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

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

Post on 05-Sep-2016

212 views

Category:

Documents

0 download

TRANSCRIPT

pgo,*,ersitiveraccAvailable online 22 January 2008social and economic endeavour.One of the most signicant developments in operationalresearch in recent years has been the rapid advance in bothenergy 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* Corresponding author.E-mail address: sadegheih@yazduni.ac.ir (A. Sadegheih).Available online at www.sciencedirect.comEnergy Conversion and Managemen1. IntroductionNetwork 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 eld of scientic,the methodology and application of network optimizationmodels. Many of the network ow problems (for example,transportation, minimum cost ow and transmission net-work planning, etc.) can be formulated as dierent 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 ve fundamental components as follows: (i)AbstractIn 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. GAs give an excellent trade o between solution quality and computing time and exibility for takinginto account specic constraints in real situations. TS has emerged as a new, highly ecient, search paradigm for nding quality solutionsto combinatorial problems. It is characterized by gathering knowledge during the search and subsequently proting 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 ow 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; Articial intelligence; Iterative improvement methodsSystem network planning exprogramming, genetic alA. Sadegheih aaDepartment of Industrial Engineering, UnivbE-Business and Operations Management Division, University of LReceived 16 April 2007;0196-8904/$ - see front matter 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.enconman.2007.12.004ansion using mathematicalrithms and tabu searchP.R. Drake by of Yazd, P.O. Box 89195-741, Yazd, Iranpool Management School, University of Liverpool, Liverpool, UKepted 6 December 2007www.elsevier.com/locate/enconmant 49 (2008) 15571566sionplanning are based on existing systems, future load, gener-ation scenarios, right of way constraints, costs, line capa-bilities, etc. Transmission planning is an important partNomenclatureNotation used in modelCk cost of generating a unit of power at bus bar kCij capital cost of state j of proposed line ie(i) = k set of lines that end at bus bar kEMi maximum power ow of existing line iK a large positive integer numberLBE number of basic loops containing existing linesonlyLBP number of basic loops containing existing linesplus one proposed lineLE() set of existing lines forming basic loop, whichcontains existing lines onlyLij linearized cost coecient representing transmis-sion losses cost of state j of proposed line iLP() set of existing lines forming basic loop , whichcontains one proposed lineMPk minimum number of proposed lines connectedto bus bar kNB number of bus barsNE total number of existing linesNG set of generation bus bars1558 A. Sadegheih, P.R. Drake / Energy Converof power system planning. Its task is to determine an opti-mal network conguration 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 networks characteristics, such as loadow, 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 [128]. 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 justica-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. TheNP total number of proposed linesNS(i) number of states of proposed line iPEi oriented power ow on existing line i from itsstart to its endPEi oriented power ow on existing line i from itsend to its startPGk power generation at bus bar kPij oriented power ow on state j of proposed line ifrom its start to its endPij oriented power ow on state j of proposed line ifrom its end to its startPLk load at bus bar kPMk maximum power output of generator kPMij maximum power ow of state j of proposed line iP 0Mij minimum power ow of state j of proposed line iSE(k) set of existing lines connected to bus bar kSi linearized cost coecient representing transmis-sion losses cost of existing line is(i) = k set of lines that start from bus bar kSP(k) set of proposed lines connected to bus bar kand Management 49 (2008) 15571566commonly used methods reported in the literature can becategorized into mathematical programming, heuristicbased, articial 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 ow 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 denedas 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 xed by theprogram formulation. Villasana et al. [7] and Serna et al.[10] also proposed methods that used a DC linear powerow 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 rst phase, admittance addition is made,while in the second phase, VAR allocation is specied. Inrsionthis method, losses have been excluded. Kaltenbach et al.[12] proposed a model that uses a combination of linearand dynamic programming techniques to nd 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. Kirchhosrst 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 xed 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 dierent 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 ineective forlarge scale systems [19], and all of the methods reviewedare xed 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 ineec-tive lines from the network. The process is directed in agood manner such that the minimum cost network willbe obtained, containing the most eective routes with thebest number of circuits. The DC load ow model is used.The drawback of this method is that power losses are nottaken into account.A. Sadegheih, P.R. Drake / Energy ConveAlbuyeh and Skiles [23] presented a planning methodinvolving three integral parts. The rst is a network modelusing a fast decoupled load ow 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 ow approach. The method determines the num-ber of lines of each specication 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 articial 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] andarticial 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 dicult task in applying this method.Moreover, maintenance of the large knowledge base isvery dicult. 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 dicult 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 rst 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, dierentiability and uni-modality, and they arevery exible in the choice of an objective function. Fur-thermore, GAs can work on very large and complexand Management 49 (2008) 15571566 1559spaces. These properties give GAs the ability to solvemany complex real world problems. TS has emerged asZ subje(i)(iii)siona highly ecient, search paradigm for nding high qualitysolutions quickly to combinatorial problems [2831]. It ischaracterized by gathering knowledge during the searchand subsequently proting 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 eectiveness 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 (tness 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 signicantly 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 o between solutionquality and computing time and exibility for taking intoaccount specic constraints in real situations.2. Formulation of the system network planning expansionmodelIn 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 ow equations for the networkare embedded in the constraints of this mathematical1560 A. Sadegheih, P.R. Drake / Energy Convermodel to avoid sub-optimal solutions that can arise if theenforcement of such constraints is done in an indirectk2LPk Ek Ekk1ik ik ik6 K 1XNSik1Zik Zik !4Xk2LPXEk PEk PEk XNSik1XPik Pik PikP KXNSik1Zik Zik 1 !; 1; 2; . . . ;LBP5(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 constraintsfor existing lines:Xi2LEXEiPEi PEi 0 3the loop equations for loop containing one pro-posed line i:XXE P P XNSiXP P P2(ii) the loop equation = 1,2, . . ., LBE containing onlyexisting lines, upholding Kirchhos Second Lawj2SEksjkEj Ejj2SEkejkEj Ej Lk Gki2SP ksikj1ij iji2SP keikj1ij ijXP P XP P P Pi1 j1CijZij Zij LijP ij P ij XNEi1SiPEi PEi Xk2NGCkPGk 1ct to:the power balance constraint at bus bar k =1,2, . . . ,NB 1 or the power ow conservationequation at each bus bar upholding Kirchhos FirstLaw:X XNSiP P X XNSiP Pway. 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:XNP XNSi and Management 49 (2008) 15571566prevent the capacity from assuming more than onevalue.3. ThThChinlinespropoIn this example, the cost of a circuit is dened as beingdirectTh(ii)Figprogr13558rsioncost in comparison analysis.. 3 shows the results obtained with mixed integer(iii) the cost of a circuit is proportional to the line length.Therefore, the line length can be used to replace thethe maximum number of states = 4;in the light of the following factors:(i) only one line type is assumed;ly proportional to the line length.e application of the developed method has been madeof the bus bars is given in Table 2.sed lines in the network. The net generation for eaching 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 specications for the existing ande 18 bus bar exampleis example is an actual system in the western part ofa [34]. The original network has 10 bus bars and 9as illustrated in Fig. 1. The system consists of 7 exist-PEi ; PEi ; P ij ; P ij ; PGk P 0 8i 2 NE;8i; j 2 NP;NSi; 8k 2 NG 11Zij ; Zij 0; 1 8i; j 2 NP;NSi 12The objective function Z consists of the capitalinvestment cost in its discrete form, the cost of trans-mission losses and the cost of generation.XNSij1Zij Zij 6 1 6(v) the overload constraint for each existing line i:PEi PEi 6 EMi ; i 1; 2; . . . ;NE 7(vi) the overload constraint for state j of each proposedline i:P 0MijZij Zij 6 Pij Pij 6 PMijZij Zij ;i 1; 2; . . . ;NP; j 1; 2; . . . ;NSi 8(vii) the generator capacity limit at each bus bar k:PGk 6 PMk 9(viii) the availability constraint at each bus bar k thiscontrols the number of lines connected to each busbar according to parameter MPk :Xi2SPkXNSij1Zij Zij P MPk ; k 1; 2; . . . ;NB10andA. Sadegheih, P.R. Drake / Energy Conveamming. The total cost of this network plan is.98.4. GA applied to mixed integer system planningThe 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 13Each individual line capacity is encoded by sucient 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 ijs.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 Kirchhos First Law, and Eqs.(7)(11) must be satised. Eqs. (3)(5), as Kirchhos Sec-ond Law, are used to penalize solutions in the costfunction.The nal step in implementation of system planningusing a GA is the tness function. The tness value of achromosome is a measure of how well it meets the desiredobjective. In this case, the objective is the minimization ofthe networks cost function. Choosing and formulating anappropriate objective function is crucial to the ecientsolution 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 tness function in Eq. (1) with penalty func-tions is used to calculate the tness value of eachindividual. In the GA approach, the parameters that inu-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 conguration of the network using TabusearchTabu search was developed by Glover [2931]. TS hasemerged as a new, highly ecient, search paradigm fornding quality solutions to combinatorial problems. It ischaracterized by gathering knowledge during the searchand subsequently proting 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 manyand Management 49 (2008) 15571566 1561new elds. For example, successful applications of TS havebeen reported recently in solving some power systemsion38 MW1562 A. Sadegheih, P.R. Drake / Energy Converproblems, such as hydro-thermal scheduling [32], alarmprocessing [33], exible manufacturing systems, neural net-work training, optimal network [26], etc. The drawback of48910656259882154GBus-barBus-barBus-barBus-barMWMWMWKey:Bus-barPower generation at bus-barLoad at bus-barExisting transmission lineGFig. 1. The original 1017181698400 363142656259882133810GGGMWMWMWMWMMWMWMWBus-barBus-barBus-bar Bus-bBus-barBus-barBus-barKey:GBus-barPower generation at bus-barLoad at bus-barExisting transmission lineProposed transmission lineFig. 2. The expan55 MWand Management 49 (2008) 15571566this method is that its eectiveness depends very much onthe strategy for manipulation of the tabu list. In each iter-ation, the neighbourhood of the current solution is37651319927612112GGBus-barBus-barBus-barBus-barBus-barBus-barMWMWMWMWMWbus bar network.762111531312415 1419912116019055276154508200110GGGMWMWWMWMWMWMWMWMWMWMWarBus-barBus-barBus-barBus-barBus-barBus-barBus-barBus-barBus-barBus-barBus-barded network.rcui00000rsionTable 1Transmission line specications for the 18 bus bar systemLine number From bus bar To bus bar Reactance per unit Ci1 1 2 0.0176 232 1 11 0.0102 233 2 3 0.0348 234 3 4 0.0404 235 3 7 0.0325 23A. Sadegheih, P.R. Drake / Energy Conveexplored, and the best solution in the neighbourhood isselected as the new current solution. TS is dierent 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,6 4 7 0.0501 2307 4 16 0.0501 2308 5 6 0.0267 2309 5 11 0.0153 23010 5 12 0.0102 23011 6 7 0.0126 23012 6 13 0.0126 23013 6 14 0.0554 23014 7 8 0.0151 23015 7 9 0.0318 23016 7 13 0.0126 23017 7 15 0.0448 23018 8 9 0.0102 23019 9 10 0.0501 23020 9 16 0.0501 23021 10 18 0.0255 23022 11 12 0.0126 23023 11 13 0.0255 23024 12 13 0.0153 23025 14 15 0.0428 23026 16 17 0.0153 23027 17 18 0.0140 230Table 2Net generation at each bus bar for the expanded 18 bus bar networkNodes (Bus bars) Generator output (MW)1 02 3603 04 05 7606 07 08 09 010 75011 54012 013 014 54015 016 49517 018 142t capacity (MW) Existing line New circuits allowed Length (km)Yes 0 70No 1 40Yes 0 138Yes 0 155Yes 0 129and Management 49 (2008) 15571566 1563and its computational ow chart is an iterative process.To describe the workings of TS, consider a combinatorialoptimization problem in the following form:Minimize ZP 14where P is a vector of dimension n, Z(P) is the objectivefunction (cost or penalty function) and can be linear ornon-linear. The rst step of TS is to produce an initial (cur-rent) solution Pcurrent either randomly or using an existingNo 1 200No 3 200Yes 0 106No 2 60No 1 40Yes 0 50No 1 50No 2 220Yes 1 60No 1 126No 2 50No 2 178Yes 1 40Yes 2 200No 2 200No 1 100No 2 50No 1 100No 1 60No 2 170No 2 60No 1 55Load (MW) Net output (MW)55 5584 276154 15438 38639 121199 199213 21388 88259 25994 656700 160190 190110 11032 508200 200132 363400 4000 142MWus-26sion171816400 36314238GGMWMWMWBus-barBus-barBus-bar B142 258105673093971564 A. Sadegheih, P.R. Drake / Energy Conver(heuristic) method for the given problem. The second stepis to dene 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 Pcurrent DP 15Here, 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].9865625988210GMWMWMWBus-barBus-barBus-barB656Key:GBus-barPower generation at bus-barLoad at bus-barExisting transmission lineProposed tansmission linePower flowFig. 3. The optimal planning network of the 18 bu1 2 3 41611 13 7 8 9 1012 5 6 14 15 1718Fig. 4. The topological conguration of the network using geneticalgorithm.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 moves762111531312415141319912116019055276154508200110GGGMWMWMWMWMWMWMWMWMWMWMWbarBus-barBus-barBus-barBus-barBus-barBus-barBus-barBus-barBus-barus-barBus-bar612151601906930820040110s bar system using mixed integer programming.and Management 49 (2008) 15571566as 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 congurations. There-fore, there were 40 neighbours of P and 27 dierent 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 classied tabu wouldbe selected as the next solution. There are three commontabu list types: xed, dynamic and hashing function, or acombination of the three. In this paper, a xed tabu list sizewas used. A tabu list size of 15 was empirically determinedfor this problem. The result was obtained after experiment-ing with dierent numbers of iterations (see Table 3).456136rsionTable 4Tabu search solutionsLines Tabu search power ow (MW) Number of circuits12 215 1111 160 123 61 134 116 1Table 3Dierent numbers of iterations and total unit costIterations 3056 5345Total unit cost 13608.47 13608.47A. Sadegheih, P.R. Drake / Energy ConveTherefore, 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 dierences in the individual line powerows, although the fundamental structure is similar withonly one dierence in network topology, two dierencesin number of lines and one dierence in direction of ow.6. ConclusionA steady-state GA has been employed rather thangenerational replacement. This means that at each itera-tion two parent chromosomes are selected from the pop-support the argument that genetic algorithms reduce37 23 1416 154 156 69 1512 190 167 40 1614 308 278 260 2713 110 189 348 2910 607 31018 49 11415 200 11617 209 11718 191 1Total cost 13608.471 2 3 41611 13 7 8 9 1012 5 6 14 15 1718Fig. 5. The topological conguration of the network using tabu search.the need for the tabu search method, since ultimately,it targets an optimal solution, while being easily adaptedto dierent applications, and consequently, a genericengineering tool. The results presented here support theextension of this argument into the eld of topologicalconguration of the network system in particular. Thepower transmission network planning problem has beenmodeled by the mixed integer programming model. TheGA has three user dened 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 eectiveness depends very much on the strategyfor manipulation of the tabu list.References[1] Sadegheih A. Design and implementation of network planningsystem. In: 20th international power system conference, 1416November 2005, Tehran, Iran.[2] Sadegheih A. Sequence optimization and design of allocation usingGA and SA. Appl Math Comput 2007;186(2):172330.[3] Sadegheih A. A novel method for designing and optimization ofnetwork. Int J Eng Trans A: Basics 2007;20(1):1726.[4] Sadegheih A. Models in the iterative improvement and heuristiculation 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 optimalsolution. However, the tabu search approach does notprovide us with a generic engineering tool. The results7 6789 3456 467808.47 13608.47 13608.47 13608.47and Management 49 (2008) 15571566 1565methods. WSEAS Trans Adv Eng Educ 2006;3(4):25661.[5] Sadegheih A, Drake PR. Network optimization using linear pro-gramming and genetic algorithm. Neural Network World, Int J Non-Standard Comput Artif Intell 2001;11(3):22333.[6] Sadegheih A. Scheduling problem using genetic algorithm, simulatedannealing and the eects of parameter values on GA performance.Appl Math Model 2006;30(2):14754.[7] Villasana R, Garver LL, Salon SJ. Transmission network planningusing linear programming. IEEE Trans PAS 1985;PAS-104(2):34956.[8] Knight UGW. The logical design of electrical networks using linearprogramming methods. Proc IEE 1960;107A(33):30619.[9] Garver LL. Transmission network estimation using linear program-ming. IEEE Trans PAS 1970;PAS-89(7):168897.[10] Serna C, Duran J, Camargo A. A model for expansion planning oftransmission systems: A practical application example. IEEE Trans1978;PAS-97(2):6105.[11] Berg G, Sharaf TAM. Reliability constrained transmission capacityassessment. Electric Power Syst Res 1988;15:713.[12] Kaltenbach JC, Peschon J, Gehring EH. A mathematical optimiza-tion technique for the expansion of electric power transmissionsystem. IEEE Trans PAS 1970;PAS-89(1).[13] Farrag MA, El-Metwally MM. New method for transmissionplanning using mixed-integer programming. IEE Proc C, Gen TransDistrib 1988;135(4):31923.[14] Sharifnia A, Ashtiani HZ. Transmission network planning: A methodfor synthesis of minimum-cost secure networks. IEEE Trans PAS1985;PAS-93(8).[15] Adams RN, Laughton MA. Optimal planning of power networksusing mixed-integer programming. IEE Proc C, Gen Trans Distrib1974;121(2):13947.[16] Lee TV, Hick KL. Transmission expansion by branch-bound integerprogramming with optimal cost-capacity curves. IEEE Trans PAS1974;PAS-93(5).[17] Romero R, Monticelli A. A zero-one implicit enumeration methodfor optimizing investments in transmission expansion planning. IEEETrans PS 1994;9(3):138591.[18] Padiyar KR, Shanbhag RS. Comparison of methods for transmissionsystem expansion using network ow and D.C. load ow models.Electric Power Energy Syst 1989;10(1):1724.[21] El-Metwally MM, Harb AM. Transmission planning using admit-tance approach and quadratic programming. Electric Mach PowerSyst 1993;21:6983.[22] El-Sobki SM, El-Metwally MM, Farrag MA. New approach forplanning high-voltage transmission networks. IEE Proc 1986;133(5):25662.[23] Albuyeh F, Skiles JJ. A transmission network planning methodfor comparatives studies. IEEE Trans PAS 1981;PAS-100(4):167984.[24] Ekwue AO. Investigations of the transmission system expansionproblem. Electric Power Energy Syst 1984;6(3):13942.[25] Galiana FD, McGillis DT, Marin MA. Expert system in transmissionplanning. Proc IEEE 1992;80(5):71226.[26] Yoshimoto K, Yasuda K, Yokoyama R. Transmission expansionplanning using neuro-computing hybridized with genetic algorithm.In: Proceedings of the 1995 IEEE international conference onevolutionary computation, Perth, Australia, 1995. p. 12631.[27] Romero R, Gallego RA, Monticelli A. Transmission system expan-sion planning by simulated annealing. In: Proceedings of the 1995IEEE power industry computer application conference (PICA95),USA. p. 27883.[28] Wen Fushuan, Chang CS. Transmission network optimal planningusing the tabu search method. Electr Power Syst Res 1997;42(2):15363.[29] Glover F, Laguna M, Taillard E, de Werra D, editors Tabusearch. Basel, Switzerland: Science Publishers; 1993.[30] Glover F. Tabu search-part I. ORSA J Comput 1989;1(3):190206.[31] Glover F. Tabu search-part II. ORSA J Comput 1990;2(1):432.[32] Bai X, Shahidehpour S. Hydro-thermal scheduling by tabu search1566 A. Sadegheih, P.R. Drake / Energy Conversion and Management 49 (2008) 15571566[19] El-Metwally MM, Al-Hamouz ZM. Transmission network planningusing quadratic programming. Electric Mach Power Syst 1990;18(2):13748.[20] Youssef HK, Hackam R. New transmission planning model. IEEETrans PS 1989;4(1):917.and decomposition method. IEEE PWRS 1996;11(2):96874.[33] Wen Fushuan, Chang CS. A tabu search approach to alarmprocessing in power systems. IEE Proc Gen Trans Distrib1997;144(1):318.[34] Wang X, McDonald JR. Modern power system planning. McGraw-Hill International (UK) Limited; 1994.System network planning expansion using mathematical programming, genetic algorithms and tabu searchIntroductionFormulation of the system network planning expansion modelThe 18 bus bar exampleGA applied to mixed integer system planningThe topological configuration of the network using Tabu searchConclusionReferences

Recommended

View more >