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Electric Power Systems Research 119 (2015) 149–156

Contents lists available at ScienceDirect

Electric Power Systems Research

j o ur na l ho mepage: www.elsev ier .com/ locate /epsr

ong-term transmission system expansion planning withulti-objective evolutionary algorithm

ldir Silva Sousaa, Eduardo N. Asadab,∗

State University of Piauí, Center of Technology and Urbanism, Computer Science, Teresina, PI 64002-150, BrazilUniversity of São Paulo, São Carlos School of Engineering, Department of Electrical and Computer Engineering, Av. Trabalhador São-carlense,00, 13566-590 São Carlos, SP, Brazil

r t i c l e i n f o

rticle history:eceived 24 March 2014eceived in revised form 6 September 2014ccepted 8 September 2014

a b s t r a c t

This article presents a solution framework for the static long-term transmission system expansionplanning problem that improves the classical least-cost planning by taking into account the trans-mission operation flexibility on different generation dispatch scenarios. In the proposed strategy, amulti-objective algorithm is proposed to obtain promising solutions according to the satisfaction of

eywords:ong-term transmission networkxpansion planningultiple generation scenariosulti-objective optimisationetaheuristics

operation scenarios. The solution framework applies the multi-objective evolutionary algorithm SPEA2to obtain a pool of expansion alternatives with the corresponding relation of cost and operation sce-nario satisfaction that will serve as a reference for the planner to select the best ones for detailedanalysis.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

The transmission system expansion planning is a classical prob-em in power system engineering. Essentially, the planning aimst the best expansion of the electrical system which is subject toechnical, economical and environmental constraints.

The classical expansion planning model minimises the cost ofnvestment and the solution provides the location, amount of trans-

ission lines or transformers and the best moment to install themn order to allow the proper power transport, adequate reliabilitynd safety. The modelling of the objectives and the time horizonefine the details of the expansion and the complexity of the cor-esponding mathematical problem. For instance, in the regulatedower systems environment, where only one entity controls theperation and expansion, the main objective is the minimisation ofosts, while in deregulated electricity market, the objective turnsut to maximise the profits, the social welfare, among other objec-ives [1,2].

Due to the complexity of this problem, it is usually treatedeparately as long-term planning and short-term planning. The

ong-term planning considers time horizon up to 20 years and sim-lified network models are used to create solutions that will serves references for the next stage of planning. The variables regarding

∗ Corresponding author. Tel.: +55 16 3373 8706.E-mail addresses: aldirss@uespi.br (A.S. Sousa), easada@usp.br (E.N. Asada).

ttp://dx.doi.org/10.1016/j.epsr.2014.09.013378-7796/© 2014 Elsevier B.V. All rights reserved.

power sources and load growth are variables with high degree ofuncertainties. On the other hand, the short-term planning (up to5 years) works with detailed network modelling, reduced level ofuncertainty and the solutions are close to those that will be selectedas the definite expansion alternative.

The long-term planning has been used as a model to test classicalmathematical optimisation methods [3,4] and heuristics/meta-heuristics [5–8]. Multistage planning has also been considered asin [9]. Three mathematical models have been most used, the dcmodel [6], the transportation model [5] and the hybrid model [7].The ac model has also been proposed [10], however, presents highercomplexity for solving.

In this paper, the mathematical model used is based on dcmodel. Even with approximations, the resulting problem is com-plex due to nonlinear and large number of possibilities whichdepend on the number of decision variables and system load-ing. New metaheuristics such as Particle Swarm Optimisation [11],hybrid GA with fuzzy logic [12] and others have been appliedto this problem with success and recently, two meta-heuristicsdeveloped to solve multi-objective problems have been applied inpower system problems. They are the Non-Dominated Sorting GA-II (NSGA-II)[13,14] and the Strength Pareto Evolutionary Algorithm2 (SPEA2) [15].

This paper proposes improvement on the classical long-termtransmission considering the following objectives to optimise: (a)the investment cost and (b) the satisfaction of the operation sce-nario represented by different power generation patterns. The

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50 A.S. Sousa, E.N. Asada / Electric Pow

ongestion, which is one of the topics in a deregulated powerarket, has not been modelled explicitly due to the impreci-

ion of obtaining congestion costs to present values, since itepends on real generation scenarios and possible power re-ispatch actions with available generators. Another reason is theossibility to consider the security level against congestion casesy considering the most probable operation scenarios in long-termeriod.

The optimisation method developed in this study is based onhe SPEA2 [15] meta-heuristic. The power generation dispatchesre simulated from a probabilistic beta distribution model andhe scenario reduction strategy has been applied to minimise theomputational processing. More information regarding publica-ions on transmission expansion planning problem can be foundn [16].

This paper is organised as follows: In Section 2 the classi-al transmission planning problem is introduced. In Section 3 theulti-objective method derived from the classical model is pre-

ented. In Section 4, tests and results with the method are shown,n Section 5 the robustness of the method is analysed and finally inection 6 the main conclusions are drawn.

. Classical transmission planning model

The static transmission expansion planning problem is formu-ated as MINLP in which the transmission grid is represented by aC power flow model as in the following.

inϕ =∑

(i,j)∈�cijnij + ˛

∑i∈�ri (1)

s.t.

Sf + g + r = d(2)

ij − �ij(n0ij + nij)(�i − �j) = 0∀(i, j) ∈ � (3)

fij| ≤ (n0ij + nij)f ij ∀(i, j) ∈ � (4)

0 ≤ g ≤ g

0 ≤ n ≤ nij

0 ≤ r ≤ d

nij integer; fij unbounded ∀(i, j) ∈ �; �j unbounded ∀j

(4)

cij is the cost of a circuit that can be added to right-of-way i − j;ij is the susceptance of the circuit i − j; n0

ijis the number of cir-

uits in the initial topology (base case); nij is the number of circuitsdded in the right-of-way i − j; fij is the active power flow in i − j;

ij is the maximum active power flow in i − j; v is the total invest-ent in monetary unit; S is the branch-node incidence matrix; g is

vector with elements gk (generation in bus k); g is the maximumeneration limit; nij is the maximum number of circuits that cane added in right-of-way i − j; � is the set of all right-of-ways; �

s the set of load buses; r is the vector of artificial generation withlements rk; ̨ is a penalty factor in $/MW.

The first term of the objective function is the cost of transmissionines and the second one (r) the slack variables which also represent∑

rtificial generators. Feasible solutions require ri = 0. The con-traint (2) models the power balance in each node. The constraint3) is an equivalent expression for Kirchhoff Voltage Law and is theource of the non-linearity.

ems Research 119 (2015) 149–156

3. Transmission expansion planning considering multiplegeneration scenarios

The proposed multi-objective model is presented as follows:

Minϕ =∑

(i,j)∈�cijnij (5)

Max =∑L

q=1lq

L(6)

s.t.

Sf q + gq + rq = d

f qij− �ij(n0

ij+ nij)(�

qi− �q

j) = 0

|f qij| ≤ (n0

ij+ nij)f ij

0 ≤ gqi≤ gq

i

gqi∼ˇ(6, 2, 0, gi)

0 ≤ rqi≤ di

0 ≤ n ≤ nij

nij integer; f qijunbounded ; �q

junbounded ∀j; (i, j) ∈ �

(6)

where cij, � ij, n0ij, nij, fij, gk, g, d, nij , � and � are the same of the pre-

vious dc model; f qij

is the active power flow in i − j in scenario q; fq is

the vector of power flows in scenario q; f ij is the maximum activepower flow in i − j; ϕ is the cost of the expansion plan in monetaryunit; S is the branch-node incidence matrix; gq is a vector with ele-ments gk (generation in bus k) in scenario q; gq

iis the maximum

generation limit of generator i in scenario q; ̌ is the beta distribu-tion function; rq is the vector of artificial generation with elementsri in scenario q;

The first objective (ϕ) is to minimise the total cost of new trans-mission lines and the second objective ( ) refers to maximisingthe number of scenarios q without load shedding (r). The variablelq defined in (7) provides the information whether an expansionplan is feasible for a generation scenario q or not.

lq =

⎧⎨⎩

1, if∑i∈�rqi= 0

0, otherwise

(7)

In the proposed strategy the TEP is solved in two stages. In thefirst stage different dispatch scenarios for which the expansion planmust satisfy are created. In the second stage a multi-objective algo-rithm to obtain the cost and scenario satisfaction curve is applied.The multi-objective strategy aims at searching for solutions thatsatisfy the maximum number of the L simulated generation sce-narios with the least investment cost.

The initial scenario (i.e., the scenario in which all generatorsare allowed to operate from 0 to g) should be among the L sim-ulated scenarios. Once the generation scenarios are created, thesearch for plans that satisfy the largest number of scenarios whileinvesting the minimum of transmission lines is carried out by themulti-objective algorithm.

3.1. Scenario generation

Each solution proposal is verified by testing its feasibility for all

scenarios. When precise information about the operation is avail-able, such as in short-term planning, the best strategy is to analysea limited set of known critical scenarios. However, for long termplanning due to the uncertainty, different operation scenarios must

A.S. Sousa, E.N. Asada / Electric Power Syst

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Fig. 1. Beta probability density function.

e considered in order to not exclude expansion plans that mightecome important in the future.

In this work, the generation has been modelled approximatelyccording to a stochastic model to represent hydro generation. Forhis specific case, the generation pattern is modelled as randomariable with beta probability density function, as shown in Fig. 1.he reason for using beta distribution has been to concentrate theajority of generation close to the rated capacity and to minimise

he occurrence of extreme low generations. L feasible scenarios areandomly generated, with each generator selling electric powernto the pool following the beta probability density function consid-ring the capacity of the generator. The generator i in the scenario

has its operation margins defined as follows (instead of simplyinimum and maximum values).

≤ gqi≤ gq

i(8)

here gqi∼ˇ(6, 2, 0, gi)

A scenario q is considered feasible if∑

i∈�g gqi≥

∑i∈�di. Where

g is the set of generation buses and � is the set of load buses.Note that with this specific distribution, the probability of the

eneration to stay between 60% and 100% of its maximum capac-ty is 84.14%. In other words, the probability of a given generatoro operate 60% below its maximum capacity is 15.86%. However, it

ust be noticed that this strategy should be flexible, so the planneran change the parameters of the beta distribution or even to usether probability distribution which is better suited to the trans-ission system under analysis. In case that the scenarios are well

efined and with reduced number, they can be modeled explicitlyo the mathematical model [17].

.2. SPEA2

The SPEA2: improving the strength Pareto evolutionary algorithms an elitist multi-objective evolutionary algorithm proposed in15]. It maintains an external population P to store a fixed num-er of the non-dominated solutions that have been found duringhe search. At every generation, newly found non-dominated solu-ions are compared with the existing external population and theesulting non-dominated solutions are preserved [18]. Most multi-bjective algorithms use the concept of dominance in order toetermine if a solution x1 is better than a solution x2. With this

bjective in mind, all the objective functions considering both solu-ions are compared. If at least one objective function for which theolution x1 is better than the solution x2 and to the other objectives1 is not worse than x2, the following statements hold true: x1 is

ems Research 119 (2015) 149–156 151

better than x2; x1 dominates x2 and x1 is non-dominated by x2. Theset of all non-dominated solutions is called Pareto-optimal set.

The search for the Pareto-optimal set involves itself two (possi-bly conflicting) objectives: to minimise the distance to the optimalfront and to maximise the diversity of the generated solutions (interms of objective or parameter values). Therefore, SPEA2 putsemphasis on two phases of the evolutionary algorithms: mat-ing selection and environmental selection. The first issue is aboutguiding the search towards the Pareto-optimal front. The secondissue addresses the objective of generating uniformly distributedsolutions along the Pareto-optimal front. Two critical phases ofthe SPEA2 are fitness assignment and environmental selection,described in next section.

3.2.1. Fitness assignmentIn the SPEA2, each individual i in both populations Pt and Pt , a

strength value S(i) representing the number of solutions it domi-nates are assigned.

S(i) = |{j : j ∈ Pt ∪ Pt ∧ i j}| (9)

where i j means that i dominates j and ∧ is the logical and.Eq. (10) is applied to determine the raw fitness R(i) of an indi-

vidual i. The raw fitness is given by the strength of its dominatorsin both external file and current population.

R(i) =∑

j∈Pt∪Pt ,ji

S(j) (10)

It is important to note that fitness R(i) = 0 corresponds to anon-dominated individual, while high value of R(i) means that iis dominated by many individuals.

3.3. Diversity control

To avoid solutions with poor diversity along the Pareto front, adensity estimator is added to the fitness of all individuals. The den-sity estimator is calculated as the method proposed in [15], whichis a derivation of the kth-neighborhood density estimate of [19]:for each individual i the distances to all individuals j in both popu-lation Pt and Pt are calculated and stored in a list. After sorting thelist in ascending order, the k-th element gives the distance of theelement i, denoted as �i. In this article, the Euclidean distance hasbeen considered. The value of k is set as the square root of the pop-

ulation size plus the external file size (k =√N + N). Afterwards,

the density D(i) corresponding to i is defined by (11).

D(i) = 1

�ki+ 2

(11)

In (11), the constant two is added in the denominator to ensureD(i) < 1. Finally, the fitness F(i) of each individual i is defined accord-ing to (12).

F(i) = R(i) + D(i) (12)

F(i) is less than one for non-dominated solutions.

3.3.1. Environmental selectionTo maintain the non-dominated solutions found during the

search, all non-dominated solutions are copied into the externalfile (individuals with F(i) < 1, as mentioned earlier). This process

can be defined formally in (13).

Pt+1 = {i : i ∈ Pt ∪ Pt ∧ F(i) < 1} (13)

Three situations can occur from (13), as listed in the following.

152 A.S. Sousa, E.N. Asada / Electric Power Syst

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Fig. 2. Coding of a candidate solution.

1) Pt+1 = N: the number of non-dominated solutions is equal tothe maximum size of the external file. In this case, nothing hasto be done and the next generation begins.

2) Pt+1 < N: the number of non-dominated solutions is notenough to fill up the external file. In this case, it is assignedRt = Pt ∪ Pt and the population Rt is ordered in increasing orderof fitness. After these steps, the first N individuals from Rt areselected for the next generation.

3) Pt+1 > N: the number of non-dominated solutions is greaterthan the maximum size of the external file. In this situation,a truncation method is applied. The individuals are orderedby F(i) value and those with highest value of F(i) are removedfrom Pt+1 until Pt+1 = N. The truncation method is explained indetail as follows. The individual with the lowest unique valueof �k

i(this individual is in a very populated region) should be

removed. Having more than one individual with the lowestvalue of �k

i(individuals tied with the lowest value), the one

having the lowest value for �li, where l = k − 1 is selected. The

variable l is iteratively decremented by one until the individualwith the lowest value for �l

ibe unique. Algorithm 1 outlines the

main steps of SPEA2 [15].

lgorithm 1. SPEA2nitialisation : Generate an initial population P0 and an empty external fileP0 = ∅. Set t = 0.

itnessassignment : Calculate fitness values of individuals in Pt and Pt asdefined in Section 3.2.1.

nvironmentalselection : Copy all individuals from Pt and Pt to Pt+1. If thesize of Pt+1 exceeds N call the truncation method. If the size of Pt+1 isless than N then fill Pt+1 with dominated individuals of Pt and Pt , asdefined in Section 3.3.1.

toppingcriterion : If t is equal to the maximum number of iterations oranother stopping criterion is satisfied then stop.atingselection : Perform binary tournament selection with replacementon Pt+1 to generate the mating pool.

ariation : Apply recombination and mutation operators to the matingpool and set Pt+1 to the resulting population. Increment t and go back tostep Fitnessassignment.

.4. Characteristics of SPEA2 algorithm for TEP

The evolutionary algorithm developed in this work is defined asollows.

.4.1. Coding of candidate solutionsThe decimal coding (Fig. 2) is used to represent a solution pro-

osal. The vector has dimension equal to the number of candidateaths (nl) where the new lines are going to be added. Elements inhe solution vector represent the number of lines existing in a pathhe value varies from 0 to nij .

.4.2. Initial populationA solution that does not present load curtailment is considered

easible. Therefore, a constructive heuristic algorithm (CHA) basedn modifications to the CHA proposed in [6] has been used to pop-late the feasible pool of solutions. Algorithm 2 outlines the mainteps of the modified CHA.

The base topology is made feasible and it is added to the pop-lation. One individual is randomly selected between the first

ndividual and the last individual from the population. The muta-ion operator and Algorithm 2 is applied to the selected individual.he resulting feasible individual is added to the population. Thisrocess is repeated until the population is filled up.

ems Research 119 (2015) 149–156

Algorithm 2. Modified Constructive Heuristic Algorithm (CHA)input: Unfeasible individual (ı)Step1 : Assume the base topology as current topology and use the dc

model.Step2 : Solve the linear programming (LP) corresponding to the dc model.

If the configuration is feasible go to Step 5. Otherwise go to Step 3.Step3 : Calculate the sensitivity index (SI) according to (14) for each path

i − j.

SIij ={

12

(�∗i− �∗

j)2�ij

}(14)

where �∗i

is the value of �i in the optimal solution of the PL.Step4 : Perform the tournament among elements in SI such that SIij > 0. In

the current approach four elements participate in the tournament. Thepath with the maximum SIij wins the tournament. A line is added to thispath. Go back to Step 2.

Step5 : Once ı is feasible for the base scenario, it is applied to all otherscenarios in order to assign value to the objective ϕ. Stop.

3.4.3. Crossover and mutationThe one point crossover operator has been used in this work.

Two individuals are selected from the mating pool in order to gen-erate two off-springs. In this work, the crossover point is randomlyselected. For mutation, a path with at least one new line (badd) anda path without new lines (bempty) are randomly selected. The muta-tion is applied with one line being removed from path badd and oneline being added to path bempty. nl is the number of candidate paths.The mutation phase is repeated for a determined number of times(nmut).

Algorithm 2 should be applied to all individuals generatedthrough crossover and mutation stages in order to ensure feasibil-ity to the base scenario. The new population is formed by crossrate% of individuals created by crossover and mutrate % of individualscreated by mutation operators.

4. Tests and results

In order to evaluate the proposed method, three systems havebeen used: the Garver six-bus test system [5], the IEEE 24-Bus testsystem (IEEE24) [17] and the Southern Brazilian System (BR46)with generation rescheduling [20]. The algorithm has been pro-grammed in Fortran 90 language. Minos 5.5 solver [21] has beenused to solve the LP problems. All tests have been executed onIntel® PC CoreTM2 Duo, CPU T9550 @2.66 GHz and Linux operat-ing system. In this paper the posterior method to obtain the paretooptimal region is considered. The decision making regarding thebest solution is not analysed because the emphasis is to find areliable pareto front, however ranking methods such as those pre-sented in [14] could be used within an adequate context of analysis.The parameters used in the SPEA2 algorithm are N = 70 (populationsize), N = 30 (archive size), nmut = 3% (mutation rate) of nl (numberof candidate paths), crossrate = 97% (% of individuals created fromcrossover) and mutrate = 3% (% of individuals created by mutation).These parameters have been based on [15] after various tests.

4.1. Generation of scenarios

The methodology to solve the transmission expansion planningproblem with multiple generation scenarios has two phases. Thefirst phase consists of creating L feasible generation scenarios. Thesecond phase consists of searching for solutions that meet the max-imum number of the L simulated generation scenarios with thelowest investment cost. As the number of all possible feasible gen-eration scenarios is infinite, in the proposed methodology only asample of the feasible generation scenario is tested.

It is important to mention that the sample should representthe set of all possible feasible scenarios. Here, the coefficientof variation is used to define the number of scenarios, L. Thecoefficient of variation (ϑ) is a measure of dispersion of data in

A.S. Sousa, E.N. Asada / Electric Power Systems Research 119 (2015) 149–156 153

Fϕ

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ig. 3. SPEA2: Pareto-optimal front to Garver 6-Bus system with SPEA2 algorithm. is the expansion cost and is the amount of scenarios met (1.0 means 100 %).

elation to the mean (ratio of the standard deviation to the mean).t is useful for comparing the variability of two or more samplesf data for different variables or for the same variables because its dimensionless measure. Algorithm 3 is used to define the size ofhe sample. All scenarios in X should be feasible (the summationf generation capacity is larger than or equal to the peak demand).

lgorithm 3. Creation of samplesreate a sample X with 100 scenarios.ssign i = 0 and L = 100.alculate ϑbo for each bus b in sample X.epeatAssign i = i + 1.

Assign k =⌈

0.1 × L⌉

.

Create a sample Xi with k scenarios.Assign X ← Xi ∪ X.Calculate ϑb

ifor each bus b in sample X.

Assign L = L + k.ntil (|norm(ϑi) − norm(ϑi−1)| > )

The value 1.0 × 10−4 has been assigned to the parameter forll tested systems. The total of 500 iterations of SPEA2 have beenun for all systems. In order to check the convergence, the non-ominated solutions have been stored at each 100 iterations.

.1.1. 6-Bus systemThis system has 15 right-of-ways for new circuit additions and

llows generation rescheduling. The peak demand is 760 MW andhe generation capacity is 1110 MW [5]. Algorithm 3 converged to

sample with 573 scenarios.Considering only the objective ϕ the optimal solution resulted in

he investment of $110. The expansion is represented by the num-er of additions in a given path. For instance, the optimal solution

s given in the following:

2−6 = 3, n3−5 = 1.

Which means that path 2-6 got 3 transmission lines and path-5 one transmission line. The least-cost solution that meets allcenarios ( = 1.0) result in an investment of $220 with addition of

transmission lines distributed as the following,

2−3 = 1, n2−6 = 3, n3−5 = 1, n4−6 = 3.

he Pareto front to this system is shown in Fig. 3. It has beenbtained in less than 100 iterations. It is possible to notice thatxpansion alternatives that satisfy over 70 % of the simulated gen-ration scenarios are obtained with cost over $170.

.1.2. IEEE 24-Bus systemThis is a test system with 24 buses and 41 right-of-ways for

ew circuit additions. The peak demand is 8550 MW and the

Fig. 4. SPEA2: Pareto-optimal fronts for the IEEE-24 Bus system. Fronts obtained for100, 200, 300, 400 and 500 iterations. The costs (ϕ) are multiplied by 104.

generation capacity is 10,215 MW [17]. For this system, the Algo-rithm 3 converged with a sample of 1, 689 scenarios.

Considering only the objective ϕ the optimal solution resultedin an investment of $ 152 × 104, which corresponds to the additionof 5 transmission lines distributed as follows:

n6−10 = 1, n7−8 = 2, n10−12 = 1, n14−16 = 1.

The least cost solution that satisfies all scenarios( = 1) has theinvestment cost $1, 057 × 104 with the following circuit additions:

n1−2 = 1, n3−24 = 1, n4−9 = 1, n5−10 = 1, n6−10 = 2, n7−8 = 2, n8−9 = 1, n8−10 = 1,

n9−12 = 1, n10−11 = 2, n10−12 = 1, n11−13 = 1, n12−23 = 1, n14−16 = 1, n15−21 = 1,

n15−24 = 1, n16−17 = 1, n17−18 = 1, n20−23 = 1, n2−8 = 1.

The earlier solution has been found in less than 200 iterations. Littlechanges occur between the Pareto front obtained with iteration 100to iteration 500, as shown in Fig. 4. In terms of scenario satisfaction,investment over $400 shows a steep increase on the satisfaction ofthe simulated scenarios.

4.1.3. Southern Brazilian system (BR46)The Southern Brazilian System is a medium size system that rep-

resents part of southern Brazilian interconnected network. Thereare 46 buses and 79 right-of-ways for new circuits. The totaldemand of this system is 6800 MW and the generation capacityis 10,545 MW [20]. Algorithm 3 has generated a sample with 2087scenarios.

Considering only the objective ϕ, the optimal solution is $70,289 × 103 with 8 circuit additions with the following distribution:

n13−20 = 1, n20−23 = 1, n20−21 = 2, n42−43 = 1, n46−6 = 1, n5−6 = 2 .Differently, the least cost solution that satisfied all scenarios

( = 1 is $246, 983 × 103 and the solution is not necessarily a com-plement of the previous least cost solution as can be observed inthe following solution:

n14−18 = 1, n32−43 = 1, n18−19 = 1, n20−21 = 2, n42−43 = 2, n46−6 = 1, n16−28 = 1,

n19−25 = 1, n31−32 = 1, n28−31 = 2, n31−41 = 1, n24−25 = 2, n40−41 = 1, n5−6 = 2.

The earlier solution has been found in less than 300 iterations.From Fig. 5 it is possible to observe that the algorithm converged

around iterations 400 and 500 for the Southern Brazilian System. Itis also possible to notice significant improvement on system flexi-bility for investments over 160 × 103.

154 A.S. Sousa, E.N. Asada / Electric Power Systems Research 119 (2015) 149–156

Fig. 5. SPEA2: Pareto-optimal fronts for the Southern Brazilian System (BR46).Fronts obtained for 100, 200, 300, 400 and 500 iterations. The cost (ϕ) is multipliedby 103.

Table 16-Bus system: Scenarios distribution for sample size N = 573 and N = 1.0 × 106.

N Surplus (%)

Low Moderate High

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o

Table 26-Bus system: scenarios not met by classical solution of $110 for N = 573 andN = 1.0 × 106 scenarios. Highest LC means the highest load curtailment (in percent).Mean LC shows the mean of the load curtailments in relation to the peak demandof the system.

Surplus (%)

N Low Moderate High Highest LC% Mean LC%

573 81.48 37.41 0.00 29.78 19.551.0 × 106 79.44 44.32 0.00 31.54 27.19

Table 3Six-bus system: Scenarios not satisfied by solution of $220.00 for N = 573 andN = 1.0 × 106. Highest LC means the highest load curtailment (in %). Mean LC is themean of the load curtailments in relation to the peak demand of the system.

Surplus (%)

N Low Moderate High Highest LC% Mean LC %

573 0.00 0.00 0.00 0.00 0.001.0 × 106 0.004 0.00 0.00 1.86 1.75

Table 4IEEE 24-Bus system: Scenarios distribution for sample size N = 1689 andN = 1.0 × 106.

N Surplus %

Low Moderate High

$1, 062.00 × 104) considering samples with size N = 1, 689 andN = 1.0 × 106. As can be noted, almost all scenarios are satisfied bythis expansion plan.

Table 5IEEE 24-Bus system: Scenarios not met considering the solution $ 152 × 104 forN = 1689 and N = 1.0 × 106. Highest LC means the highest load curtailment (in %).Mean LC is the mean of the load curtailments in relation to the classical least costsolution.

Surplus %

573 53.32 37.86 8.821.0 × 106 53.17 37.61 9.22

. Analysis on the results

The algorithm has been successful to obtain the known globalptimal solution for all tested systems (considering only one oper-tion scenario). The extreme solution, which corresponds to thexpansion that satisfies all possible generation scenarios, has alsoeen obtained and in most of the situations resulted in cost pro-ibitive expansion. More important than this type of solution, theareto-optimal front provides the planner means to select thexpansion alternatives observing the cost and the expected sce-arios.

In this section the robustness of the proposed methodologys verified. Firstly the sample created from Algorithm 3 is tested

hether it represents all operation scenarios. Then the optimalolution taking into account only the objective ϕ and the solutionhich considers both with ϕ and are verified for one million

andomly generated scenarios. The scenarios have been classifiedccording to three equally divided patterns of available generationurplus in comparison to the peak demand.

.1. 6-Bus system

The 6-Bus system has 32% of surplus in the generation. The sur-lus have been divided in three classes of generation scenarios:cenarios with low generation surplus (0% ≤ surplus < 11 %); sce-arios with moderate generation surplus (11% ≤ surplus ≤ 21 %) andcenarios with high generation surplus (21% < surplus ≤ 32 %). Withhe Algorithm 3, the three classes of scenarios with sample size

= 573 and with size N = 1.0 × 106 are created as shown in Table 1.Table 2 shows the percentage of scenarios not satisfied by the

opology with cost $110 for samples size N = 573 and N = 1.0 × 106.s an example, 81.48% of the low generation surplus scenariosresent load curtailment and the highest load curtailment is 29.78%or sample size 573. The sample size 573 is adequate since negli-

ible difference is observed when using a sample of one million ofcenarios.

Table 3 shows the amount of scenarios not satisfied by the topol-gy with cost $220 for samples sizes of N = 573 and N = 1.0 × 106.

1689 96.63 3.32 0.061.0 × 106 96.75 3.25 0.01

The multi-objective solution was successful and satisfied almostall scenarios.

5.2. IEEE 24-Bus system (IEEE24-Bus)

The IEEE 24-Bus system presents 16% of surplus in the gen-eration. The scenarios are divided as follows: scenarios withlow generation surplus (0% ≤ surplus < 5 %); scenarios with mod-erate generation surplus (5% ≤ surplus ≤ 11 %) and scenarios withhigh generation surplus (11% < surplus ≤ 16 %). Algorithm 3 hasbeen used to create these scenarios with sample sizes 1689 andN = 1.0 × 106, as shown in Table 4 and almost all scenarios are inthe low surplus level.

Table 5 shows the amount of scenarios not met for each gen-eration level and the load curtailment of the classical least costsolution with cost $ 152 × 104 for scenarios with sample size N = 1,689 and N = 1.0 × 106. It is possible to verify that for low and moder-ate surplus, the classical optimal solution is limited to satisfy onlyscenarios where large generation capacity is available and resultsin only 0.06 % of total scenarios.

Table 6 shows the amount of scenarios not met by thesolution that satisfies all generation scenarios (investment of

N Low Moderate High Highest LC% Mean LC%

1689 100.00 100,00 0.00 12.14 9.761.0 × 106 99.98 99.19 66.67 14.73 10.68

A.S. Sousa, E.N. Asada / Electric Power Systems Research 119 (2015) 149–156 155

Table 6IEEE 24-Bus system: Scenarios not met considering the solution with investment$1, 062.00 × 104 for sample size N = 1, 689 and N = 1.0 × 106.

Surplus %

N Low Moderate High Highest LC% Mean LC%

1689 0.00 0.00 0.00 0.00 0.001.0 × 106 0.02 0.00 0.00 0.87 0.51

Table 7Southern Brazilian System: Scenarios distribution for sample size N = 2, 087 andN = 1.0 × 106.

N Surplus %

Low Moderate High

2087 91.71 8.24 0.051.0 × 106 91.14 8.83 0.03

Table 8Southern Brazilian System: Scenarios not met considering the solution with invest-ment 70, 289 × 103 for N = 2, 087 and N = 1.0 × 106.

Surplus %

N Low Moderate High Highest LC% Mean LC%

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Table 10Number of LP and elapsed time for SPEA2 (s).

System CR NLP Time (s)

6-Bus 001 − 100 305, 031 390.3IEEE24 200 − 300 21, 753, 172 10, 457, 31BR46 200 − 300 37, 810, 843 39, 489.55

Figs. 6 and 7 show the pareto front of SPEA2 and NSGA-II forIEEE24 and BR46 system. Both solutions show close profile andbetter performance is observed for SPEA2.

2087 100.00 90.91 0.00 25.84 16.421.0 × 106 100.00 99.99 0.00 30.98 23.07

.3. Southern Brazilian system (BR46)

The Southern Brazilian System has 35% of generation surplus.he scenarios are divided as follows: scenarios with low generationurplus (0% ≤ surplus < 12 %); scenarios with moderate generationurplus (12% ≤ surplus ≤ 23 %) and scenarios with high generationurplus (23% < surplus ≤ 35 %). The Algorithm 3 has been used to cre-te these scenarios with sample sizes N = 2087 and N = 1.0 × 106, ashown in Table 7. Similarly to the earlier case, most of scenariosre low surplus scenarios, while the high surplus is less or equal to.05% of the total scenarios.

Table 8 shows the amount of scenarios not met by the classi-al least cost solution (investment of 70, 289 × 103) consideringamples with size N = 2087 and N = 1.0 × 106. This classical solutions also very limited and cannot satisfy variations on the operationcenario. Load curtailment of almost 30% is observed for this expan-ion proposal. On the other hand, the expansion proposal that costsore than three times of the previous solution is able to satisfy all

peration scenarios. 2, 087 scenarios resulted in a representativeample.

Table 9 shows the percentage of scenarios not met by theolution that satisfies all generation scenarios (investment of35, 681 × 103) considering samples with size N = 2087 and

= 1.0 × 106. When considering the sample with 2087 scenarioshere are no violations.

.4. The performance of SPEA2

Table 10 shows information about the performance of SPEA2lgorithm for the three systems. The column convergence range

able 9outhern Brazilian System: Scenarios not met by the solution with investment of235, 681.00 × 103 for N = 2, 087 and N = 1.0 × 106.

Surplus %

N Low Moderate High Highest LC% Mean LC%

2087 0.00 0.00 0.00 0.00 0.001.0 × 106 0.26 0.00 0.00 1.54 1.02

Fig. 6. Pareto-optimal fronts for the IEEE24 bus system obtained with NSGA-II andSPEA2 meta-heuristics.

(CR) shows the range of the number of iterations until the con-vergence of the algorithm. As many meta-heuristic that requirestesting a solution proposal (which results in solving an LP problem),the column NLP shows the maximum number of linear program-ming problems solved in the range. For instance, in the first row inTable 10, 305, 031 is the number of LP solved in 100 iterations. Weuse this number as a performance index for the meta-heuristics.Column Time shows the average elapsed time in seconds for theconvergence of the algorithm. The algorithm was set to run for 500iterations for all systems. The best Pareto front was stored at each100 iterations and after completing, the Pareto fronts have beencompared to determine the CR of the algorithm.

To assess the overall quality of SPEA2, the NSGA-II has beenprogrammed and presented the following statistics of number ofiterations and computation time (Table 11).

Fig. 7. Comparison of Pareto-optimal fronts for the BR46 system obtained withNSGA-II and SPEA2 meta-heuristics.

156 A.S. Sousa, E.N. Asada / Electric Power Syst

Table 11Number of LP and elapsed time for NSGA-II (s).

System CR NLP Time (s)

6-Bus 001 − 100 303, 510 386.3

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IEEE24 400 − 500 36, 151, 929 17, 461, 30BR46 400 − 500 62, 834, 369 64, 105.36

. Conclusion

This paper proposes an extended mathematical model to theong-term transmission system expansion planning that takesnto account different generation dispatch schemes. The topol-gy obtained from the classical centralised transmission expansionlanning satisfies a negligible amount of generation scenarios. Theroposed method expands the model which allows obtaining thexpansion options with correlation to operation flexibility giveny the amount of scenarios met by the expansion. The scenariosre created based on beta distribution and the proposed scenarioeduction strategy was adequate to represent the whole distribu-ion with fewer samples. The extended method also allows usingny algorithm that provides reliable scenarios generation. Its mainse is for the initial screening of a large number of options forosterior detailed planning phase.

The solution method relies on SPEA2 algorithm and hashown better performance in comparison with NSGA-II, howeveroth present similar solution quality. One possible explanationegarding the difference in performance is due the use of exter-al file by the SPEA2, which is not present in NSGA-II. Since thexternal file contains only non-dominated solutions and the recom-ination and mutation are done based on this file, the quality ofhe offsprings have low degree of infeasibility, resulting in lessumber of iterations to obtain feasible solutions. From the testshown in this paper, similarly to methods that use or create dif-erent operation scenarios and includes the evaluation of systemeliability, huge computational processing is required. The use ofarallel computation is an important strategy to reduce the com-utation time, since the scenarios can be calculated separately. Anfficient parallel algorithm is a trend to be covered in the comingapers.

cknowledgement

The authors thank the State of São Paulo Research FoundationFAPESP), Brazil, for the financial support.

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ems Research 119 (2015) 149–156

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