in multistage planningproblems and the process of resolution becomes more diffi-cult because the decisions for building substations andfeeders in one of the planning stages have an influence onsuch decisions in the remaining stages. This paper presentsa model for multistage planning problems. The model isbased on mixed-integer programming to provide anoptimum solution for the distribution system expansion.Practical problems have been solved by implementing themodel using a standard mathematical programmingsystem [1-3].

2 Classification of the selected models

In the models reviewed, the duration of the plan is definedas a single or several stages, where a single stage usuallymeans a single year of the plan. In the case of severalstages, the 'pseudodynamic methodology' considers theoverall duration of the plan without reference to the actualdemands in intermediate years of the plan. In this method-ology, each annual expansion problem is solved by a con-catenated optimisation process after the overall

optimisation has been achieved. In the completely-dynamic methodology, the building decisions for years inthe plan are reached simultaneously in the optimisation.Table 1 gives a comparison of the models based on theircharacteristics, e.g. whether or not substations and feedersare included separately or simultaneously in the opti-misation process. Also included are the radiality andvoltage drop considerations for the models that representthe feeder network explicitly as well as other aspects. Table2 gives a classification of the models selected based on themathematical techniques used (e.g. mixed-integer program-ming, dynamic programming etc.) together with the com-putational futures used (e.g. standard packages or specialalgorithms) to solve the practical distribution system plan-ning problems. The comparison of the models selectedincludes the following aspects.

2.1 Treatment of the distribution systemThe distribution system has been divided into two sub-systems for planning purposes, namely, the subsystem ofsubstations [4-10] and subsystem of feeders [11, 12].Therefore the sizes and locations of substations areplanned and later attention is focused on determining theoptimum size and structure of the feeder network.However, from the mathematical point of view, such anapproach will not necessarily yield a truly optimal solutionbecause each subsystem is solved independently of theother [13]. From a practical point of view, Wolff [14] says'there is a growing awareness that optimising the individ-ual elements of a system (feeders for a distribution system)is not the same as optimising the system as a whole.' Themodels of Masud [4, 5] and Adams [8] consider the feedernetwork only in terms of substation load transfers, whichmay be too simple a representation of feeders. A moreelaborate treatment of feeder networks consists in theapproach of load times distance which can be found inmodels developed by Afuso [6, 7] and Crawford [9, 10].On the other hand, some models [15-21] use the globalplanning problem approach for the complete distributionsystem and achieve more accuracy in its representation forthe optimization purposes. However, the ways of express-ing the objective function and the ways of solving theoptimal sizing and location problems are various. Forexample, Thompson and Wall [15], Sun et al. [16], Gonenand Foote [17, 18], Hindi and Brameller [19], Fawzi et al.[20] and El-Kady [21] solve the substation locationproblem and Gonen and Foote [17, 18], Hindi and Bra-meller [19] and El-Kady [21] also report the sizing sub-station problem explicitly in the objective function andtake into account the addition of new transformers to thesubstations [17, 18, 21]. However, models Hindi and Bra-meller [19] and El-Kady [21] do not have the costs ofenergy and demand losses for the substations, althoughthey should be included in the objective function. Theoptimal routing and sizing problem for feeders areincluded simultaneously in the models of Wall andThompson [12, 15], Gonen and Foote [17, 18], Hindi andBrameller [19] and Fawzi et al. [20]. Only the decisionvariables of the routing problem are included in the objec-tive function of the model given by Sun et al. [16]. Itappears that El-Kady [21] has attempted to include theoptimal feeder sizing aspects but the way of including thecorresponding variables in the objective function is notshown. The cost of losses are modelled with variousdegrees of accuracy by all authors [12, 15-21]. Reference[21] presents an approximate cost approach for step wisefeeder losses (by using the 0-1 integer variables directly inthe objective function), where decision variables for the

398 IEE PROCEEDINGS, Vol. 133, Pt. C, No. 7, NOVEMBER 1986

Table 1: Classification of the selected models

Period of One stagethe study

Several stages PseudodynamicCompletelydynamic

Power Feeders and substationsdistribution simultaneously

systemSubstations Fixed costs

Variable costs*Optimal locationproblemOptimal sizingproblemArea loadtransfersLoad x distance

Feeders Fixed costsVariable costs*Optimal routingproblemOptimal sizingproblemRadialityVoltage drops

Ada

ms

[11]

Yes

Yes

No

YesYesYes

Yes

NoNo

to1

12

,'W

all a

nd

Tho

mps

on [

Yes

Yes

YesYesYes

No

NoYesYes

Yes

* • *

No

Sun

eta

l. [1

Yes

Yes

Yes

YesYesYes

No

YesYesYes

No

NoNo

CO*

Gon

en a

nd

Foo

te [

17, 1

Yes[17]

Yes[18]

Yes

YesYesYes

YesG

YesYesYes

Yes

NoNo

57

Hin

di a

ndB

ram

elle

r [1

Yes

Yes

YesNoYes

Yes

YesNPPYes

Yes

***No

[20]

Faw

zi e

t al.

\

Yes

Yes

YesYesYes

No

NoYesYes

Yes

* * •

* *

El-K

ady

[21

Yes

OS

Yes

YesNoYes

YesG

YesAPYes

S

ASROSN

Mas

ud [

4]

Yes

No

YesNoNo

YesGYes

Mas

ud [

5]

Yes

No

NoNoNo

YesGYes

(—i r**<o *~

Af u

so e

t al.

Ada

ms,

et a

>

Yes

No

YesNoYes

YesG

Yes

Ada

ms

[8]

Yes

No

NFNFYes

Yes

Yes

Cra

wfo

rdan

d H

olt

[9,

Yes

No

NoNoEP

EP

Yes

OS = only in subnetworks* = energy and demand losses as variable costs* * = not modelled but checked in the computational procedure* * * = enforced in the computational procedureASR = algorithm sets radiality automaticallyNF = no formulationG = growingEP = empirical procedureAP = rough approach suggestedS = seems to deal w i th sizing but method of including variables in objective funct ion is not presentedNPP = suggested but not used in practical problemsN = formulation seems to be incorrect

optimal sizing of feeders do not seem to be considered. Themodel developed by Gonen and Foote [17] includes allthe aspects discussed so far. It is a complete and com-prehensive model due to the fact that it incorporates themajor planning decisions and has a detailed objective func-tion. However, the radiality and voltage drop consider-ations are not included in this model.

2.2 Planning periodMost of the models developed consider only a single timestage as the planning period [4, 9-12, 15-17, 19-21]. Suchmodels are known as static models [22]. Frequently, inpractical applications, the algorithm used can see the plan-ning problem as a sequence of yearly expansions so thateach yearly expansion is a separate expansion problem.However, this may not lead to the optimisation of thewhole plan because, in general, partial optimal solutionscannot guarantee an overall optimal solution. Therefore,there have been efforts to develop more complex methodsto solve the distribution system planning problem inseveral stages rather than in a single stage. Such modelsare known as dynamic models [22]. For example, Adams[8] and Afuso [6, 7] have developed models to study thesubstation planning problem (without explicit representa-tion of the feeder network) in several planning stages

(years) using dynamic programming. Adams has alsodeveloped a model [11] to study the time-phased networkplanning problem. The model has been used by Adams tostudy subtransmission systems rather than distributionsystems. Recently, El-Kady [21] has developed a model tostudy the distribution system planning problem over asingle time stage that corresponds to the overall timespanof the plan. After the study has been carried out, theresulting network is partitioned heuristically in severalsmall subnetworks (e.g. based on the planner's judgement).Afterwards, a time-phased analysis is done based on theprocedure to obtain optimal plans for each subnetwork.However, as has been pointed out previously by El-Kady[21], such partitioned analysis is not equivalent to theoverall unpartitioned one which would provide the ulti-mate accuracy. Sun et al. [16] have developed a single-stage model and pseudodynamic methodology fordistribution system planning, where the overall plan isdivided into two phases: in the first phase, the single-stagemodel is applied to achieve a solution to meet the requireddemand at the end of the total study period consideringthe complete set of network components (i.e., feeders, sub-stations, etc.) which would be built during the studyperiod. The period of the study is used as a single stage inthe optimisation process. In the second phase, the single-


Table 2: Classification of the selected models

Mathematicaltechniques

ComputationalFeatures

Remarks

Mixed-integerlinearprogramming

Dynamicprogramming

Others

Mathematicalprogrammingsystems

Specialalgorithms

Ada

ms

[11

]

Yes

YesXDLC M 1 0

^^o

CN

Wal

l and

Tho

mps

on [

Yes

CD

aCOcre

"O co

SJTJ ~C J;3 °o &>

JO reTJ TJ

ch a

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c rere Q.

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TJ

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l<he

use

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Sun

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1

Yes

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Q." t -

</}

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ch a

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tate

c rere a.CO o

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m

Q.CD

! (

00

Gon

en a

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Foo

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17

,1

Yes

Yes [17]MPSX-MIP

Hin

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Bra

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ler

[1

Yes

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EQ .

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coC

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Faw

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El-K

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[21

Yes

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ch a

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Mas

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4]

Zero-oneprogramming

Unknown

Mas

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5]

Zero-oneprogramming

Nopracticalproblems

1—1 •

Af u

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dam

s, e

ra;

Yes

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[8]

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stage model is applied to each intermediate network,expanding from the preceding year, choosing networkcomponents from the set of equipment that was selectedfrom the first phase. Therefore, the planning procedure isbased on this successively concatenated optimisationprocess and provides a solution that is not necessarily theoptimal one. Gonen and Foote [18] developed a model

that is completely dynamic because the building timingdecisions are included in the objective function. However,the model has not been tested with multistage practicalexamples.

2.3 Radiality and voltage-drop considerationsThe planner can be interested in achieving a feeder


network solution which has radial characteristics, depend-ing on the geographical location of the network and thecompany's criteria and concern for the security of the elec-trical service. (Frequently, the topological structure of dis-tribution feeders resembles the form of an arborescentgraph [22, 23].) The radiality constraints have not beenincluded in some of the mathematical models studied (e.g.References 11 and 16-18). Others (e.g. References 12, 19and 20) use special rules or heuristic methods in their algo-rithms to enforce the radiality condition into their solu-tions. Reference 21 reports that the algorithm of the modelautomatically enforces the radiality condition in the tree ofthe branch and bound search. However, as pointed out byHindi and Brameller (in their discussion of Reference 12),sometimes such heuristic rules for radialising solutionscould lead to suboptimal solutions. The radial solutionsgiven in the selected literature usually show topologicalstructures in the form of arborescent branches emanatingfrom each substation (with the exception of Reference 21,where, in particular, no arborescent branches are shown).Most models reviewed [11, 12, 16-19] do not includevoltage drop constraints in the optimisation procedure.The method proposed in Reference 20 checks for voltagedrops in its computational procedure but it does not havethe necessary voltage drop constraints in the model itself.The method proposed in Reference 21 treats the overallplanning period of several years as a single time periodwithout considering voltage drops. After the resultingnetwork is partitioned into smaller subnetworks heuristi-cally, each subnetwork is analysed by using a time-phasedoptimisation procedure that includes voltage drop calcu-lations only for some of its nodes. However, the actuallevels of power flow in feeders are not considered in thesecalculations. Instead, a stepwise relationship betweenpower flow levels and impedances of standard conductorsizes is used in the voltage drop calculations. If the powerflow between two nodes is zero, the voltage drop equationsproposed in Reference 21 would force the voltage levels ofthe two nodes to be the same which is usually not the case.

2.4 Mathematical techniques usedThe basic mathematical technique used to model the dis-tribution planning problem is mixed-integer programming[4-7, 11, 15-21]. It is well suited to distribution planningbecause the decisions to build or not to build can be rep-resented by the 0-1 integer variables. However, there arealso other techniques that have been used successfully. Forexample, Adams [8] and Afuso [6, 7] used dynamic pro-gramming and Crawford [9, 10] used a technique thatcombines the shortest path technique with the transporta-tion algorithms. It is interesting to note that in most of themodels reviewed [6-10, 12, 15, 16, 19-21] practical prob-lems are solved by using special algorithms, e.g. a branch-and-bound algorithm [6, 7, 12, 15, 16, 19-21], sometimescoupled with transhipment techniques. Models developedby Gonen and Foote [17, 24] and Adams [11] use stan-dard mathematical programming systems.

3.1 Cost functionThe distribution system expansion problem involves a costfunction that represents the present value of the decisionsto build substations and feeders throughout the studyperiod. It includes the investment cost, operating andmaintenance expenses, book depreciation etc., includingtheir after tax effects, the cost of energy and demand lossesthat take place in the distribution system as well as theresulting ones in the transmission and generation systemsthat are connected to it. The present value associated withbuilding a substation at time t is the sum of the presentvalue of fixed and variable costs. The fixed costs mainlyinclude the investment cost and the cost of energy anddemand losses in the iron of transformers, whereas thevariable costs include the costs of demand and energylosses in the copper of transformers. Thus, the resultingtotal present value is a quadratic function of the annualpeak power flows (Xjz)r supplied by the substation plus aconstant (KSjt)x.. Therefore, the present value of a futuresubstation to be built with size X e Nx- at node j e NSF attime t can be expressed as

jdv = (KSjt)x. + X (K'Sjt)r(Xjz)l (1)

where t = 0, 1, 2,..., Th_l and T = t, t + 1,..., Th_x or, bylinearisation approximately

(PWSjt)x, = (KSjt)x. (2)

Similarly, the present value associated with building afeeder at time t is the sum of the present value of a fixedcost (which represents mainly the investment cost) andvariable costs (which represent costs of energy and demandlosses). The present value of a future feeder of size X e Nxto be built on route (i, k) e NFF at time t can be expressedas

{PWFikt)x = (Kfikt),Th-i

(K'fikzUXikT)f

or, by linearisation, approximately

(3)

(4)

3.2 Objective functionThe optimisation problem, which corresponds to themultistage planning of the distribution system, involves theminimisation of an objective function. Such a function isthe result of a mixed-integer programming model of thecost function mentioned above by using continuous powerflow variables and integer 0-1 variables that represent thedecisions of building substations and feeders. The objectivefunction of the mixed-integer programming model is

3 Planning model

In this section, a mixed-integer model is presented to solvethe optimal sizing, timing and location of distribution sub-station and feeder expansion problems. The model allowsexplicit constraints of radiality and voltage drops in its for-mulation to be included, as are presented in Appendix 8.

f! Z (= O jeNSE

Y Z Z» = 0 ( i , k ) e N F F * e N

x {(FiktUYikt)x + (CiktU(Xikt)x

Y Z (ciktu(xikt)E + (Xt = 0 (i, k) e NPE

(5)

The first summation of eqn. 5 represents the component ofthe cost function due to the building of new substations.

1EE PROCEEDINGS, Vol. 133, Pt. C, No. 7, NOVEMBER 1986 401

Defining

and

and introducing the variable (YJt)x, for all the proposedsubstation locations, eqn. 2 becomes the terms shown inthe first square bracket of eqn. 5. In the objective function,the variable costs of existing substations are expressed bythe second summation of eqn. 5. The third summation ofthis equation represents the component due to the buildingof new feeders. Defining

Vai \1(Kfikt)x = (Fikt)x and (K

and introducing the variable (Yikt)x, eqn. 4 becomes thethird summation in eqn. 5. Note that bidirectional powerflow variables (Xikt)x and (Xkit)x are considered betweennodes i and k. The last summation in the objective func-tion represents the variable costs of existing feeders.

3.3 Mathematical constraintsThe objective function of the model is subject to mathe-matical constraints, to be minimised in the optimisationprocess. Such constraints relate technical conditions thathave to be met according to the requirements of the multi-stage expansion problem of the distribution system. Theconstraints are the following:

(a) Power demand constraints: The power demands ofthe load centres must be met throughout the planningperiod. Kirchhoff's current law is applied to each node foreach year. Therefore

Z Z l(XtJdx ~i eNj Xe Nx

= Sjt - (Xjt)x' (6)

for all j e N(b) Power capacity constraints: Limits for the maximum

capacity of the power that can be carried by each feederand supplied by each substation are specified according totheir proposed sizes. For a future substation,

(7)

(8)

(9)

(10)

(ID

(12)

0 ^ (Xjt)x. < (£/,-)

for all j e NSF and for all X e Nx.For an existing substation,

0 ^ (Xjt)E ^ (Uj)E for all jeNSE

For a future feeder on route (i, k),

and

0 ^ (Xkit)x ^ (Uik)

for all (i, k) e NFF and for all X e Nx.For an existing feeder over route (i, k),

0

and

(Xikt)E ^ (Uik)E for all (i, k) e NFE

0 ^ (Xkit)E ^ (Uik)E for all (i, k) e NFE

(c) Logical constraints: These are the mathematical con-straints linking the planning decision variables for buildingsubstations or feeders. For future substations, the follow-ing constraints guarantee that only one substation of agiven size can be built at a given location during the

period of the plan:Th-l

Z Z (Yjdv* fo ra l l ;6N S f (13)

The total number of future substations can be limitedusing the following constraint:

Y Z Z (Yjx)x. ^ nSFt = 0 je NSF *•' eNy

(14)

For future feeders, the following constraints guarantee thatonly one feeder of a given size can be built on feeder route(i, k) during the period of the plan:

Th-l

E It = 0 XeN

1 for all (i, k) € NFF (15)

The total number of feeders to be built can be limited byusing the following constraint:

Z(i,k)eNFF

Z (UFF

(16)

(d) Voltage drop constraints: The maximum allowablevalues of voltage drops between substations and demandcentres can be included as explicit mathematical con-straints [3], as shown in Appendix 8.

(e) Radiality constraints: If the radiality of the networkis a prerequisite, these constraints can be added to themodel [3] to guarantee such radiality in the optimal solu-tion. The constraints are shown in Appendix 8.

The model developed also includes reconductoring deci-sion variables for feeders and size increasing decision vari-ables for substations in the years of the plan [3].Constraints (a), (b) and (c) are defined as the basic con-straints. Therefore the basic model is defined as the onewhich has the objective function that is subject to only thebasic constraints.

4 Application of the model

The model developed has been tested to solve distributionsystem expansion problems using the mathematical pro-gramming system MPSX/370 R1.6 and an IBM 3081 com-puter at the University of Missouri-Columbia [1-3]. Thedata have been provided by Hidroelectrica Espanola SA,Madrid, Spain. The general inflation rate considered insuch problems is based on Reference 25. This section pre-sents some of the example problems.

Case 1: Consider the future feeder to be built between(demand and/or transhipment) nodes i and k on route(i, k). The transhipment node is the node which can receiveand transfer electric power without consuming it. Therequired voltage drop constraints are:

Dikt

Dkit

J - l(Vk)t + Dkit-]

= A Z£>up[l - **«]

for all (i, k) e NFF.If both nodes i and k are demand nodes at time t,

Dup = (Vno

Otherwise

up v "nom)

(17)

(18)

(19)

(20)

(21)


If node i is a demand node at time t, (Vk)t for all a e (Nt)k (27)

Otherwise

- >r E <* -,)inom/l / j via,/ I

La 6 (Nr), J

(K),

(K.wX^iJ ^ (K), for all a e (Nf),.

If node A: is a demand node at time t,

V < (V} < VHow = \vk)t " rnom

Otherwise

La 6 (N,)k J

(22)

(23)

(24)

(25)

(26)

Fig. 1 shows an existing distribution system to beexpanded over a period of nine years and the routes pro-posed for a 20 kV feeder expansion as well as designatedlocations for two future 132/20 kV substations C and D.The planners proposed LAI 10 and LA56 feeder sizes inmost of the future routes, and the substation sizes given inTable 3. Table 4 gives the annual peak power demandrequirements of the distribution system over the nine yearperiod.

Case 2: Consider the existing feeder (with conductor sizeXE) between (demand and/or transhipment) nodes i and kon route (i, k). The required voltage drop constraints are:

[TO, + D,J

node A

node 14

P -rfnodei XT'/ / \ node 12

node 2

node 3

O,d \nodeC

xE ~ (Xkit)XE]GikXEt (28)

0 ^ Dikt ^ Dup[l - (zikt)E-] (29)

0 ^ Dkit ^ Dup[\ - (z/fct)£] (30)

for all (i, k) e NFE. Also include eqns. 20-27.

Case 3: Consider the future feeder to be built between afuture substation located at node j and a demand or tran-shipment node k located on route (j, k). The requiredvoltage drop constraints are:

0 ^ Djkt ^ Dup[l - zjkt-]

0 ^ Dkjt ^ Dup[\ - zjkj

for all; e NSF and (j, k) e NFF

Table 3: Proposed substation sizes

= I [(^,)A - (^,)JGjUr (31)

(32)

(33)

node B

Fig. 1 Initial system and proposed routes for the expansion

initial systemproposed routes

Substation

ABCD

Existingcapacity, MVA

2020

Proposedcapacity tobe built, MVA

3020

Proposedcapacity tobe added, MVA

20

Table 4: Annual peak power demand requirements, MVA

Node

ABCD123456789

1011121314151617181920

InitialSystem

7.3611.30

——4.630.723.370.410.261.074.100.65————————————

Yeari

7.7012.06

——4.050.782.580.320.281.174.040.721.141.56——————————

Year 2

8.0912.87

——4.430.862.830.350.311.283.090.671.251.701.67—————————

Year 3

8.5213.7410.02

—4.850.703.090.380.341.053.380.731.031.861.780.881.12———————

Year 4

9.0014.6810.02

—3.450.773.380.410.370.923.700.601.122.041.910.931.153.051.62—2.16———

Year 5

9.5315.6710.02

—3.780.842.780.340.331.004.050.651.232.232.050.991.182.521.620.942.201.89——

Year 6

10.1116.739.004.594.130.923.040.370.361.103.330.721.351.832.211.051.222.661.620.992.241.941.45—

Year 7

10.7417.869.004.594.521.013.330.410.391.203.640.781.472.002.381.131.262.811.621.052.291.991.553.79

Year 8

11.4219.06

9.004.594.951.113.640.450.431.323.980.861.612.192.581.201.312.971.621.132.352.041.673.79

Year 9

12.1720.33

9.004.595.421.213.980.490.471.444.360.941.772.402.801.291.353.161.621.222.402.101.813.79


If node k is a demand node at time t and if the demandcentre of the substation has an associated demand that isnot zero at time t, include eqn. 20. Otherwise include eqn.21.

If the demand centre of the substation has an associateddemand that is not zero at time t,

',), ^ Vnnm (34)

Also include

Vlow

Otherwise

(Kom)\\_a

(Vlow)(zja

5.51 MVA

/

\\9

Q5

I (zjj\>{V)te (N,)j J

,) ^ (Vj)t for all a e {Nt)j

A 018.13 MVA

/ \4.92 MVA

/ \\ 2

I.46MVA \ A . 1 4 M V A

\0.32MVA T

\ . , . 11.56 MVAo ^ ion—'

10.28 MVA 7 D

V6

\1.45 MVA>

4.04 MVA /

8CJ

^ /4 .76 MVA

B I/M 18.27 MVA

Fig. 2 Optimal solution for first year

19.51 MVA

6.03 MVA

1.67 MVA

1.59 MVA\

B l^M 19.89 MVA

Fig. 3 Optimal solution for second year

404

(35)

(36)

- t(37)

(38)

for all j e NSF and (j, k) e NFF. For node k include eqn.25, or eqns. 26 and 27, depending on the characteristics ofthe node at time t.

Case 2 includes case 1 and also 20 MVA capacity to beadded to substation B. Case 3 includes case 2 and, further-more, considers a 30 MVA substation to be built in sub-station D. Figs 2-10 show the optimal solution of case 1.

The basic model has been also used to solve case 2 andcase 3. Table 5 gives the resulting values of the objectivefunction for the studied cases.

15.98MVA

5.88 MVA 1.58 MVA

20.51 MVA

i.78 MVA

130.73 MVA

B | / ^ | 16.99 MVA

Fig. 4 Optimal solution for third year

A IA118.69 MVA

7.99 MVA/ \ 1.70 MVA

3.38 MVA

23.08MVA

1.91 MVA

13

B | A | 19.51 MVA

Fig. 5 Optimal solution for fourth year

IEE PROCEEDINGS, Vol. 133, Pt. C, No. 7, NOVEMBER 1986

A LA] 20 MVA

0.72 MVA

20 MVA

1.11 MVA

V30.94MVA\ 3.89 MVA

1 6 ° " ^ ^ 10 x c

3.17 MVA"^47^25.84MVA

2.05 MVA

B O 20 MVA

Fig. 6 Optimal solution for fifth year

A j/NJ 17.56MVA

5.48MVA/ \L97MVA

14

?2.66 MVA/CH8

4.60 MVA I

n P43^U 1 S\

10.84MVAO/^

1.22 MVA)Q20

1.22 MVA\

\613

/ 12

1 d °\ 135MVA

b 9

/>5 CX\ 0.37 M

10.07 MVA\ 1.62 MV/

\ 6

^ 1 . 0 3 MVA

JD21.05 MVA

3

- ^ i ^ M V ? -307 yPzSxy'

0.72 MVA/ (

7a 8

\<^2.24MVA

.3.04MVA

\ c

Ifl.2\ MVA

11

29 MVA 1 6 , 13 M V A \ 3.64 MVA

1 3 v .0.94 MVA

B 0 2 0 MVA

Fig. 9 Optimal solution for eighth year

21.86 MVA

2.50 MVA

5.26 MVAD

19.30 MVA

5.14 MVA

1.35 MVA

.22MVA\3.98MVAC

0.49 M V A ^ ! 6 2 Q M V A ^ ^ ] 2 8 9 5 M V A

2.80 MVA

13'0.33 MVA

B 12] 20 MVA

Fig. 10 Optimal solution for ninth year

B E l 20 MVA

Fig. 7 Optimal solution for sixth year

18.87 MVA

2.14 MVA

D16.18 MVA

5.05 MVA

1.05MVA\3.33MVA

C0.41 M V A ^ v ^ 3 ^ y « .^^17^125.76 MVA

1.35 MVA 3.17 MV>1.62 MVA

20 \ o - 1915

6 0.78 MVA1.26 MVA\ AP.15MVA

13

B IS] 20 MVA

Fig. 8 Optimal solution for seventh year

Table 5: Optimal values of the objective function

Case Value, million pesetas

1 933.102 921.233 875.79

Note that the solutions of case 2 and case 3 represent asaving of 11.87 million pesetas (1.27%) and 57.31 millionpesetas (6.14%) with respect to case 1. This illustrates thatmore complex alternatives (e.g. more substation sizesand/or more possible additions to substation capacity canlead to a cheaper expansion of the distribution system.

Case 4: Consider a future feeder to be built between anexisting substation located at node j and a demand ortranshipment node k located on route (j, k). The requiredvoltage drop constraints are:

l(Xjkt)x - (Xkjt)^GjkXt (39)- l(Vk)t +

for all j e NSE and for all (j, k) e NFF. Also include eqn.33. If node k is a demand node at time t, include eqn. 20.Otherwise include eqn. 21. Also, for node k, include eqn.25, or eqns. 26 and 27, depending on the characteristics ofsuch node at time t.


Case 5: Consider the existing feeder (with conductor sizeXE) between an existing substation located at node j and ademand or transhipment node k located on route (j, k).The required voltage drop constraints are:

(Ynom) - [ T O , + Dkjt] = [_(Xjkt)XE - (Xkjt)XE]GjkxEt (40)

0 < Dkjt ^ D[l - (zyk,)£] (41)

for a\\j e NSE,(j, k)e NFE

As before, if node k is a demand node at time t, includeeqns. 20 and 25. Otherwise, include eqns. 21, 26 and 27.

Case 6: Consider the existing feeder (with conductor sizeXE) between a future substation located at node j and ademand or transhipment node k located on route (j, k).The required voltage drop constraints are:

- [_(Vk)t

0 ^ Djkt ^ D u p [ l - (zjkt)E-]

0 ^ Dkjt ^ Dupll - {zjkt)E-\

(42)

(43)

(44)

for all j e NSF and for all (jf, k) e NFEIf the node k is a demand node at time t and if the

demand centre of the substation has an associated demandthat is not equal to zero at time t, include eqn. 20. Other-wise, include eqn. 21. If the demand centre of the sub-station has an associated demand that is not equal to zeroat time t, include eqn. 34. Otherwise, include eqns. 35 and36. Also include eqns. 37 and 38. For node k include eqn.25, or eqns. 26 and 27, depending on the characteristics ofsuch a node at time t.

5 Conclusions

The optimal sizing and location of distribution substationsand feeders can be studied by mathematical models using asingle planning stage with different levels of accuracydepending on models. Only a few models have been devel-oped to take into account the optimal timing for buildingdistribution substations and feeders. These models arebased on either pseudodynamic or completely dynamicmethodology. Some of them do not take into accountsizing and location considerations.

It has been found that the radiality considerations havenot usually been mathematically formulated in the models,but they are artificially enforced in the optimisationprocess. Furthermore, in the reviewed literature, only a fewmodels contain the voltage drop constraints so that suchconstraints can be included directly in the optimisationprocess.

A multistage planning model has been presented in thispaper. The model developed takes into account the deci-sion variables for substation and feeder building simulta-neously and provides the optimal solution for the sizing,location and timing decisions. Based on the calculations ofpresent value, a cost function has been formulated to beused as an objective function in a mixed-integer program-ming model. The objective function can be minimisedsubject to technical constraints (i.e., Kirchhoff's currentlaw, power capacity limits and logical constraints). Fur-thermore, the model developed allows to explicit con-straints of radiality and voltage drops to be included in itsformulation.

The model has been tested with real distribution systemexpansion problems. Some of them have been included.The results have illustrated that more complex planning

alternatives (in terms of more possible substation sizesand/or more capacity additions for substations), can leadto cheaper distribution system expansions [1, 3]. This canachieve savings in the overall expansion cost, as well assavings in the man-hours involved in the planning of theexpansion.

6 Acknowledgments

The authors would like to thank the Hidroelectrica Espaii-61a SA, Madrid, Spain, for providing data for the testproblems as well as the Ministerio de Educacion yCiencia/Fulbright Program for supporting the research.

7 References

1 GONEN, T., and RAMIREZ-ROSADO, I.J.: 'Optimal expansion ofpower distribution systems. Part I: Basic Modeling'. 12th IASTEDInternational Conference on Applied Simulation and Modelling.Montreal, Canada, 3rd-5th June 1985

2 GONEN, T., and RAMIREZ-ROSADO, I.J.: 'Optimal expansion ofpower distribution systems. Part II: Voltage drop and radiality con-straints'. 12th IASTED International Conference on Applied Simula-tion and Modelling, Montreal, Canada, 3rd-5th June 1985.

3 RAMIREZ-ROSADO, I.J.: 'Completely-dynamic optimal expansionof electrical power distribution system planning'. Ph.D. Thesis, Uni-versity of Missouri-Columbia, 1985

4 MASUD, E.: 'An interactive procedure for sizing and timing distribu-tion substations using optimization techniques', IEEE Trans., 1974,PAS-93, (5), pp. 1281-1286

5 MASUD, E.: 'Distribution planning: state-of-the-art and extensionsto substation sizing', Electr. Power Syst. Res., 1978,1, pp. 203-212

6 AFUSO, A., GEREZ, V., and RODRfQUEZ, A.: 'An integratedsystem for distribution planning'. Proceedings of the 4th IEEE Inter-national Symposium on Large Engineering Systems, Calgary, Alberta,Canada, 9th-l lth June 1982

7 ADAMS, R.N., AFUSO, A., RODRfGUEZ, A., and GEREZ, V.: 'Amethodology for distribution system planning'. 8th Power SystemsComputation Conference, Helsinki, Finland, 19th-24th August 1984

8 ADAMS, R.N., and LAUGHTON, M.A.: 'A dynamic programming/network flow procedure for distribution system planning'. Pro-ceedings of the 8th Power Industry Computer applications (PICA)Conference, Minneapolis, MN, USA, 3rd-6th June 1973, pp. 348-354

9 CRAWFORD, D.M., and HOLT, S.B.: 'A mathematical optimizationtechnique or locating and sizing distribution substations, and derivingtheir optimal service areas', IEEE Trans., 1975, PAS-94, (2), pp. 230-235

10 HOLT, S.B., and CRAWFORD, D.M.: 'Distribution substation plan-ning using optimization methods'. IEEE Tutorial Course, 1976,pp. 69-76.

11 ADAMS, R.N., and LAUGHTON, M.A.: 'Optimal planning of powernetworks using mixed-integer programming. Pt 1-Static and time-phased network synthesis', Proc. IEE, 1974,121, (2), pp. 139-147

12 WALL, D.L., THOMPSON, G.L., and NORTHCOTE-GREEN,J.E.D.: 'An optimization model for planning radial distribution net-works', IEEE Trans., 1979, PAS-98, (3), pp. 1061-1068

13 LASDON, L.S.: 'Optimization theory for large systems' (Macmillan,New York, 1970)

14 WOLFF, R.F.: 'The new electronic frontier-distribution design',Electr. World, May 1982, pp. 65-80

15 THOMPSON, G.L., and WALL, D.L.: 'A branch and bound modelfor choosing optimal substation locations', IEEE Trans., 1981, PAS-100, (5), pp. 2683-2688

16 SUN, D.I., FARRIS, D.R., COTE, P.J., SHOULTS, R.R., and CHEN,M.S.: 'Optimal distribution substation and primary feeder planningvia the fixed charge network formulation', ibid., 1982, PAS-101, (3),pp. 602-609

17 GONEN, T., and FOOTE, B.L.: 'Distribution-system planning usingmixed-integer Programming', IEE Proc. C, Gen. Trans. & Distrib.,1981,128, (2), pp. 70-79

18 GONEN, T., and FOOTE, B.L.: 'Mathematical dynamic opti-mization model for electrical distribution system planning', Electr.Power & Energy Syst., 1982, 4, (2), pp. 129-136

19 HINDI, K.S., and BRAMELLER, A.: 'Design of low-voltage distribu-tion networks: a mathematical programming method', Proc. IEE,1977,124, (1), pp. 54-58

20 FAWZI, T.H., ALI, K.F., and EL-SOBKI, S.M.: 'A new planningmodel for distribution systems', IEEE Trans., 1983, PAS-102, (9),pp. 3010-3017


21 EL-KADY, M.A.: 'Computer-aided planning of distribution sub-station and primary feeders', ibid., 1984, PAS-103, (6), pp. 1183-1189

22 GONEN, T : 'Electric power distribution system engineering'(McGraw-Hill, New York, 1985)

23 PELISSIER, R.: 'Les reseaux d'energie electrique. Architecture etdeveloppements des reseaux. Vol. 3' (Bordas, Paris, France, 1975)

24 GONEN, T., FOOTE, B.L., and THOMPSON, J.C.: 'Developmentof advanced methods for planning electric energy distributionsystems'. US Department of Energy, Report COO-4830-3 Oct. 1979(Available from the National Technical Information Service, USDepartment of Commerce, Springfield, VA, USA)

25 'Programa Economico a Medio Plazo: Escenarios Macroeconomicospara la Economia Espanola'. Secretaria General de Economia y Pla-nificacion, Madrid, Spain, 1984

8 Appendixes

8.1 Development of voltage drop constraints

8.1.1 Feeder voltage drop: The percent voltage drop of afeeder over its entire length can be expressed as [22]

%AF =

where

l(R cos <D + X sin 0)100 (45)

S = 3-phase apparent power, MVA/ = feeder length, km

R = resistance of the feeder, Q/kmX = reactance of the feeder, Q/km

cos O = power factor,Vnom = line-to-line nominal voltage, kV

8.1.2 Voltage drop constraints: The constraints (that theinteger variables should always meet) are:

4 ZUeN

+

l(Xikt)xNx

for all (i, k) e NFF, t = 0, 1, 2 , . . . , Th

(zikth > aEl(Xikt)E + (Xkit)E-] + bE

(ZJE ^ HL(Xikt)E + (Xkit)E-]

(46)

(47)

(48)

(49)

for all (i, k) e NFE

Explanations about the feasible regions determined bythese constraints can be found in Appendix 8.3.

8.2 Development of radiality constraintsThe radiality constraints are:

Z ((i, k) 6 NFE

Z(i, k) e NFF

-["SE+iojNsFxijYjt)x,t = 0,l,2,...,Th_1 (50)

8.3 Feasible regions for integer variablesFor illustrative proposals, consider that Z>ieN,i+ (ATfcfJJ is defined as a variable x. Also consider that

(zikt) is defined as a variable z. Then constraints 46 and 47dictate the feasible region for the variable z as shown inFig. 11. As z is an integer 0-1 variable, z can take the value0 from x = 0 to x = H~l, or z can take the value 1 fromx = H'1 to x = (Uik)m. Note that H is a large positiveconstant which causes its inverse H~l to be very close tozero.

8.4 Example of the application of the voltage dropand radiality constraints

These constraints have been tested to solve test problemsusing the MPSX package [3]. The data have been provid-ed by the Hidroelectrica Espanola SA, Madrid, Spain. One

x=0 .x=H"1

Fig. 11 Feasible region

x=(U i k ) m

node A

node 1

node A - t ) node 5

J_ j nodeB

Fig. 12 Initial system and proposed routes for the expansion.

20 kV ISh5.30MVA 20kV 018.42 MVA7 . 4 3 M V A / \ 6 . 6 0 M V A 8 . 5 5 M V A / \ 7 . 5 9 M V A

19.26kv/ \.19.29kV 19.15kv/ \ i9.18kV

3.32 MVAy

18.91 kVc

2.07 MVA18.97kV 18.74kV

2.38MVA<!>19.82kV

2.07 MVA

2OkV|Al2.O7MVA

>19.80kV

2.38 MVA

20kV(Sl2.38MVA

b

20 kV tTq 19.93kV 20 kV 0 21.89MVA

8.51 MVA/ \7.92MVA 9.78 MVA/ \ 9 1 1 MVA

19.16 KV i ^15 MVA 1 9 .o3kv/ ^19.02kV

3.53 MVA

19.83kV 1 1 9

18.81kV 18.65 kV2.19 MVA

.73 MVA

18.63kV2.53 MVA19.68 kV19.81 kV

1.59 MVA\ 13.27MVA 1.83 MVA\ 3.77MVA

20kVO5.99MVA 20kVE110.11 MVA

c d

Fig. 13 Optimal solutiona first year b second year c third year d fourth year


of these problems is presented here. Fig. 12 shows the grows throughout a 4-year period. Fig. 13 shows theinitial system, the routes proposed for future feeders and optimal solution obtained by using the basic model andthe proposed location for a new substation B. The pro- the voltage drop and radiality constraints simultaneously,posed sizes are 10 and 15 MVA for substation B and The optimal substation size is 15 MVA and the objectiveLAI 10 for the future feeders. The distribution system function value is 776.41 million pesetas.


review of distribution system planning models: a model for optimal multistage planning

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