system network planning expansion using mathematical programming, genetic algorithms and tabu search

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Energy Conversion and Management 49 (2008) 1557–1566

System network planning expansion using mathematicalprogramming, genetic algorithms and tabu search

A. Sadegheih a,*, P.R. Drake b

a Department of Industrial Engineering, University of Yazd, P.O. Box 89195-741, Yazd, Iranb E-Business and Operations Management Division, University of Liverpool Management School, University of Liverpool, Liverpool, UK

Received 16 April 2007; accepted 6 December 2007Available online 22 January 2008

Abstract

In this paper, system network planning expansion is formulated for mixed integer programming, a genetic algorithm (GA) and tabusearch (TS). Compared with other optimization methods, GAs are suitable for traversing large search spaces, since they can do this rel-atively rapidly and because the use of mutation diverts the method away from local minima, which will tend to become more common asthe search space increases in size. GA’s give an excellent trade off between solution quality and computing time and flexibility for takinginto account specific constraints in real situations. TS has emerged as a new, highly efficient, search paradigm for finding quality solutionsto combinatorial problems. It is characterized by gathering knowledge during the search and subsequently profiting from this knowledge.The attractiveness of the technique comes from its ability to escape local optimality. The cost function of this problem consists of thecapital investment cost in discrete form, the cost of transmission losses and the power generation costs. The DC load flow equationsfor the network are embedded in the constraints of the mathematical model to avoid sub-optimal solutions that can arise if the enforce-ment of such constraints is done in an indirect way. The solution of the model gives the best line additions and also provides informationregarding the optimal generation at each generation point. This method of solution is demonstrated on the expansion of a 10 bus barsystem to 18 bus bars. Finally, a steady-state genetic algorithm is employed rather than generational replacement, also uniform crossoveris used.� 2007 Elsevier Ltd. All rights reserved.

Keywords: System planning; Tabu search; Genetic algorithm; Mathematical programming; Artificial intelligence; Iterative improvement methods

1. Introduction

Network representations also are widely used for prob-lems in such diverse areas as manufacturing systems analy-sis, logistics, project planning, facilities location andresource management. In fact, a network representationprovides such a powerful visual and conceptual aid for por-traying the relationships between the components of sys-tems that it is used in virtually every field of scientific,social and economic endeavour.

One of the most significant developments in operationalresearch in recent years has been the rapid advance in both

0196-8904/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2007.12.004

* Corresponding author.E-mail address: [email protected] (A. Sadegheih).

the methodology and application of network optimizationmodels. Many of the network flow problems (for example,transportation, minimum cost flow and transmission net-work planning, etc.) can be formulated as different formsof mathematical programming, e.g. linear, non-linear andinteger.

The structure of a typical electrical power or energy sys-tem is very large and complex. Nevertheless, it can bedivided into five fundamental components as follows: (i)energy source; (ii) energy converter; (iii) transmission sys-tem; (iv) distribution system; (v) load. The objective of sys-tem planning is to optimize the facilities necessary toprovide an adequate electrical energy supply at the lowestreasonable cost. Transmission planning is closely relatedto generation planning. The objectives of transmission

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Nomenclature

Notation used in model

Ck cost of generating a unit of power at bus bar kCij capital cost of state j of proposed line i

e(i) = k set of lines that end at bus bar k

EMi maximum power flow of existing line i

K a large positive integer numberLBE number of basic loops containing existing lines

onlyLBP number of basic loops containing existing lines

plus one proposed lineLE(‘) set of existing lines forming basic loop, which

contains existing lines onlyLij linearized cost coefficient representing transmis-

sion losses cost of state j of proposed line i

LP(‘) set of existing lines forming basic loop ‘, whichcontains one proposed line

MP k minimum number of proposed lines connectedto bus bar k

NB number of bus barsNE total number of existing linesNG set of generation bus bars

NP total number of proposed linesNS(i) number of states of proposed line iPþEi

oriented power flow on existing line i from its‘start’ to its ‘end’

P�Eioriented power flow on existing line i from its‘end’ to its ‘start’

P Gk power generation at bus bar k

Pþij oriented power flow on state j of proposed line i

from its ‘start’ to its ‘end’P�ij oriented power flow on state j of proposed line i

from its ‘end’ to its ‘start’P Lk load at bus bar k

P Mk maximum power output of generator k

P Mij maximum power flow of state j of proposed line i

P 0Mijminimum power flow of state j of proposed line i

SE(k) set of existing lines connected to bus bar k

Si linearized cost coefficient representing transmis-sion losses cost of existing line i

s(i) = k set of lines that start from bus bar k

SP(k) set of proposed lines connected to bus bar k

1558 A. Sadegheih, P.R. Drake / Energy Conversion and Management 49 (2008) 1557–1566

planning are based on existing systems, future load, gener-ation scenarios, right of way constraints, costs, line capa-bilities, etc. Transmission planning is an important partof power system planning. Its task is to determine an opti-mal network configuration according to load growth and ageneration planning scheme for the planning period so asto meet the requirements of delivering electricity safelyand economically.

In general, transmission planning should answer the fol-lowing questions: (i) where to build a new transmissionline? (ii) when to build it? (iii) what type of transmissionline to build?

Transmission network planning is generally divided intotwo stages: (i) scheme formation; (ii) scheme evaluation.

At the scheme formation stage, the topology andcapacity of the transmission lines are determined,whereas at the scheme evaluation stage, an analysis isperformed of the network’s characteristics, such as loadflow, short circuit current capacity, stability analysis, reli-ability, etc. There are many good and well establishedmethods of analysis that can be applied at the schemeevaluation stage. However, satisfactory methods forscheme formation are still evolving, and they are the sub-ject of much current research [1–28]. Scheme formationcan be a complex task, subject to many constraintsand a non-linear object function. ‘Optimal’ networkdesigns are important because they can result in largecost savings. There is clearly a need and strong justifica-tion for the development of methods for the design ofnetworks that are as near to ‘optimal’ as is possible. Thispaper aims to provide such methods.

System network planning expansion is a complex math-ematical optimization problem because it involves, typi-cally, a large number of problem variables. Thecommonly used methods reported in the literature can becategorized into mathematical programming, heuristicbased, artificial intelligence and iterative improvementmethods.

As long ago as 1960, Knight [8] used such a method inwhich, starting from the geographical positions of the sub-stations required to interconnect, a set of equations isobtained and solved by linear programming to obtain aminimum cost power transmission network design. Thedrawback of this method is that the load flow constraintsare not taken into consideration. Garver [9] proposed amethod that starts by converting the electrical networkexpansion problem into a linear programming problem.The mathematical programming technique used in solvingthe linear network model minimizes a loss function definedas power times a guide number summed over all networklinks. The overload path with the largest overload isselected for circuit addition. The drawback of this methodis that the model has no user interaction and is fixed by theprogram formulation. Villasana et al. [7] and Serna et al.[10] also proposed methods that used a DC linear powerflow model and a transportation model, respectively. Inboth methods, the model is intractable.

Berg and Sharaf [11] proposed a method using theadmittance approach and linear programming for planningtransmission capacity additions. The method consists oftwo phases. In the first phase, admittance addition is made,while in the second phase, VAR allocation is specified. In

A. Sadegheih, P.R. Drake / Energy Conversion and Management 49 (2008) 1557–1566 1559

this method, losses have been excluded. Kaltenbach et al.[12] proposed a model that uses a combination of linearand dynamic programming techniques to find the mini-mum cost capacity addition to accommodate a givenchange in demand and generation. The drawback of thismethod is that a very large number of decision variablesare required.

Farrag and El-Metwally [13] proposed a method usingmixed integer programming in which the objective functioncontains both capital cost represented in its discrete formand the transmission loss cost in a linear form. Kirchhoff’sfirst and second laws are included in the constraints inaddition to the line security constraints. In this method,the loss term is linearized, and a large number of decisionvariables are required. Sharifnia and Ashtiani [14] pro-posed a method for synthesis of a minimum cost secure net-work. In this method, the loss terms are linearized in theconstraints, and a large number of decision variables arerequired. Adam and Laughton [15] proposed a method thatis based on an interpretation of fixed cost transportationtype models and includes both network security (in thetransmission network) and cost of loss (in the distributionnetwork). The drawback of this method is that the lossterm is in a linearized form, and it requires a large numberof decision variables due to the use of the mixed integer lin-ear programming technique as the solution tool.

Lee and Hick [16] proposed a method that is based onthe static expansion of networks using the zero-one integerprogramming technique, and Romero and Monticelli [17]proposed a zero-one implicit enumeration method for opti-mizing investments in transmission expansion planning.These methods require a large number of decision variablesand are computationally very expensive. Padiyar andShanbhag [18] made a comparison of the computationtimes required by four different optimization techniques:the transportation model; linear; zero-one and non-linearprogramming. The use of zero-one and non-linear pro-gramming techniques requires high CPU times comparedto the other methods, which makes them ineffective forlarge scale systems [19], and all of the methods reviewedare fixed by program formulation.

Yousef and Hackam [20] proposed a model capable ofdealing with both static and dynamic modes of transmis-sion planning using non-linear programming. The costfunction includes the investment and transmission losscosts. Again, this method requires long computation timesand a large number of decision variables [21].

El-Sobki and El-Metwally [22] proposed a heuristicmethod that is a systematic procedure to cancel the ineffec-tive lines from the network. The process is directed in agood manner such that the minimum cost network willbe obtained, containing the most effective routes with thebest number of circuits. The DC load flow model is used.The drawback of this method is that power losses are nottaken into account.

Albuyeh and Skiles [23] presented a planning methodinvolving three integral parts. The first is a network model

using a fast decoupled load flow relating the changes inactive and reactive powers to changes in bus angles andvoltages, respectively. In the second part, a selection con-tingency analysis is employed to determine the maximumoverload on each branch and the maximum voltage devia-tion for each bus. Finally, the line cost, maximum overloadand a sensitivity matrix are combined into two formulae todetermine the branch to be added and the susceptance ofthat branch. The procedure is repeated until the contin-gency analysis shows no overload. In this method, losseshave been included as a linear term.

Ekwue [24] proposed a method derived on the basis of aDC load flow approach. The method determines the num-ber of lines of each specification to be added to a networkto eliminate system overloads at minimum cost. A staticoptimization procedure, based on the steepest descent algo-rithm, is then used to determine the new admittances to beimplemented along these rights of way. In this method, themodel is only applicable to already connected systems andnot expansion as considered here.

In general, a characteristic of heuristic techniques isthat, strictly speaking, an optimal solution is not sought,instead the goal is a ‘‘good” solution. While this may beseen as an advantage from the practical point of view, itis a distinct disadvantage if there are good alternative tech-niques that target the optimal solution.

With the development of artificial intelligence (AI) the-ory and techniques, some AI based approaches to trans-mission network planning have been proposed in recentyears. These include the use of expert systems [25] andartificial neural network (ANN) based [26] methods.The main advantage of the expert system based methodlies in its ability to simulate the experience of planningexperts in a formal way. However, knowledge acquisitionis always a very difficult task in applying this method.Moreover, maintenance of the large knowledge base isvery difficult. Research into the application of the ANNto the planning of transmission networks is in the preli-minary stages, and much work remains to be done. Thepotential advantage of the ANN is its inherent parallelprocessing nature.

In recent years, there has been a lot of interest in theapplication of simulated annealing (SA) and tabu search(TS) to solving some difficult or poorly characterized opti-mization problems of a multi-modal or combinatorial nat-ure. SA is powerful in obtaining good solutions to largescale optimization problems and has been applied to theplanning of transmission networks [27]. In this reference,the transmission network planning is first formulated asa mixed integer, non-linear programming problem andthen solved using SA. The strength of GAs is that theyare free from limitations about the search space, e.g. con-tinuity, differentiability and uni-modality, and they arevery flexible in the choice of an objective function. Fur-thermore, GAs can work on very large and complexspaces. These properties give GAs the ability to solvemany complex real world problems. TS has emerged as


a highly efficient, search paradigm for finding high qualitysolutions quickly to combinatorial problems [28–31]. It ischaracterized by gathering knowledge during the searchand subsequently profiting from this knowledge. TS hasbeen applied successfully to many complicated combina-torial optimization problems in many areas includingpower systems [32,33], The drawback of this method isthat its effectiveness depends very much on the strategyfor manipulation of the tabu list. Obviously, how to spec-ify the size of the tabu list in the searching process playsan important role in the search for good solutions. Ingeneral, the tabu list size should grow with the size of agiven problem.

From the above review, in this paper, the application ofmixed integer programming, a genetic algorithm and TSare proposed to solve the system network planningproblem.

GAs are based, in concept, on natural genetic and evo-lutionary mechanisms working on populations of solutionsin contrast to other search techniques that work on a singlesolution. Searching, not on the real parameter solutionspace but on a bit string encoding of it, they mimic naturalchromosome genetics by applying genetics like operators insearch of the global optimum. An important aspect of GAsis that, although they do not require any prior knowledgeor any space limitations, such as smoothness, convexityor uni-modality of the function to be optimized, they exhi-bit very good performance in the majority of applications[34]. They only require an evaluation function to assign aquality value (fitness value) to every solution produced.Another interesting feature is that they are inherently par-allel (solutions are individuals unrelated with each other),and therefore, their implementation on parallel machinesreduces significantly the CPU time required.

Compared with other optimization methods, GAs aresuitable for traversing large search spaces since they cando this relatively rapidly and because the use of mutationdiverts the method away from local minima, which willtend to become more common as the search space increasesin size. GAs give an excellent trade off between solutionquality and computing time and flexibility for taking intoaccount specific constraints in real situations.

2. Formulation of the system network planning expansion

model

In this paper, system network planning is formulatedand a novel model of the problem of minimum cost expan-sion of power transmission networks is solved by mixedinteger programming. The model explicitly takes into con-sideration the capital investment cost in its discrete formand the cost of transmission losses. The model is also for-mulated to be applied with or without the cost of powergeneration. The DC load flow equations for the networkare embedded in the constraints of this mathematicalmodel to avoid sub-optimal solutions that can arise if theenforcement of such constraints is done in an indirect

way. The solution of the model gives the best line additionsand also provides information regarding the optimal gener-ation (MW) at each generation point. This new formula-tion is demonstrated on an example expansion problem.

Minimize:

Z ¼XNP

i¼1

XNSðiÞ

j¼1

ðCijðZþij þ Z�ij Þ þ LijðPþij þ P�ij ÞÞ

þXNE

i¼1

SiðPþEiþ P�Ei

Þ þX

k2NG

CkP Gk ð1Þ

subject to:

(i) the power balance constraint at bus bar k =1,2, . . . ,NB � 1 or the power flow conservationequation at each bus bar upholding Kirchhoff’s FirstLaw:

Xi2SP ðkÞsðiÞ¼k

XNSðiÞ

j¼1

ðP�ij � Pþij Þ þX

i2SP ðkÞeðiÞ¼k

XNSðiÞ

j¼1

ðPþij � P�ij Þ

þX

j2SEðkÞsðjÞ¼k

ðP�Ej� PþEj

Þ þX

j2SEðkÞeðjÞ¼k

ðPþEj� P�Ej

Þ ¼ P Lk � P Gk

ð2Þ

(ii) the loop equation ‘ = 1,2, . . ., LBE containing onlyexisting lines, upholding Kirchhoff’s Second Lawfor existing lines:X
i2LEð‘Þ
X EiðPþEi� P�Ei

Þ ¼ 0 ð3Þ

(iii) the loop equations for loop ‘ containing one pro-posed line i:

Xk2LPð‘Þ

X Ek ðPþEk� P�Ek

Þ þXNSðiÞ

k¼1

X P ik ðPþik � P�ikÞ

6 K 1�XNSðiÞ

k¼1

ðZþik þ Z�ikÞ !

ð4Þ

Xk2LPð‘Þ

X Ek ðPþEk� P�Ek

Þ þXNSðiÞ

k¼1

X P ik ðPþik � P�ikÞ

P KXNSðiÞ

k¼1

ðZþik þ Z�ikÞ � 1

!; ‘ ¼ 1; 2; . . . ;LBP

ð5Þ

(iv) the exclusivity constraint for each proposed line i.This constraint forces the program to select one stateonly for each proposed line or delete all its states. Theexclusivity constraints result from the fact that thecapacity of any line can take on only one value. Thatvalue, however, may be any of the discrete capacitiesin the cost-capacity curve. The exclusivity constraintsprevent the capacity from assuming more than onevalue.


XNSðiÞ

j¼1

ðZþij þ Z�ij Þ 6 1 ð6Þ

(v) the overload constraint for each existing line i:

PþEiþ P�Ei

6 EMi ; i ¼ 1; 2; . . . ;NE ð7Þ

(vi) the overload constraint for state j of each proposedline i:

P 0MijðZþij þ Z�ij Þ 6 Pþij þ P�ij 6 P MijðZþij þ Z�ij Þ;

i ¼ 1; 2; . . . ;NP; j ¼ 1; 2; . . . ;NSðiÞ ð8Þ

(vii) the generator capacity limit at each bus bar k:

P Gk 6 P Mk ð9Þ
(viii) the availability constraint at each bus bar k – this
controls the number of lines connected to each busbar according to parameter MP k :

Xi2SPðkÞ

XNSðiÞ

j¼1

ðZþij þ Z�ij ÞP MP k ; k ¼ 1; 2; . . . ;NB

ð10Þand

PþEi; P�Ei

; Pþij ; P�ij ; P Gk P 0 8i 2 NE;8i; j 2 NP;

NSðiÞ; 8k 2 NG ð11ÞZþij ; Z

�ij ¼ 0; 1 8i; j 2 NP;NSðiÞ ð12Þ

The objective function Z consists of the capitalinvestment cost in its discrete form, the cost of trans-mission losses and the cost of generation.

3. The 18 bus bar example

This example is an actual system in the western part ofChina [34]. The original network has 10 bus bars and 9lines as illustrated in Fig. 1. The system consists of 7 exist-ing load buses and 3 existing generator buses.

The system is to be expanded to 18 bus bars as shown inFig. 2 with 4 new load buses added and 4 new generatorbuses. Table 1 gives the specifications for the existing andproposed lines in the network. The net generation for eachof the bus bars is given in Table 2.

In this example, the cost of a circuit is defined as beingdirectly proportional to the line length.

The application of the developed method has been madein the light of the following factors:

(i) only one line type is assumed;(ii) the maximum number of states = 4;

(iii) the cost of a circuit is proportional to the line length.Therefore, the line length can be used to replace thecost in comparison analysis.

Fig. 3 shows the results obtained with mixed integerprogramming. The total cost of this network plan is13558.98.

4. GA applied to mixed integer system planning

The chromosome structure used to represent a particu-lar set of possible transmission line power capacities formixed integer transmission network planning using GAhas 27 state variables (genes) Pij as follows:

P 1;2; P 1;11; P 2;3; . . . ; P 17;18 ð13ÞEach individual line capacity is encoded by sufficient bits tocover its allowable range of values. The bit strings for eachPij are concatenated to form a chromosome. The initialpopulation is generated randomly, that is, each bit in eachchromosome is set randomly to either 1 or 0. Whenever anew chromosome is generated, it is checked to see that,in decoded form, it produces valid values for the P ij’s.When an invalid value is produced, the chromosome is dis-carded and another one is generated.

The spreadsheet model is developed for solving thisproblem. In the next step for solving the system planningusing a GA, Eq. (2), as Kirchhoff’s First Law, and Eqs.(7)–(11) must be satisfied. Eqs. (3)–(5), as Kirchhoff’s Sec-ond Law, are used to penalize solutions in the costfunction.

The final step in implementation of system planningusing a GA is the fitness function. The fitness value of achromosome is a measure of how well it meets the desiredobjective. In this case, the objective is the minimization ofthe network’s cost function. Choosing and formulating anappropriate objective function is crucial to the efficientsolution of any given genetic algorithm problem. Whendesigning an objective function for an optimization prob-lem with constraints, penalty functions can be introducedand applied to individuals that violate the imposed con-straints. The fitness function in Eq. (1) with penalty func-tions is used to calculate the fitness value of eachindividual. In the GA approach, the parameters that influ-ence its performance include population size, crossover rateand mutation rate. A population size of 50, crossover rateof 0.5 and mutation rate of 0.006 for system network plan-ning are used. Fig. 4 shows the results obtained with thegenetic algorithm. The total cost of this network plan is13558.98. This result is the same as that obtained withmixed integer programming.

5. The topological configuration of the network using Tabu

search

Tabu search was developed by Glover [29–31]. TS hasemerged as a new, highly efficient, search paradigm forfinding quality solutions to combinatorial problems. It ischaracterized by gathering knowledge during the searchand subsequently profiting from this knowledge. Theattractiveness of the technique comes from its ability toescape local optimality. TS has now become an establishedoptimization approach that is rapidly spreading to manynew fields. For example, successful applications of TS havebeen reported recently in solving some power system

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656

259

88

213199

38

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55

1

2

G

G

GBus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

MW

MW

MW

MW

MW

MW

MW

MW

MW

MW

Key:

Bus-bar

Power generation at bus-bar

Load at bus-bar

Existing transmission line

G

Fig. 1. The original 10 bus bar network.

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MW

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MWMW

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MWMW

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MWMWMW

Bus-bar

Bus-bar

Bus-bar Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-barBus-bar

Bus-bar

Key:

G

Bus-bar


Load at bus-bar


Proposed transmission line

Fig. 2. The expanded network.


problems, such as hydro-thermal scheduling [32], alarmprocessing [33], flexible manufacturing systems, neural net-work training, optimal network [26], etc. The drawback of

this method is that its effectiveness depends very much onthe strategy for manipulation of the tabu list. In each iter-ation, the neighbourhood of the current solution is

Table 1Transmission line specifications for the 18 bus bar system

Line number From bus bar To bus bar Reactance per unit Circuit capacity (MW) Existing line New circuits allowed Length (km)

1 1 2 0.0176 230 Yes 0 702 1 11 0.0102 230 No 1 403 2 3 0.0348 230 Yes 0 1384 3 4 0.0404 230 Yes 0 1555 3 7 0.0325 230 Yes 0 1296 4 7 0.0501 230 No 1 2007 4 16 0.0501 230 No 3 2008 5 6 0.0267 230 Yes 0 1069 5 11 0.0153 230 No 2 60

10 5 12 0.0102 230 No 1 4011 6 7 0.0126 230 Yes 0 5012 6 13 0.0126 230 No 1 5013 6 14 0.0554 230 No 2 22014 7 8 0.0151 230 Yes 1 6015 7 9 0.0318 230 No 1 12616 7 13 0.0126 230 No 2 5017 7 15 0.0448 230 No 2 17818 8 9 0.0102 230 Yes 1 4019 9 10 0.0501 230 Yes 2 20020 9 16 0.0501 230 No 2 20021 10 18 0.0255 230 No 1 10022 11 12 0.0126 230 No 2 5023 11 13 0.0255 230 No 1 10024 12 13 0.0153 230 No 1 6025 14 15 0.0428 230 No 2 17026 16 17 0.0153 230 No 2 6027 17 18 0.0140 230 No 1 55

Table 2Net generation at each bus bar for the expanded 18 bus bar network

Nodes (Bus bars) Generator output (MW) Load (MW) Net output (MW)

1 0 55 �552 360 84 2763 0 154 �1544 0 38 �385 760 639 1216 0 199 �1997 0 213 �2138 0 88 �889 0 259 �259

10 750 94 65611 540 700 �16012 0 190 �19013 0 110 �11014 540 32 50815 0 200 �20016 495 132 36317 0 400 �40018 142 0 142


explored, and the best solution in the neighbourhood isselected as the new current solution. TS is different fromother local search techniques in that the procedure doesnot stop when no improvement is possible. The best solu-tion in the neighbourhood is selected as the current solu-tion, even if it is not better than the current solution.This strategy allows escape from local optima and, conse-quently, exploration of a larger proportion of the solutionspace. TS is a restricted neighbourhood search technique,

and its computational flow chart is an iterative process.To describe the workings of TS, consider a combinatorialoptimization problem in the following form:

Minimize ZðP Þ ð14Þwhere P is a vector of dimension n, Z(P) is the objectivefunction (cost or penalty function) and can be linear ornon-linear. The first step of TS is to produce an initial (cur-rent) solution Pcurrent either randomly or using an existing

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G

MW

MW

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MW

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MW

MW

MW

MW

MWMW

MW

MW

MW

MWMWMW

Bus-bar

Bus-bar

Bus-bar Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-bar

Bus-barBus-bar

Bus-bar

142 258

105

67

61

26

215

160

190

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308

200

40

110

309

397

656Key:

G

Bus-bar


Load at bus-bar


Proposed tansmission line

Power flow

Fig. 3. The optimal planning network of the 18 bus bar system using mixed integer programming.

1 2 3 4

16

11 13 7 8 9 10

12 5 6 14 15 17

18

Fig. 4. The topological configuration of the network using geneticalgorithm.


(heuristic) method for the given problem. The second stepis to define a set of moves that may be applied to the cur-rent solution to produce a set of trial solutions. For exam-ple, the move can take the form of:

Ptrial ¼ Pcurrent � DP ð15ÞHere, Dp is a vector with the same dimension as P In fact, amove produces a neighbourhood search. Among all thetrial solutions thus produced, TS seeks the one that mostimproves the objective function. In certain situations, ifthere are no improving moves, a local optimum does exist.In this case, TS chooses the one that least degrades theobjective function. A move is a transition from one solu-tion to another. Moves are generally expected to producean improvement to the solutions. Many kinds of movesare currently available [30,31].

Another important aspect of TS is the tabu list. Thetabu list stores characteristics of previous moves so thatthese characteristics can be used to classify certain movesas tabu (to be avoided) for a number of iterations. Byaccepting moves that do not produce improvements, it ispossible to return to solutions that have already beenfound, which is known as cycling. Obviously, how to spec-ify the size of the tabu list in the searching process plays animportant role in the search for good solutions. In general,the tabu list size should grow with the size of a given prob-lem. Up to now, the tabu list size has been determinedexperimentally [28]. In addition, for how many iterationsa move should be retained in the tabu list is also an impor-tant issue. Many methods to implement and manage thetabu list have been developed. For strategies for the shortterm memory and long term memory in TS, see [30,29].

For this problem, a 27 state variables string was used torepresent all possible topological configurations. There-fore, there were 40 neighbours of P and 27 different possi-ble moves. The neighbourhood was constructed withadjacent solutions created from a current solution bychanging a bit between 0 and 1.

The objective function, which, in this case, is total unitcost, Z(P) was applied to the neighbourhood of P. Then,the last improving move that had been classified tabu wouldbe selected as the next solution. There are three commontabu list types: fixed, dynamic and hashing function, or acombination of the three. In this paper, a fixed tabu list sizewas used. A tabu list size of 15 was empirically determinedfor this problem. The result was obtained after experiment-ing with different numbers of iterations (see Table 3).

Table 4Tabu search solutions

Lines Tabu search power flow (MW) Number of circuits

1–2 �215 11–11 160 12–3 61 13–4 �116 13–7 23 14–16 �154 15–6 �69 15–12 190 16–7 40 16–14 �308 27–8 �260 27–13 110 18–9 �348 29–10 �607 310–18 49 114–15 200 116–17 209 117–18 �191 1

Total cost 13608.47

1 2 3 4

16

11 13 7 8 9 10

12 5 6 14 15 17

18

Fig. 5. The topological configuration of the network using tabu search.

Table 3Different numbers of iterations and total unit cost

Iterations 3056 5345 4567 6789 3456 4678

Total unit cost 13608.47 13608.47 13608.47 13608.47 13608.47 13608.47


Therefore, the tabu search algorithm obtained a valueof 13608.47. It can be seen that an improvement of0.36% is achieved by the genetic algorithm and mixedinteger programming.

Table 4 and Figs. 4 and 5 show that there are some sub-stantial differences in the individual line powerflows, although the fundamental structure is similar withonly one difference in network topology, two differencesin number of lines and one difference in direction of flow.

6. Conclusion

A steady-state GA has been employed rather thangenerational replacement. This means that at each itera-tion two parent chromosomes are selected from the pop-

ulation for reproduction. These parents produce a childwhich is added into the existing population and theweakest member of the population is then deleted. Incontrast, a generational GA produces a whole new pop-ulation of children at each iteration. It could be arguedthat the results demonstrate the validity of the tabusearch method since it achieves a ‘very near optimal’solution. However, the tabu search approach does notprovide us with a generic engineering tool. The resultssupport the argument that genetic algorithms reducethe need for the tabu search method, since ultimately,it targets an optimal solution, while being easily adaptedto different applications, and consequently, a genericengineering tool. The results presented here support theextension of this argument into the field of topologicalconfiguration of the network system in particular. Thepower transmission network planning problem has beenmodeled by the mixed integer programming model. TheGA has three user defined parameters – population size,mutation rate and crossover rate. The results wereobtained with a population size of 50, crossover rate of0.5 and mutation rate of 0.006 – a utilitarian set ofparameters. Further experiments showed that the valueof the crossover rate does not appear to be highly criti-cal, with a value between 0.2 and 0.8 giving the bestresults in terms of rate of convergence. However, initialexperiments showed that the rate of convergence canbe very sensitive to the mutation rate. Overall, tabusearch needed longer computation times than those ofthe genetic algorithm. The drawback of tabu search isthat its effectiveness depends very much on the strategyfor manipulation of the tabu list.

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