la

o

lom

Keywords:Transmission planning

doloriosone s

using the 6-bus Garver system and the IEEE-24 bus system. The TEP is based on the DC model of the net-

the rerk, inndin

network and all associated infrastructure, and consequently isthe base for the electric market. In the case of the generation, thetransmission network permits different dispatch scenarios andallows competition among the agents.

Under the previous premises, it becomes necessary to build atransmission network capable of taking advantage of future gener-

problem.

The problem can be solved using a static approach [1multistage model [48]. The Static approach considers ongeneration-demand scenario, and the multistage or dmodel takes into account several generation-demand periods oftime.

Different mathematical representations have been proposed tosolve the TEP. The main implemented models in order of complex-ity, are: transportation [9], hybrid [10], DC [4,11] and AC [12,13].

For solving the previous the mathematical models, differenttechniques and methods of solution have been used, such as linear

Corresponding author. Tel.: +57 3103290375; fax: +57 13417900.E-mail address: carcorrea@unisalle.edu.co (C.A. Correa Florez).

Electrical Power and Energy Systems 62 (2014) 398409

Contents lists availab

Electrical Power an

.e lability and security, and accompanied by remuneration toequipment owners.

Planning also promotes network access for generators, as wellas customers. The bridge to allow this access is the transmission

Mixed-Integer Non-linear Programming (MINLP)

1.1. Modeling and solving the TEPhttp://dx.doi.org/10.1016/j.ijepes.2014.04.0630142-0615/ 2014 Elsevier Ltd. All rights reserved.3] or aly oneynamicferent aspects should be taken into consideration with the purposeof facing the new challenges that have arisen in the previous years.

Restructuring process in the electricity sector has led to a stron-ger interaction of technical and market aspects. Theoretically,these changes allow competition, promote higher quality and leadto better prices of the service. Planning and expansion in compet-itive markets should be characterized by low costs, quality, reli-

test for planners, in order to model and nd a suitable transmissionsystem with plenty of capacity, and guaranteeing social welfare.

The mathematical model for planning the transmission systemconsiders current system topology, the forecast of generation anddemand, power balance equations, among others, and results inlinear and non-linear algebraic expressions containing real andinteger variables. Given the nature of model, it is considered as aGeneration scenariosMultiobjective optimizationMarketPareto front

1. Introduction

The TEP consists on determiningto reinforce the transmission netwomum cost without load shedding. Forwork and non-linear interior point method is used to initialize the population.A set of Pareto optimal expansion plans with different levels of cost and load shedding is found for each

system, showing the robustness of the proposed approach. 2014 Elsevier Ltd. All rights reserved.

quired investment planorder to achieve mini-g an adequate plan, dif-

ation, supplying forecasted load, and avoiding potential congestioncosts, which are at the end transferred to customers. The planningprocess and models must take into account investment and con-gestion costs, by analyzing possible dispatching scenarios resultingfrom market rules. The resulting power ow patterns become aAvailable online 2 June 2014strategies for the optimization process, implementing a new hybrid modied NSGA-II/ChuBeasley algo-rithm and taking into account variable demand and generation. The proposed methodology is validatedMulti-objective transmission expansion pgeneration scenarios

Carlos A. Correa Florez a,, Ricardo A. Bolaos OcampaUniversidad De La Salle, Cra 2 10-70, Bogota, ColombiabXM Filial de ISA, Calle 12 Sur 18-168, Medellin, Antioquia, ColombiacUniversidad Tecnologica de Pereira, Complejo Educativo La Julita, Pereira, Risaralda, Co

a r t i c l e i n f o

Article history:Received 2 September 2013Received in revised form 10 April 2014Accepted 30 April 2014

a b s t r a c t

This paper shows a methoMultiple Generation Scenacaused by realistic operaticonditions or fuel prices. Th

journal homepage: wwwnning considering multiple

b, Antonio H. Escobar Zuluaga c

bia

gy for solving the Transmission Expansion Planning (TEP) problem when(MGS) are considered. MGS are a result of the multiple load ow patternsof the network, such as market rules, availability of generators, weatherolution to this problem is carried out by using multiobjective evolutionary

le at ScienceDirect

d Energy Systems

sevier .com/locate / i jepes

Bound [21]. Besides classical techniques, metaheuristic methods

generational cycle t

er anhave also been satisfactorily used as an alternative, for instance,references [1,2228] show how TEP is solved using SimulatedAnnealing, Tabu Search, Genetic Algorithms and Particle SwarmOptimization. Other recent metaheuristic optimization techniques,such as frog leaping, immune systems, ant colony, chaos and beecolony algorithms, have also been used as referenced in [3].

1.2. Planning the transmission network in a market environment

Deregulation in electricity markets have led to new challengesprogramming [9,14,15], dynamic programming [16], non-linearprogramming [17], mixed-integer programming [18], Benders[19,20], and also decomposition techniques such as Branch-and-

List of symbols

cij cost of circuit between buses i jfij power ow between buses i jcij susceptance between buses i jnij number of added circuits between buses i jn0ij number of circuits in the base case between buses i jf ij maximum power ow between buses i jnij maximum number of circuits between buses i jS branch-node incidence matrixg generation vectord demand vectorw;wk ctitious generation vector for the base case and for the

k-th generation scenariof vector of power owshi voltage angle at bus iX set of candidate branchesX1 subset of generators in the lower limitX2 subset of generators in the upper limitNb number of busesa penalization factor of load sheddingd savings of supplying additional demand

C.A. Correa Florez et al. / Electrical Powin the planning process. Under a market environment, the networkexpansion must ensure equity in access for all system participants,which leads to additional complications in the model. The follow-ing paragraphs summarize some of the proposed approaches toface these new challenges.

Reference [29] develops a multi-period model which takes intoaccount nodal prices, line congestion, nancial investment param-eters and their relation with the amortization during the planningperiod. The model is validated on the Spanish network and differ-ent scenarios of demand and contingencies are used.

A multiobjective methodology is presented in [30], incorporat-ing investment cost, congestion cost and reliability level, which areto be minimized. A multi-period model is solved and the NSGA-IIalgorithm is used to return a set of non-dominated solutions.

Author in [31] used an improved differential evolution model toaddress the TEP. Market considerations were included by calculat-ing annual generation cost for different technologies and by addingannuitized cost of transmission. The author also performs a com-parison of the results of differential evolution and the genetic algo-rithm for the IEEE 30-bus system.

The work presented in [32] solves the TEP by including the cur-tailment cost for bilateral transactions and the one associated tocustomers for spot market, besides the investment for new trans-mission equipment. This way, the network is reinforced in such away that congestion constraints are alleviated in order to allowmarket transactions. Benders decomposition is used and the meth-odology is validated on the South-Brazilian system.Particle Swarm Optimization was used in [33] to obtain optimalplans for Garver and IEEE 24-bus systems. The problem considers amulti-year model and the expansion process depends on theinvestment costs, and the social welfare. A performance compari-son of different swarm approaches is also carried out.

Another market approach is shown in [34]. The objective func-tion includes operation cost, load curtailment and investment costfor different load levels, with the idea of providing equity to allmarket participants. The model considers multiple stages and thesolution is found by a genetic algorithm for Garver and IEEE 24-bus system.

The approach presented in [35] used a congestion surplus indexthrough lagrange multipliers, in addition to the transmissioninvestment and the expected energy not supplied. All of the objec-tives are to be minimized and the Strength Pareto Evolutionary

Qt offspring population of the optimization algorithm inthe generational cycle t

vmaxm ; vminm maximum and minimum value of the objective func-tion m

vImj1

m ;vImj1m neighbor solutions for conguration j

ri rank of solution iqdiv number of different bits for diversity checkqmut number of bits for mutationdtotal total system demandng number of generatorsgs number of generation scenariosxk variable x evaluated in the k-th generation scenarioLmax maximum allowed load shedding valueBPC basic planning constraintsMPC multiple generation scenarios constraintsPt parent population of the optimization algorithm in the

d Energy Systems 62 (2014) 398409 399Algorithm (SPEA) was used to obtain a set of Pareto optimalsolutions.

An alternative for treating the described problem is shown in[36]. In this work, a procedure for network reinforcement in aderegulated environment is designed, different patterns for powerow are considered and a decision scheme is incorporated tominimize the risk of the selected plan. The authors design andselect a number of generation scenarios with a probability ofoccurrence for a future year. This problem was also faced in[37] considering network security (N-1 contingency criteria).The way of solving this problem using a mono-objective approachis shown in [38].

1.3. About the present work

This paper proposes an approach for the TEP when full openaccess for generators is considered. As a result, multiple powerow patterns need to be analyzed in order to obtain a set of invest-ment proposals. An enhanced multiobjective algorithm is used toobtain a set of Pareto optimal expansion plans with different levelof investment and future load shedding. The solutions provide ade-quate operative conditions for any load ow pattern resulting fromany dispatch scenario, ensuring low potential values of load shed-ding. This is achieved by considering the feasible Multiple Genera-tion Scenarios (MGS) and also taking into account demand andgeneration as a variable in a narrow range. The proposed method

s:t: Sf g w d 2

r anfij cijn0ij nijhi hj 0 3fij 6 n0ij nijf ij 40 6 g 6 g 50 6 w 6 d 60 6 nij 6 nij 7ij 2 X;nij integer 8

Eqs. (2) and (3) represent the rst and second Kirchhoff laws,respectively.

This problem is usually divided into two sub-problems. Therst, is the investment problem and has the objective of determin-ing the expansion plans that should be evaluated. The investmentplans might have certain level of infeasibility, which is evaluatedby the second subproblem: the operative [2,24,26,37]. The latterresults in a Linear Programming (LP) problemwhen the investmentproposal is known, which is solved in this paper using an InteriorPoint Method. For more information regarding the implementationof this method, readers are advised to examinate reference [40].

These two problems are iteratively solved until a feasible min-imum cost plan is found.

2.1. TEP considering multiple generation scenarios

In order to take advantage of future generation and to supplynecessary power for future loads, the transmission network mustis validated using the Garver system and the IEEE 24-bus RTS testsystem.

The main contributions of this paper are listed below:

Multiple load ow patterns are included in the model to reecta more realistic planning process, by means of multiple genera-tion scenarios and a multiobjective approach. Instead of one expansion proposal, several Pareto optimalexpansion plans are obtained for both test systems. This meth-odology differs from most traditional planning schemes. An original multiobjective algorithm is presented as a tool forthe community related to the electrical power systems andoperational research elds. Performance analysis are carriedout to demonstrate its convenience. Variable generation and demand in each bus is considered inthe model and comparative analysis with xed values is carriedout to show the impacts on the investment costs.

1.4. Organization of the paper

The present work is organized as follows: Section 2 presents themathematical formulation considering MGS. Section 3 shows theproposed multiobjective algorithm used for solving the TEP. Next,simulation tests and results are detailed in Section 4 for Garverand IEEE 24-bus system. Finally, conclusions are drawn inSection 5.

2. Mathematical model

When DC load owmodel is used for representing the transmis-sion network, the mathematical formulation of the static TEP con-sidering load shedding is the following [39]:

min v Xij2X

cijnij aXi2Nb

wi 1

400 C.A. Correa Florez et al. / Electrical Powebe reinforced based on the existence of deregulated markets. Toface this new scheme, different load ow patterns must be takeninto account according to the dispatch scenarios created by themarket rules, by the changes in generation and demand, and bythe availability of primary energy sources [4143].

When considering the multiple load ow patterns, the differentcombinations of generated power in the electric system should beconsidered. These combinations depend on the cost of the MWh ofeach plant, weather conditions, the hourly demand, bids, and ingeneral, the market rules of the specic location. Network shouldbe able to deliver power without load shedding, hence, operativeconditions must be evaluated for all possible generation scenarios,which is the purpose of the presented methodology. Along with theMGS, planners in a specic country should consider detailed mar-ket rules.

For a system with demand (d) and generation (g) level, a gener-ation scenario is dened as any dispatch (within the generationlimits) capable of meeting total demand, as follows:

Xngi1

gi dtotal 9

Given the real nature of the active power, innite feasible sce-narios can be found. The concept of feasible extreme scenario isused to generate a representative set of scenarios, which consistson setting the generators at the upper or lower limit. However,constraint (9) might not be met for a large number of combina-tions. To face this problem, practical extreme scenarios are consid-ered, by dispatching ng 1 generators in their upper/lower limitswhile the remaining generator dispatches the necessary power toestablish generation-demand balance according to (9) [38].

According to the previous ideas, a feasible extreme scenariomust comply with the following constraints:Xi2X1

gi Xi2X2

gi 6 dtotal 10Xi2X1

gi Xi2X2

gi gmaxq P dtotal 11

It is important to point that the number of possible scenarios islarge, and can be calculated as ng 2ng1. Depending on knowledgeof the specic test system and the market, the number of scenarioscan be reduced to avoid prohibitive computational time, however,the present work considers all feasible scenarios to test the pro-posed algorithm and methodology under the most extremeconditions.

2.2. Mathematical formulation with MGS

The formulation of the TEP when market is considered throughMGS, and using the DC model, is the following:

min v Xij2X

cijnij aXgsk1

Xi2Nb

wki 12

s:t: Sf k gk wk d 13f kij cijn0ij nijhki hkj 0 14f kij 6 n0ij nijf ij 150 6 gk 6 gk 160 6 wk 6 d 170 6 nij 6 nij 18ij 2 X 19k 1;2; . . . ; gs 20nij integer 21

d Energy Systems 62 (2014) 398409The objective with this modeling is to obtain an expansion planthat meets the demand under any generation scenario. It is worthto notice that this mathematical model adds even more complex-

from the analyzed population.

er anity, given that the number of variables increases when MGS areincluded. This problem can be solved under a mono-objective ormultiobjective approach, given that load shedding and investmentare conicting objectives.

2.3. Considering variable demand

Traditionalplanningschemes considerxeddemandasa result ofload forecasting for a given horizon. It is possible to include demandin the problem as a variable due to load forecasting uncertainty. Inthis work, load in all buses is considered as a variable, allowing thedemand to vary within certain range. The objective function for theoperative subproblem has to be adjusted including demand:

min v aXgsk1

Xi2Nb

wki dXi2Nb

di 22

It is considered that a > d, indicating that load shedding is moreseverely penalized than supplied demand. These two factorsimpact the objective function as a linear combination of load shed-ding and demand, so any combination that meets a > d impliesthat more importance is given to alleviating 1 MW of load shed-ding than supplying one additional megawatt of load. The minussign in (22) shows that when the investment plan allows meetinga larger demand, the objective function tends to reduce its value,and the plan becomes attractive.

A new constraint must be also added to the model to considerdemand variations:

dmin 6 d 6 dmax 23In general, a narrow range for variations of demand should be

enough, since load forecasting studies in power systems usuallyprovide high, medium and low demand scenarios.

2.4. Multiobjective formulation

Investment in the TEP is conicting with load shedding levels.In the proposed multiobjective formulation, low load shedding val-ues are accepted to form a Pareto optimal set of expansion planswith different cost levels. This allows multiple choices for decisionmakers regarding the selection of one plan, according to higherlevel information.

The complete mathematical formulation is expressed by:

min v1; v2f g 24s:t: v1

Xij2X

cijnij aXi2Nb

wi 25

v2 maxXi2Nb

wki

( )26

Xi2Nb

wki < Lmax 27

n nij 2 BPC 28n nij 2 MPC 29k 1;2; . . . ; gs 30

Lmax is the maximum allowed load shedding, which depends onthe total demand of the specic system.

Eq. (25) calculates the rst objective function to be minimized,which is the cost of the expansion plan, and penalized if load shed-ding in the base case (without MGS) is different from zero. It isimportant to clarify that when certain expansion plan nij is ana-lyzed, objective function (26) measures the most critical out of

C.A. Correa Florez et al. / Electrical Powthe k generation scenarios. This is, objective function v2 is thehighest load shedding in MW, which is obtained after calculatingand comparing the operative conditions for each scenario. FromWhen any individual from the set of non-dominated solutionsdominates any other solution remaining in the population, thenon-dominated set is called Pareto Front.

For a multiobjective problem, the optimal Pareto front shouldbe found, that is, nding the best non-dominated set of solutions.the previous ideas, it can be inferred that for each investment pro-posal (nij), k operative subproblems need to be run in order toobtain the corresponding load shedding levels, although only themaximum is selected at the end.

For tests with variable demand, constraint (23) should be added.

3. Solution of the multiobjective formulation

When planning the transmission system to eliminate conges-tion under any generation scenario, the associated cost increases.Under a single-objective formulation, to achieve zero load shed-ding, expansion plans present high costs. Hence, it becomes impor-tant to explore other expansion plans with lower cost, allowingcertain levels of non-supplied load. The present approach proposesa multiobjective planning scheme that allows low levels of infeasi-bility, idea expressed in Eq. (26). This multiobjective proposal pre-sents a rst objective to be minimized, which is the cost of theexpansion plan, and the second objective measures the load shed-ding of the most critical scenario. These two objectives areexpressed respectively in Eqs. (26) and (25).

It is clear that these two objectives are conicting, given thatlow investment in the transmission system, tends to generate con-siderable load shedding, and vice versa. This characteristic justiesthe importance of using a multiobjective approach.

To implement the multiobjective algorithm, investment pro-posals (nij) are evaluated using formulation (24)(30), to obtainthe values of both objectives: v1 and v2. In order to obtain a setof solutions with minimum levels of cost and load shedding, a mul-tiobjective algorithm has to be implemented, as explained in thenext subsections.

3.1. Concept of dominance

Most of the multiobjective algorithms use the concept of dom-inance, which consists on comparing two solutions to determinewhich one dominates the other. In the case of this work, the objec-tive functions must be minimized, so it is said that solution x1

dominates x2 if these conditions are met [44]:

vmx1 6 vmx2 for m 1;2; . . . ;M. vmx1 6 vmx2 for at least one m0 2 m 1;2; . . . ;M.

When the rst condition is not met by any of the two solutions,it cannot be stated which of them dominates the other. When thishappens the solutions are non-dominated.

This concept can also be extended to nd a set of non-domi-nated solutions belonging to a population. Reference [44] showsin detail the procedure for nding the set of non-dominated solu-tions when N individuals andM objective functions are considered.

3.2. Pareto optimality and ranking of solutions

When treating a multiple objective problem, the concept ofoptimal solution changes. For a multiobjective problem presentingconicting objectives a set of trade-off solutions should be found,and that set must be formed by non-dominated solutions taken

d Energy Systems 62 (2014) 398409 401When analyzing a set of solutions, a sorting is carried out in orderto determine the number of Pareto fronts in a population, and toassign each solution an attribute called the ranking (r). This process

create a new population Rt of size 2NP. Next, objective functionsof R are evaluated and classied through a non-dominated sorting

Algorithm 2. NSGA-II Algorithm [47]

cycle; after this, it is combined with the Parents, obtaining theset of solutions Rt . This fact generates an important computationaleffort due to the need of calculating NP objective functions for theset Qt in each cycle. For reducing computational effort and improv-ing the performance of the multiobjective approach, some of thefeatures of the CBGA are included. The CBGA creates only one off-spring per cycle and maintains the population size constant, hence,reduces the number of times the objective function is calculated.The solution found in each cycle is included as a parent, based onPareto-optimal theory as described in the next paragraphs.

3.4.1. InitializationThe process starts solving the non-linear problem of TEP, which

is a relaxed version of problem (12)(21) [40]. After this, real val-

r ant

in different Pareto fronts. Once the sorting process is terminated, anew population is generated from the solutions in the best non-dominated fronts. This new population is created using the solu-tions in the best Pareto fronts until NP solutions conform the newset. When the size of the last Pareto front entering the new popula-tion exceeds the number of remaining slots, those with larger dis-tance to their neighbors are selected in order to preserve diversity.

To obtain an idea of the density of solutions around a solution i,the average distance to two surrounding solutions is calculated,based on the values of the objective functions. This distance is usedas an estimation of the perimeter of the cuboid, formed by usingthe closest neighbors as vertices, as shown in Fig. 1.

The crowding distance (dImj ) calculation for each solution j,according to an index I, can be found using the followingexpression:

dImj dImj vImj1m v

Imj1m

vmaxm vminm31

The distances consider all of the objective functions, and innitevalue is assigned to the extremesolutions in the analyzedPareto front,given that they have the best value in one of the objective functions.

Then, when each neighbor of the j th solution is taken intoaccount, the objective functions are sorted in ascending ordescending order so that each distance can be evaluated. TheCrowding distance assignment algorithm is shown in Algorithm 1:

Algorithm 1. Crowding-sort (F; )dIm1 1dIml 1for j 2 : l 1 dodImj by using Eq. (31)

end forend for

The use of the distance of a solution is the key for preservingdiversity in the NSGA-II, which is very important in populationbased algorithms. This methodology tends to privilege less sur-rounded solutions to promote them into the next generationalis done based on the value of the objective functions, which in thecase of this work, result from solving the problem (24)(30) toobtain v1 and v2.

The solutions in the best Pareto front in a population areassigned ranking r1, and so on until the worst front. This attributehelps in determining the quality of a solution by its presence in adetermined Pareto front, and is key to understand how the pro-posed genetic operators work, as shown in Sections 3.4.7 and 3.4.3.

3.3. Elitist non-dominated sorting genetic algorithm: the basic NSGA-II

This evolutionary algorithm was proposed in the year 2000[45,46]. In the NSGA-II, the offspring set Qt of size NP, is createdfrom the parents population Pt also of size NP. The offspring popu-lation is created using tournament selection, crossover and muta-tion. After this process, both populations are merged together to

402 C.A. Correa Florez et al. / Electrical Powecycles. In the case of TEP, this allows searching expansion plansin wider areas of the search space, and disregard investment pro-posals with similar values of objective functions.Data Branches, Buses, Demand, GenerationP0 RandomF Non-Dominated sorting (P0)Distances Crowding-sort (Fi;

problems. Although the continuous solution shows an interesting

Another feature of the proposed algorithm is an improvement

Qoriginal Q

er anindication of important paths, it is not totally secure that all ofthe paths would be present in the nal solution.

So said, this solution is used to generate only a few individuals,and the decision of adding a line where nij 0 is taken randomly,and this way the individual has line additions in some of the pathsmeeting nij 0.

After this step, the individual generation is carried out blockingthe paths with nij 0 and solving another non-linear problem. Thisleads to discovering other important paths that are not present inthe base case and that have also certain importance in the planningprocess. The generation of the remaining individuals is then a cyc-lic process of blocking paths, running the non-linear problem andassigning additions, repeated a number of times depending onthe population size.

3.4.2. Diversity vericationAfter the population is created, diversity check is carried out

among the individuals, by comparing each one of the solutions,and ensuring that they are different in at least qdiv bits.

The previous procedure ensures a controlled initialization toavoid large number of lines in the initial population and also spreadsthe solutions in the search space, which is even more important inmultiobjective approaches and population based algorithms.

3.4.3. SelectionIn the selection process two crowding distance tournaments are

carried out in order to select two parents. Since each solution isevaluated using the formulation (24)(30), then two importantattributes can be calculated for each one of them by means of v1and v2: ranking (ri, presence in a specic Pareto front) and distance(di, measure of diversity). In each tournament kk parents are com-peting, and one of them is selected according to the crowding tour-nament selection operator, which is based on the rank ri of theselected parents and the associated crowding distance di. Thepseudocode of the proposed procedure is shown in Algorithm 3:

Algorithm 3. Crowding Tournament Selection [47]

for i 1 : 2 doQ1;Q2; . . . ;Qkk Random (P)j index (minr1; r2; . . . ; rkk)Qbest Qjif j Qbest j 1 thenParenti Qbest

elseo index (mind1; d2; . . . ; dkk)Parentkk Qo

end ifi i 1

end for

It is important to note for each tournament, that the solutionwith better rank is selected as a parent, or the less surroundedone (larger distance) when the rank of competing parents is thesame. In conclusion, this operator tends to select the better rankedsolutions from the Pareto optimality standpoint and the mostdiverse ones, which makes it an elitist operator.

3.4.4. Crossovertransfer ratio, and are also relevant for alleviating load shedding

C.A. Correa Florez et al. / Electrical PowThis work uses single point crossover for parents combination. Itis important to point that in this enhanced approach, no crossoverprobability is predened given that the population (P) remains thefor j 1 : Branches doQOrderedj QoriginalOrderedj 1if Q infeasible thenQOrderedj QoriginalOrderedj

end ifj j 1

end forend if

3.4.7. Promotion

To include an offspring Q into the population, a number of cri-teria must be met in order to ensure that good quality solutionsare promoted to the next generational cycles. In this case, both Par-eto optimal and diversity criteria are taken into account as shownin Algorithm 5.

Before this procedure is carried out, the population Pt and theoffspring Q are merged together in order to perform the rankingof the complete population Rt . This is done by analyzing the infor-mation of all objective functions v1 and v2 of Rt , which is in turnobtained after solving (24)(30). In general, this stage includesthe offspring into the next generational cycle if it is diverse andbelongs to a Pareto front that is best than the current worst, inthe attempt of constantly improve the quality of the population.If the offspring is located in the rst Pareto front (r1) and it differsfrom all other solutions, it is also included. Is important to notethat the only attribute that is taken into account in this stage isprocedure consisting in analyzing the solution outcome from themutation stage. This offspring is subject to a circuit redundancyanalysis in order to determine if Pareto optimality can beimproved, by temporarily retiring circuits and check if Q is still fea-sible. The drawback of this process is the increase of the computa-tional effort, but the trade-off is the possibility of leading thealgorithm towards high quality regions. The outline of this stageis described in Algorithm 4.

Algorithm 4. Improvement

if Q infeasible thenOrdered sort circuit costs in descending ordersame and only one individual is allowed to enter the population ifdiversity and Pareto optimal conditions are met. After crossover ofparents, two offspring solutions are generated and analyzed in orderto keep only one, according to the rank and distance methodologyfollowed in the selection stage. This is done by comparing both off-spring with the entire population Pt and sorting this extended tem-porary set. After this, the offspring with the best features from thePareto optimality and distance logic standpoint, is selected, andthe other one is disregarded. This ensures that the offspring leadsto interesting Pareto and distance based optimal regions.

3.4.5. MutationIn this stage qmut branches are randomly chosen in order to add

or remove circuits. The decision of adding or removing a circuit isalso based on a random parameter. In the case of this paper, a 50/50% probability was chosen.

3.4.6. Improvement

d Energy Systems 62 (2014) 398409 403the raking, and not the distance. Diversity of the offspring isdirectly measured by the number of different bits when comparedwith the rest of the population, as explained in Section 3.4.2.

Non Dom-Crowding

r anOffspringPromotionmseth

A

FiofnebetaorinatedandCrowding

Sorting

Mutation

Improvement

TournamentSelection and

Crossover

Non Dom-inated andCrowding

SortingNon-linearcontrolledPopulation

Initialization

DiversityCheck

(div)andImprovement

Start

404 C.A. Correa Florez et al. / Electrical PoweFor keeping the population size constant, when the candidate Qeets promotion criteria, the worst individual in the population islected to be replaced. This selection is simple: The solution withe lower distance in the worst Pareto front.

lgorithm 5. Promotion

if Qt 2 Pt thenPt1 Pt

else if rQ 1 thenPt1 includeQt

else if rQ < rankmax and Q diverse then

Pt1 includeQtelsePt1 Pt

end if

The structure of the complete proposed algorithm is shown ing. 2, and the stop criteria consists on a predetermined numberPLs without improving the best Pareto front, as detailed in thext section for each test system. Besides the decrease in the num-r of operative problems calculated in each iteration, the compu-tional complexity is reduced to OM1 N=22, whereas in theiginal NSGA-II is OMN2.

Stop CriteriaMet?

stop

yes

no

Fig. 2. Scheme of the enhanced NSGA-II.4. Tests and results

The problem formulated in (24)(30), is solved using theenhanced multiobjective algorithm described in the previous sec-tion and programmed in Matlab R2011a environment. Two testsystems from the specialized literature were used: the 6-bus sys-tem proposed by Garver [9], and the IEEE 24-bus system. Networkdata for these systems can be found in [19,10,36,48].

First, xed demand model is investigated and Pareto fronts areshown for both systems, along with the obtained expansion plans.Then, a set of Pareto optimal expansion plans is shown for 5%uncertainty in demand, demonstrating the increase in supplieddemand and decrease in cost. Lmax was set to 10% and 5% for Garverand IEEE 24-bus system, respectively. The algorithmwas initializedwith the scheme proposed in Section 3.4.1.

4.1. Garver 6-bus system with xed demand

This network has 6 buses, 15 branches, a total demand of760 MW, and a maximum of 5 parallel circuits to be added.

For this test system there are 4 feasible scenarios according to(10)-(11), which are detailed in Table 1. Scenario one, correspondsto generation in bus 1 at the minimum level, in bus 6 at the max-imum, and generator in bus 3 is free to match the demand of760 MW. Using the same logic, the other three generation scenar-ios are created with combinations of maximum, minimum levels,and one free generator.

As shown in the table, generation in buses 3 and 6 are impor-tant to create feasible scenarios. All of them have non-zero gener-ation value in those specic buses. It is clear that each scenarioequals the necessary demand, 760 MW.

The parameters used for this system were the following: 50individual population, qdiv 5;qmut 4 and kk 2, and stop crite-ria is set to 5000 PLs without improving the best Pareto front.

The enhanced algorithm found the Pareto front depicted inFig. 3, and the corresponding circuit additions for the seven plansare shown in Table 2. The most critical scenario for each planranges from 70 to zero MW, and the cost varies from 200 to268 103 USD respectively.

The extreme point with zero load shedding (268 103) is theone reported in [37] and improves the one reported in [49] by2 103 USD, for single objective planning. This shows that the pre-

Table 1Generation scenarios for Garver system.

Scenario Generation (MW)

Bus 1 Bus 3 Bus 6

1 0 160 6002 150 10 6003 0 360 4004 150 360 250

d Energy Systems 62 (2014) 398409sented multiobjective approach contains that specic expansionplan and six additional options. The basic TEP solution is alsoobtained in the Pareto front, which corresponds to an investmentof 200 103 USD.

4.2. Garver 6-bus system with variable demand

When 5% variation in demand and generation is considered andthe same algorithm parameters are used, the obtained Pareto frontis the one in Fig. 4 with the corresponding circuit additions inTable 3. It is concluded after comparing both cases, that relaxinggeneration and demand within certain limit, leads to a decreasein the cost of the expansion plans. For the zero load shedding case,

200 210 220 230 240 250 260 2700

20

40

60

80

Cost [USD103]

Load

shed

ding

220 35.1 n23 1;n26 4;n35 1;n46 2218 46.79 n26 3;n35 1;n36 1;n46 2200 55.5 n26 4;n35 1;n46 2190 69.14 n15 1;n26 3;n35 1;n46 2170 80.62 n26 3;n35 1;n46 2

700 800 900 1,000 1,100 1,200 1,300 1,4000

200

400

Cost [USD106]

Load

shed

ding

Fig. 5. Pareto front for IEEE 24-bus system with xed demand.

Table 4Alternative optima for the seven zero load shedding plans with Fixed demand. IEEE24-bus system.

Branch Circuit additions

1 2 3 4 5 6 7

0102 1 1 1 1 1 1 10105 1 1 1 1 10324 1 1 1 1 1 1 10409 1 1 1 1 1 1 1

C.A. Correa Florez et al. / Electrical Power anFor this case the parameters were: population size of 100 indi-viduals, qmut 4; kk 2;qdiv 9 and Lmax set to 5% of the totalthe cost is reduced 30 103 USD and the total supplied loadincreases to 798 MW.

4.3. IEEE 24-bus system with xed demand

Fig. 3. Pareto front for Garver system without demand uncertainty.

Table 2Expansion plans for Garver system with xed demand.

Cost (103 U$) rg (MW) Circuit additions

268 0.0 n26 4;n35 2;n36 1;n46 2260 13.2 n15 4;n23 2;n26 1;n35 2;n46 2240 18.4 n23 1;n26 4;n35 2;n46 2238 26.1 n26 3;n35 2;n36 1;n46 2231 45.3 n26 3;n35 1;n46 2;n56 1220 58.1 n23 1;n26 4;n35 1;n46 2200 70.0 n26 4;n35 1;n46 2demand, 427.5 MW, and stop criteria is set to 1 million PLs withoutimproving the best Pareto front. This system has 178 feasible sce-narios, which leads to higher computational effort. The bestobtained Pareto front is shown in Fig. 5.

Besides The zero load shedding plan (1330 106 USD), whichwas also reported in [38], there are 38 additional expansion planswith different levels of load shedding and cost. Under the maxi-mum load shedding permitted, the investment of the less expen-sive plan is 756 106 USD with a maximum load shedding of418.99 MW. It is interesting to point that the zero load sheddingplan has seven alternative optima which have not been previouslyreported and are shown in Table 4. In addition, the solutionobtained in the present work for zero load shedding, improvesthe one reported in [49], which has a cost of 1477 106 USD.The solution in references [37,36] presents lower investment costs,

170 180 190 200 210 220 230 2400

20

40

60

80

Cost [USD103]

Load

shed

ding

Fig. 4. Pareto front for Garver system with variable demand.Table 3Expansion plans for Garver system with variable demand.

Cost (103 U$) rg (MW) Circuit additions

238 0.0 n26 3;n35 2;n36 1;n46 2231 16.01 n25 1;n26 4;n35 1;n46 2

d Energy Systems 62 (2014) 398409 405given that those authors did not take into account the 178 scenar-ios but only 4, hence, the solution cannot be directly comparedwith the one obtained here for zero load shedding.

Boxplot in Fig. 6, shows a more graphical idea for load sheddingdistribution of each expansion plan. For all cases, minimum loadshedding is zero. However, for congurations 29 there is a highnumber of scenarios with zero load shedding, given that most ofdata are outliers.

0510 2 2 2 2 1 2 10610 2 2 2 2 2 2 20708 2 2 2 2 2 2 20809 2 2 1 1 1 1 10810 1 1 2 2 3 2 20911 1 10912 1 1 1 1 11011 2 1 1 1 2 2 11012 1 2 2 2 1 1 21113 1 1 2 1 11114 1 1 1 1 1 1 11213 1 11223 1 1 1 1 1 1 11416 2 2 2 2 1 2 21516 21521 1 1 1 1 2 1 11524 1 1 1 1 1 1 11617 2 2 2 2 1 2 11619 1 1 1 1 1 11718 1 1 1 1 2 1 11821 12023 1 1 1 1 1 1 12122 1 1 1 1 1 1 10108 1 1 1 1 1 10208 1 1

192021222324252627282930313233343536373839

onfig

urat

ion

406 C.A. Correa Florez et al. / Electrical Power anIn addition, conguration 16 has higher values of vmean2 andinfeasible scenarios, in spite of having less vmax2 . The selected met-ric only considers maximum values of load shedding, hence thisplan results more attractive when compared to other congura-tions with less vmean and infeasible scenarios, such as plans 17

0 50 100 150 200 250 300 350 400123456789

101112131415161718C

Load shedding (MW)

Fig. 6. Load shedding boxplot for IEEE 24-bus system with xed demand.2

25 or 27. The previous idea leads to the possibility of exploring dif-ferent metrics for objective function 2, and reveal interesting infor-mation for some congurations.

Moreover, there are congurations with a median of zero (quar-tile Q2). This means that a high number of scenarios have zero loadshedding (50% or more).

It is also concluded that conguration 15 has high load shed-ding values, distributed in quartiles Q1 and Q3. Therefore, thereis a high density of data with non-zero load shedding and the med-ian is also higher when compared to surrounding congurations.

On the other hand, when the impact of the 178 scenarios is ana-lyzed, it can be concluded that the most critical scenarios arerelated to low generation levels in buses 1, 2, 7, 16 an 21. Thisinformation is relevant to discard scenarios in real life systemsgiven their low probability of occurrence.

500 600 700 800 900 1,0000

200

400

Cost [USD106]

Load

shed

ding

Fig. 7. Pareto front for IEEE 24-bus system with variable demand.4.4. IEEE 24-bus system with variable demand

In this test, 5% variation in demand at each bus is taken intoaccount, and an upper bound of 1:05g for generation.

The parameters used were the following: 100 individual popu-lation, kk 2;qmut 4;qdiv 10 and stop criteria is set to 1 millionPLs without improving the best Pareto front.

One of the extreme points of the Pareto front in Fig. 7, has a costof 1004 106 USD and zero load shedding, with the followingadditions:

n0102 1; n0105 1; n0309 1n0324 1; n0409 1; n0510 1n0610 2; n0708 3; n0809 2n0810 1; n0911 1; n0912 1n1011 1; n1012 1; n1114 2n1213 1; n1416 1; n1524 1n1617 1; n1619 1; n0108 1n1423 1Besides this expansion plan, 33 additional congurations are

found.When the Pareto fronts for xed demand and uncertainty are

compared, it is concluded that the latter leads to an importantdecrease in investment.

Boxplot in Fig. 8 depicts load shedding for each congurationaccording to each generation scenario. In all cases, minimum loadshedding is zero. For solutions 215, there is large number of sce-narios with zero load shedding, given that all data are outliers.

For solutions 1619 the median is zero, which means that alarge number of generation scenarios are feasible. Again, thereare interesting expansion plans that lead to cost reduction in spiteof having certain level of infeasibility. This is evidenced by the mul-tiobjective approach.

Regarding the generation scenarios, it is found that the low gen-eration levels in buses 1, 2 and 7 lead to infeasible operation forseveral obtained expansion plans. This information is importantfor reduction of scenarios by only considering the most criticalones, thus reducing computational effort.

4.5. Impacts of the MGS versus the basic planning scheme

To analyze the impact of the MGS on the traditional planningschemes, the following lines will address the problem of neglectingscenarios during the planning process, using as a reference the zeroload shedding expansion plans for comparison purposes.

4.5.1. Garver systemWhen the basic planning problem modeled in (1)(4), (8) is

solved with generation rescheduling, the obtained plan has a costof 110 103 USD with additions: n35 1 and n46 3 [39]. Thisexpansion plan is only suitable under a unique generation scenario,but as discussed before, generation levels in each plant depend ondifferent facts and a small change in generation levels, could leadto infeasibility. If this expansion plan is subject to operative analy-sis for the 4 feasible generation scenarios in Table 1, the resultantload shedding values are 300, 300, 120 and 38.54 MW,respectively.

It can be concluded from this data, that the basic planningscheme it not suitable for future operation of the power system,due to the multiple load patterns when generators are dispatched,which is the case in real life systems. This fact explains the impor-

d Energy Systems 62 (2014) 398409tance of solving the TEP with MGS.In addition, there is an evident increase in the cost of the expan-

sion plan under MGS, which also justies the importance of consid-

262728293031323334 Table 6

Comparison for all planning schemes, IEEE 24-bus system.

Parameter Generation (MW)

Basic MGS withuncertainty

MGS w/ouncertainty

Cost (106 USD) 152 1004 1330

C.A. Correa Florez et al. / Electrical Power and Energy Systems 62 (2014) 398409 407ering demand uncertainty to partially alleviate extra costs, asalready shown for Garver system. Table 5 shows the trade-offsbetween the cost, load shedding, and supplied power, for eachplanning scheme. These trade-offs justify the multiobjectiveapproach presented in this work, given that multiple expansionplans can be found and the extra-costs of considering MGS can

0 50 100 150 200 250 300 350 400123456789

10111213141516171819202122232425

Con

figur

atio

n

Load shedding (MW)

Fig. 8. Load shedding boxplot for IEEE 24-bus system with variable demand.be also mitigated, as already discussed.

4.5.2. IEEE 24-bus systemWhen only basic planning constraints are taken into account,

the obtained expansion plan has a cost of 152 106 USD. Thisexpansion plan is related to the following additions:n0610 1;n0708 2;n1012 1 and n1416 1 [26]. If the 178 gen-eration scenarios are analyzed with these reinforcements, the min-imum, average and maximum load shedding values are 144, 825and 1488 MW respectively, which turns this expansion plan intoinfeasible. This clearly shows that including MGS in the analysisis necessary for a proper planning of the future network in orderto face different generation levels.

Table 6 summarizes the main information for all planningschemes and the differences for the most important variables: cost,load shedding and supplied demand.

4.6. Computational performance of the proposed algorithm

To determine the competence of the enhanced algorithm, dif-ferent tests were carried out for both networks and xed demand

Table 5Comparison for all planning schemes, Garver system.

Parameter Generation

Basic

Cost (103 USD) 110Accumulated load shedding under MGS (MW) 758Supplied power 760cases. The performance of the proposed algorithm versus the basicNSGA-II, is measured with the number of PLs to achieve good qual-ity solutions.

4.6.1. Garver systemTo carry out tests with the basic NSGA-II, the initialization

scheme explained in sub Section 3.4.1 was used, and both multiob-jective algorithms used a population of 50 individuals. Each algo-rithm was run 10 times and stopped when the 7 point Paretofront in Fig. 3 was found. For this test system, the Pareto frontwas obtained in all trials for both algorithms, but there was animportant difference in the number of PLs they took to achievethe complete set of solutions. As depicted in Table 7, the basicNSGA-II takes in average, more than 6 times to nd the Paretofront, when compared with the enhanced NSGA-II.

As explained before, the basic NSGA-II needs to calculate morePLs in order to achieve good quality results. This is explained by thefact that in each iteration, an offspring population Q of size NP iscreated, increasing the number of times the operative problemhas to be solved, and also increasing the computational effort. Onthe contrary, the enhanced algorithm exploits the best of theChuBeasley logic, by keeping the population size constant, andreducing drastically the PLs to be calculated. Furthermore, theenhanced methodology controls diversity and assures that allexpansion plans in the population are different, which increasesthe search capability of the algorithm, leading the solutionstowards good quality regions, as demonstrated with the results.

Even comparing the best performance of the basic scheme withthe worst for the enhanced methodology, the latter turns out tostand out, with 14500 PLs versus 45000. From these data, it is con-cluded that in 100% of the cases, the authors proposal convergessignicantly faster to the optimal Pareto front.

4.6.2. IEEE 24-bus systemThis test system presents a muchmore complex and demanding

challenge than the previous one. Mainly for being a larger system,and more importantly, for having 178 scenarios to be evaluated,hence, the computational effort is critical due to the combinatorialexplosion.

The algorithms were run 10 times each, and the enhancedmethodology was able to nd the 39 Pareto points in Fig. 5 for

Average load shedding under MGS (MW) 825 0 0Supplied power 8550 8977.5 8550all trials, within a range of 1014 million PLs. On the other hand,the basic NSGA-II losses Pareto-optimality capabilities, since noneof the trials could nd the entire set of 39 points. In addition, thealgorithm was stopped after 100 million PLs, given that no

(MW)

MGS with uncertainty MGS w/o uncertainty

238 268

0 0798 760

A69

r animprovement was evident up to this point, and also because ofcomputational effort being already prohibitive.

Table 8 shows the summary of the trials performed, the differ-ences of PL computation and Pareto points obtained.

For all cases, the number of PLs with the enhanced NSGA-II isless when compared to the basic scheme and the number of plansis also higher, concluding that performance and optimality areimproved. Again, this shows the advantages of the proposedmethod to handle NP-hard problems such as the TEP includingMGS.

5. Conclusions

A multiobjective approach has been proposed to address theproblem of TEP when MGS are taken into account. The model con-siders cost as one of the objective functions, and low levels of loadshedding as objective function 2, both to be minimized.

A new multiobjective algorithm is proposed to solve the prob-lem. This approach includes features of the NSGA-II and the CBGA,such as crowding distance, elitism, and diversity. This new algo-rithm allows to nd a set of Pareto optimal expansion plans forxed and variable demand. Solutions under these considerationsare found for the 6-bus network proposed by Garver and the IEEE24-bus test system. The proposed algorithm stands out over thebasic NSGA-II, substantially improving computational effort andoptimality.

When market constraints are considered, the cost of the expan-sion plan increases. The multiobjective approach returns a set ofsolutions with lower levels of cost and allows to identify potentialsavings that are not present under a single objective approach. Thetrade-off solutions are to be analyzed by a decision maker to selecta plan with higher level information.

Pareto optimal plans are analyzed to obtain information for loadshedding within the generation scenarios. This information showsthat there are interesting plans to be considered, if other metrics

Table 7Comparison of algorithms to obtain the 7 point Pareto front, Garver system.

Algorithm Parameters

Basic NSGA-II qdiv 5, qmut 4Enhanced NSGA-II qdiv 5, qmut 4, kk = 2

Table 8Performance comparison for the basic and enhanced NSGA-II, IEEE 24-bus system.

Algorithm Parameters

Basic NSGA-II qdiv 9, qmut 5Enhanced NSGA-II qdiv 9, qmut 5, kk = 2

408 C.A. Correa Florez et al. / Electrical Poweare used to calculate objective function 2.Future work can include comparison of these metrics and also

generation scenario reduction by selecting the most critical andrealistic ones. This could lead to decrease computational effort.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.ijepes.2014.04.063.

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Multi-objective transmission expansion planning considering multiple generation scenarios1 Introduction1.1 Modeling and solving the TEP1.2 Planning the transmission network in a market environment1.3 About the present work1.4 Organization of the paper

2 Mathematical model2.1 TEP considering multiple generation scenarios2.2 Mathematical formulation with MGS2.3 Considering variable demand2.4 Multiobjective formulation

3 Solution of the multiobjective formulation3.1 Concept of dominance3.2 Pareto optimality and ranking of solutions3.3 Elitist non-dominated sorting genetic algorithm: the basic NSGA-II3.4 Enhanced multiobjective algorithm3.4.1 Initialization3.4.2 Diversity verification3.4.3 Selection3.4.4 Crossover3.4.5 Mutation3.4.6 Improvement3.4.7 Promotion

4 Tests and results4.1 Garver 6-bus system with fixed demand4.2 Garver 6-bus system with variable demand4.3 IEEE 24-bus system with fixed demand4.4 IEEE 24-bus system with variable demand4.5 Impacts of the MGS versus the basic planning scheme4.5.1 Garver system4.5.2 IEEE 24-bus system

4.6 Computational performance of the proposed algorithm4.6.1 Garver system4.6.2 IEEE 24-bus system

5 ConclusionsAppendix A Supplementary dataReferences