1Distribution Network Planning Using a ConstructiveHeuristic Algorithm

Marina Lavorato, Marcos J. Rider, Member, IEEE, Ariovaldo V. Garcia, Member, IEEE, andRuben Romero, Senior Member, IEEE,.

Abstract An optimization technique to solve distributionnetwork planning (DNP) problem is presented. This is a verycomplex mixed binary nonlinear programming problem. Aconstructive heuristic algorithm (CHA) aimed at obtaining anexcellent quality solution for this problem is presented. In eachstep of the CHA, a sensitivity index is used to add a circuit ora substation to the distribution network. This sensitivity indexis obtained solving the DNP problem considering the numbersof circuits and substations to be added as continuous variables(relaxed problem). The relaxed problem is a large and complexnonlinear programming and was solved through an efficientnonlinear optimization solver. A local improvement phase anda branching technique were implemented in the CHA. Results oftwo tests using a distribution network are presented in the paperin order to show the ability of the proposed algorithm.

Index Terms Distribution network planning, constructiveheuristic algorithm, mixed binary nonlinear programming, powersystems optimization, KNITRO, AMPL.

I. INTRODUCTION

THE main objective of the distribution network planning(DNP) problem is to provide a reliable and cost effectiveservice to consumers while ensuring that voltages and powerquality are within standard range. Several objective functions,including new equipment installation cost, equipment utiliza-tion rate, reliability of the target distribution system, andloss minimization should be evaluated considering increaseof network loads and newly installed loads for the planninghorizon. New optimization models, new techniques aimed tofind optimal or then good solutions for the DNP problem arestill needed, considering the location and size of substationsand circuits, construction of new circuits as well as newsubstations or alternatively the reinforcement of the existingones in order to permit a viable operation of the system in apre-defined horizon [1].

The DNP problem can be divided in three time ranges:short-range (1 to 4 years), long-range (5 to 20 years) andhorizon-year (20+ years) planning periods, that include single-period models and multi-period models [2]. Some DNP re-search has included also, the primary-secondary distributionplanning [3] and DNP with distributed generation [4] and [5].

Manuscript sent December 1, 2008. This work was supported by theBrazilian Institutions CNPq and FAPESP.

M. Lavorato, M. J. Rider and A. V. Garcia are with the Department ofElectrical Energy Systems, University of Campinas, Campinas - SP, Brazil.

R. Romero is with the Faculty of Engineering of Ilha Solteira, PaulistaState University, Ilha Solteira - SP, Brazil.

(E-mails: {lavorato, mjrider, ari}@dsee.fee.unicamp.br, ruben@dee.feis.unesp.br).

The circuits (location, size and type of conductor) andsubstation (location) short-range planning problem to a single-period in the primary distribution network, is modeled inthis paper as a mixed integer (binary) nonlinear programming(MBNLP) problem, where the binary decision variables rep-resent the construction (or not) of new circuits or new substa-tions, and the objective is to minimize the total investment cost(fixed and variable costs) subjected to technical constraints ofdistribution networks operation [6]. The distribution networkshave particular operation characteristics that are of fundamen-tal importance to the formulation of MBNLP problem, as itsradial topology and the impossibility of interconnection ofnetworks fed by different substations.

In the literature there are several approaches proposed tosolve the DNP problem. Among them, discrete optimizationlike branch-exchange algorithm [7] and branch-and-bound [3].However the size of the search space of the DNP problemleads to a big computational effort. Techniques based in meta-heuristics like genetic algorithms [8], [9], simulated annealing[10], [11], tabu search [12], ant colony system [13], and evolu-tive algorithms [14], [15] are also present in the literature. Themajority of the proposed methods solve a load flow problemto calculate the operation point of the network and to verifythe viability of each investment proposal. Heuristic algorithmsare also proposed to guarantee the radial configuration of thenetwork. Meta-heuristics are techniques that can find the bestsolutions for the DNP problem in an acceptable CPU time.

The constructive heuristic algorithm (CHA) also has beenproposed to solve the DNP problem [16], [17]. The CHA isrobust, easy to be applied and it normally converges quicklyto a local solution with a finite iterations number. In [16], [17]the DNP problem is modeled as a mixed integer quadratic pro-gramming problem, in which the objective is to minimize thecost of construction of new circuits and substations and alsothe cost of the active power loss in the circuits. A linearizedmodel is employed to represent the operation constraints whataffect the quality and precision of the results. The CHAsalso were used with success to solve other power systemoptimization problems like the reconfiguration of distributionsystems [18], power system transmission expansion planning[19] and capacitor allocation in power distribution systems[20].

In this work a CHA is proposed to solve the DNP problemmodeled as a MBNLP problem. In each CHA iteration anonlinear programming problem (NLP) is solved to obtain asensitivity index that is used to add a circuit or a substationto the distribution network. The NLP is obtained through

978-1-4244-4241-6/09/$25.00 2009 IEEE

2relaxation of the integer nature of the decision binary variableswhich are considered as continuous (but restricted) variables.The objective of the NLP problem is to minimize the operationand construction costs of the distribution system (constructionof new circuits and/or substation plus the costs associated withthe circuits active power losses) in a time range previouslydefined, and the constraints are the demand attended, voltagelevels between limits, capacity of both circuits and substationsand the radial configuration of the network. A nonlinearoptimization solver was used to solve the NLP. A branchingtechnique is implemented to avoid operation unfeasible casesand the final solution is improved through a proposed simplelocal improvement technique. Two tests using a distributionnetwork are shown. For the first test the solution obtained bythe CHA is better than the one presented in the literature thatwas obtained using meta-heuristics.

II. THE DISTRIBUTION NETWORK PLANNING MODEL

The distribution network expansion planning problem canbe formulated as follows:

min f = l

(ij)l(cij nij lij) + s

ibs

(cfi mi) +

l

(ij)l(gij(nij)(V 2i +V

2j 2ViVj cos ij)) +

sibs

(cvi(P2Si + Q

2Si)) (1)

s.t.

P calci PSi + PDi = 0 i b (2)Qcalci QSi + QDi = 0 i b (3)1V

100 Vi

V nom 1+V

100 i b (4)

P 2Si + Q2Si [(m0i + mi)Si]2 i bs (5)

P 2ij + Q2ij [(n0ij + nij)Sij ]2 (ij) f (6)

nij [0, 1] (ij) l if n0ij = 0 (7)mi [0, 1] i b if m0i = 0 (8)

(ij)l(n0ij + nij) = nb nbs (9)

where nij = n0ij + nij and the constants l and s are givenby:

l = 8760 l l cl ; s = 8760 s s

and f is the set of power flows directions

f = {ij / i b and j bi}and l, b, bs b and bi b are the sets of branches(existing and proposed), nodes, bus substation nodes (existingand proposed) and connected nodes in the node i, respectively.ij represents the circuit between nodes i and j. l and sare the interest rate for circuits and substation, respectively.cij , nij , n0ij and lij represent the circuit cost that can beadded (US$/km), the number of circuits added, the circuitexistent and the length of the circuit, respectively, in branchij. cfi , mi and m

0i represents the substation fixed cost that

can be added to the network (US$), the number of substation

added and substation existent, respectively, in the node i. land s are the interest rate to losses power and providedenergy for the substation, respectively. l and s are theload factor to the circuits and substation, respectively. cl isthe energy cost to active losses (US$/kWh) and cvi is thesubstation variable cost (US$/kVAh2) to the node i. PSi andQSi are the real and reactive power provided by substationsof the node i, respectively. P calci and Q

calci are the real and

reactive power calculated of the node i, respectively. Si isthe apparent power maximum limit provided by substationin node i. The maximum and minimum limits of voltagemagnitudes are V % of the voltage nominal value. Vi andV nom are the voltage magnitude of de node i and nominalvoltage magnitude of the distribution network. Pij and Qijare the real and reactive power flow that leaves node i toward

node j, respectively. Sij =

P 2ij + Q2ij is the apparent power

flow of the branch ij and Sij is its maximum. nb = |b| isthe number of nodes and nbs = |bs | is the number of bussubstation nodes (existing and proposed).

The objective function is the total cost defined in Eq. (1)which has three parts (one part at each line of the equation)and is based on the reference [6]. The first part represent theinvestment cost (construction of circuits and substations), thesecond part represents the corresponding annual cost of theactive losses and the third represents the annual cost of sub-stations operation. Eqs. (2) and (3) represent the conventionalequations of AC power flow and the elements of P calci andQcalci are calculated by (10) and (11), respectively.

P calci = Vijb

Vj [Gij(nij) cos ij + Bij(nij)sin ij ] (10)

Qcalci = Vijb

Vj [Gij(nij)sin ij Bij(nij) cos ij ] (11)

where ij=ij represents the difference of phase anglebetween nodes i and j. Gij() and Bij() are the nodaladmittance matrix elements considering the existing numberof circuits and the added circuit in branch ij as variable.

Eq. (4) represents the constraints of magnitude voltageof nodes and Eq. (5) represents the maximum capacity ofsubstation i. The elements of real and reactive power flowin branch ij of Eq. (6) are given by:

Pij = V 2i gij(nij)ViVj(gij(nij) cos ij+bij(nij) sin ij)

Qij = V 2i bij(nij)ViVj(gij(nij) sin ijbij(nij) cos ij)where gij() and bij() are the conductance and the suscep-tance of the branch ij considering nij as variable. The binaryinvestment variables nij and mi are the decision variablesand a feasible operation solution of the distribution networkdepends on its value. The remaining variables only representthe operating state of a feasible solution. For a feasibleinvestment proposal, defined through specified values of nijand mi, one can have several feasible operation states. Eqs.(7) and (8) represent the limits of number of circuits andsubstations that can be added to the distribution network,respectively (note that duplication of circuits and substationsis not allowed).

3Eq. (9) is essential to generate radial connected solutions.From graph theory it is known that a subgraph T is a tree if thesubgraph meet both following conditions: (1) the subgraph has(nb1) arcs and (2) it is connected. Eq. (9) guarantees the firstcondition and the second condition is guaranteed by Eqs. (2)and (3) (power balance), provided that there is power demandat each node, for the optimization technique is required togenerate a feasible solution connecting all the system nodes.This means that at the end of the process the obtained networkis connected and has radial topology.

III. CONSTRUCTIVE HEURISTIC ALGORITHM

The DNP problem as formulated in Eqs. (1) (9) is amixed binary nonlinear problem. It is a complex combinatorialproblem that can lead to combinatorial explosion on thenumber of alternatives that have to be tested. Considering thenumber of circuits and substations (nij and mi) as contin-uous variables, the problem becomes a still difficult to solvenonlinear programming problem. Although the solution of thisrelaxed DNP problem may not be an alternative for planning(fractional number of circuits and substation), it serves to builda useful sensitivity index used in the CHA.

A CHA may be viewed as an iterative process in which agood solution of a complex problem is built step by step. TheCHA is robust and has good convergence characteristics. Inthe case of the DNP problem, in each step, a substation or asingle circuit is added to the distribution network based on thesensitivity indexes shown in Eqs. (12) and (13). The iterativeprocess ends when a feasible solution is found (generallythis is a good quality solution). The previously mentionedsensitivity indexes SSI (Substation Sensitivity Index) andCSI (Circuit Sensitivity Index) are based on the maximumMVA power flow in circuits with nij = 0 and in the providedMVA power of substation with mi = 0 obtained in the solutionof the NLP problem.

SSI = maxibs

{Si, mi = 0} (12)CSI = max

(ij)l{max{Sij , Sji}, nij = 0} (13)

where Si is the apparent power provided by substation of thenode i. Sij and Sji are the apparent power flow of the branchij. The sensitivity indexes are function only of the operationalcharacteristics of the distribution network.

The NLP problem used to obtain the sensitivity indexes isobtained from Eqs. (1) (9), considering as continuous vari-ables the number of new circuits and substations (continuousbut limited between 0 and 1) and adding two new parametersto the constraints set (5) (9) as it is shown in Eqs. (14) (18).

P 2Si + Q2Si [(m0i + m+i + mi)Si]2 i bs (14)

P 2ij + Q2ij [(n0ij + n+ij + nij)Sij ]2 (ij) f (15)

0 nij 1 n+ij n0ij (ij) l (16)0 mi 1m+i m0i i bs (17)(ij)l

(n0ij + n+ij + nij) = nb nbs (18)

where n+ij e m+i represent the circuits and substations, respec-

tively, that can be added to the distribution network during theCHA iterative process. Also, nij = n0ij + n

+ij + nij is used in

the calculation of the Eqs. (1), (10) and (11), and the elementsof real and reactive power flow in branch ij of Eq. (6).

At the end of the CHA iterative process, the total cost, givenby Eq. (19), is obtained using the solution of the last NLPproblem.

v = l

(ij)l(cij n+ij lij) + s

ibs

(cfi m+i ) +

l

(ij)l(gij(n0ij+n

+ij)(V

2i +V

2j 2ViVj cos ij)) +

sibs

(cvi(P2Si + Q

2Si)) (19)

The substation and circuit feasible indexes (20) and (21),respectively, are used to define the stop criterion of the CHA.

SFI =ibs

(mi) (20)

CFI =

(ij)l(nij) (21)

The constructive heuristic algorithm, is very simple and ispresented in the sequence.

1: Assume that m+i = 0, i bs and n+ij = 0, (ij) lis the current topology.

2: Solve the NLP problem (1) (4) and (14) (18) for thecurrent topology.if SFI + CFI = 0 then

A feasible solution for the DNP problem has been foundand the total planning cost is v; go to Step 5

end ifOtherwise continue;

3: (Substations construction phase)if SFI = 0 then

Using SFI index (Eq. (12)), identify and add the mostattractive substation to the current topology; return to Step2

end ifOtherwise continue;

4: (Circuits construction phase)if CFI = 0 then

Using the CFI index (Eq. (13)) identify and add the mostattractive circuit in the current topology; This circuit mustbe connected or to the substation bus or to a node alreadyfeeded by a substation and return to Step 2;

end if5: (Local improvement phase) Rank added circuits, in a descent order using as criteriaits investment and operation cost; Remove one by one, an added circuit according to theranking obtained and solve the CHA (only for addingcircuits); If the circuit found by the CHA algorithm is the

4same that was removed one then keep it in the configuration;Otherwise the new circuit is included in the configuration.

The circuits in the final topology represent the solution of theCHA (the distribution network expansion plan).

It must be noted that in the CHA firstly all needed sub-stations were added and in the sequence the circuits areconsidered (see steps 3 and 4 of the CHA algorithm). Toimprove the convergence and to reduce the CPU time to solveNLP problems it is recommended to use the previous NLPsolution as initial point.

In CHA Step 4 the circuits are added to the network insuch a way that a tree that starts from an existing substationbus is constructed. The local improvement phase is aimed toimprove the initial solution obtained by the CHA. Note thatthe local improvement local is made only in circuits addedfor the CHA using the NLP problem results to substitute onecircuit by other.

IV. TESTS AND RESULTS

The CHA proposed to solve the DNP problem was writtenin AMPL (a mathematical programming language) [21] andthe solution of the NLP of steps 2 and 6 was solved using thenonlinear programming solver KNITRO (Nonlinear Interior-point Trust Region Optimizer) [22] (called with default op-tions).

The 23-nodes system available in [10], [13] has been used toshow the performance and robustness of the CHA for the DNPproblem. The 34.5-kV distribution network, supplied by a 10-MVA substation, feeds an oil production area with 21 loadsnodes. The proposed feasible routes are displayed in Fig. 1.The maximum allowable voltage drops is 3%. The averagepower factor is equal to 0.9. All aluminum conductors 1/0and 4/0 are used with parameters given in [23]. The capitalcost for circuits is 10 kUS$/km. The cost of energy losses is0.05 US$/kWh, loss factor equals 0.35, interest rate is 0.1 andthe planning period extends to 20 years.

Two kind of tests were performed: Test 1 - planning of adistribution network considering one existing substation; andTest 2 - planning of a distribution network considering twosubstations, an existing one and a candidate substation.

The NLP problem that is solved at each iteration of the CHAfor the tested system has the following characteristics: 117variables (24 unconstrained and 93 constrained), 1 equalitylinear constraint, 46 equality nonlinear constraints, 35 linearinequality constraints and 141 inequality nonlinear constraints.

A. Test 1 - Planning of the Distribution Network

This test has as objective to show a comparison betweenresults obtained with the proposed method and results pre-sented in the literature using meta-heuristics. Table I showsthe sequence in which the circuits were added to the networkby the CHA, the circuits costs, the active power losses costsand the total investment cost. The added circuits by the localimprovement (LI) phase are also shown. Due to the radialconfiguration constraint it is possible to preview the totalnumber of iterations that is equal to 2(nb 1)+1. In this test

1

2

3

4

5

6

7

89

10

11

1213

14

15

16

171819

20

21

22

23

Fig. 1. 23-nodes system - propose feasible routes

TABLE I

ITERATIVE PROCESS OF THE CHA FOR THE TEST 1

Iteration Added Circuit Losses Total

CHA circuits cost (US$) cost (US$) cost (US$)

0 0 37851 37851

1 n+0110 = 1 2021 37701 397222 n+1014 = 1 6318 37750 440683 n+0614 = 1 14495 36848 513434 n+1019 = 1 20444 36848 572925 n+0607 = 1 28625 31666 602916 n+0708 = 1 35491 31667 671587 n+1423 = 1 40351 31667 720188 n+1921 = 1 45901 31666 775679 n+1922 = 1 51728 31667 8339510 n+0308 = 1 78807 25912 10471911 n+1020 = 1 85780 24117 10989612 n+0523 = 1 92189 24066 11625513 n+1417 = 1 96671 24069 12073914 n+1521 = 1 102740 24065 12680515 n+1121 = 1 109134 24067 13320016 n+0309 = 1 127336 22407 14974217 n+0405 = 1 136738 20987 15772518 n+1223 = 1 143523 19946 16347019 n+1620 = 1 148542 19947 16848820 n+1113 = 1 153594 19946 17354121 n+1718 = 1 158006 19946 17795222 n+0208 = 1 158762 19947 178709

Iteration Removed/added Circuit Losses Total

LI circuits cost (US$) cost (US$) cost (US$)

1 n+0308 = 0n+0809 = 1 152250 19921 172171

2 n+1521 = 0n+1518 = 1 151892 20227 172119

51

2

3

4

5

6

7

89

10

11

1213

14

15

16

1718

19

20

21

22

23

Fig. 2. Distribution network expansion plan for Test 1

TABLE II

PRESENT WORTH COSTS FOR THE TEST 1 (US$)

Solutions Circuit Losses Total

cost cost cost

Final [13] 151892 21021 172913

Final [10] 151892 21007 172899

Initial 158762 19947 178709

Final 151892 20227 172119

the CHA solves a total of 45 NLP problems and the solutionis depicted in Fig. 2.

Table II shows a comparison between the results obtainedby references [10] and [13] and the results obtained by theproposed method. Note that the total cost of circuits addedof proposed method is equal to the obtained by the meta-heuristics [10], [13]. However the total operation cost of theproposed method is smaller due to the optimization of thenetwork operation modeled in the NLP problem (1) (4) and(14) (18). It should be noted also, that the initial solutionobtained by the CHA, without the local improvement phase isa solution of good quality for the proposed problem.

B. Test 2 - Planning Distribution Network and Substations

In this test, it was considered that the substation located atnode 1 has a maximum capacity of 4 MVA and that in node 2there is a candidate substation with a maximum capacity alsoof 4 MVA and with a construction cost of 1000 kUS$ andthe operation cost of this substation is 0.1 US$/kVAh2. Thesolution obtained by CHA is shown in Fig 3. Table III showsa summary of the results obtained with the proposed method.

Fig 4 shows a problem that may occur when solvingplanning problem with the CHA and more than one substationis present. Node k is feeded by two substations, what is notallowed. In this case the CHA creates two different problems(branching), that are separately analyzed. In one problem the

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

Fig. 3. Distribution network expansion plan for Test 2

TABLE III

SUMMARY OF RESULTS OF TEST 2 (US$)

Solutions Circuit Losses Substation Operation Total

cost cost cost cost cost

Initial 155694 14687 1000000 6493490 7663871

Final 149182 14661 1000000 6493450 7657293

circuit ik is added to the current topology and the circuitjk is removed. In the other problem circuit jk is added andcircuit ij is removed. By doing this, two problems are solvedindependently and the final solution is the one with the bettercost. Fig. 5 shows the steps sequence of CHA to find the bestsolution. Note that two branching were made in the iterativeprocess. The best solution is the Solution 1. In this test 114NLP problems were solved.

V. CONCLUSIONS

A constructive heuristic algorithm aimed to solve thepower system distribution expansion planning was presented.The CHA has some advantages like robustness, and quicklypresents viable investment proposals.

i

j

k

l

m

SikSjk

SklSkm

Sk

VA from

VA from

substation A

substation B

Fig. 4. Illustrative example

60

11

1

1

22

2

3

4

5

66

77

88

99

10 10

10

11 11

11

12 12

12

13 13

13

14 14

14

15 15

15

16 16

16

17 17

17

18 18

18

19 19

19

20 20

20

21

21 21

m+2 =1

n+0110=1

n+0208=1

n+0708=1

n+0607=1

n+1019=1

fix: n1014=0 fix: n0614=0

n+0614=1

n+1921=1 n+1921=1

n+1921=1

n+1014=1

n+1423=1

n+1423=1

n+1922=1

n+1521=1n+1521=1

n+1521=1

n+1518=1n+1518=1

n+1518=1

n+0308=1n+0308=1

n+0308=1

n+0523=1

n+0523=1

n+0523=1

n+1020=1

n+1020=1

n+1121=1

n+1121=1

n+1113=1

n+1113=1

n+1113=1

n+1718=1

n+1718=1

n+1718=1

n+0309=1

n+0309=1

n+0309=1

n+0405=1n+0405=1

n+0405=1

n+1223=1n+1223=1

n+1223=1

n+1620=1

n+1620=1

n+1417=1

n+0616=1

n+1622=1

n+1622=1

n+1122=1

n+0308=0

n+0308=0n+0308=0n

+0809=1

n+0809=1n+0809=1

n+1521=0n+1521=0

fix: n1620=0 fix: n1020=0

LI Phase

LI Phase LI Phase

Solution 1

Solution 2 Solution 3

Fig. 5. Tree of the CHA to Test 2

A nonlinear programming problem, in which the costs ofsystem operation and of construction of circuits and substa-tions are minimized, subject to constraints of to attend dedemand, voltage magnitude limits, capacity of circuits andsubstations and also the radial configuration of the networkwas solved using a robust commercial software.

In the proposed method a branching technique to avoidinfeasible operation cases and also a local improvement tech-nique are included.

Results obtained show the capability of the method to findan expansion plan to distribution networks and the topologyobtained in some tests were identical to those presented inthe literature, but the final total cost obtained in this workis smaller. Results show, also, the capacity of the method insolving problems in which construction of new substations ispossible.

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