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Abstract — The Transmission Expansion Planning (TEP)

problem is considered a very complex problem due to its combinatorial and nonconvex features. Some analytical and meta-heuristic methods have been proposed to tackle it, however, it is recognized that new efficient optimization tools are still needed. Particle Swarm Optimization has been an evolving research area in the last ten years and many interesting and successful applications in a variety of complex problems have shown the potential of this technique. In this work, two state of art Particle Swarm Optimization (PSO) based algorithms, known as Unified Particle Swarm Optimization (UPSO) and Evolutionary Particle Swarm Optimization (EPSO), are used to solve the above-mentioned problem. Comparisons, detailed analysis, guidelines and particularities are shown in order to apply the PSO technique for realistic systems. Also, results are provided for test and realistic power systems.

Index Terms— Electric Power Systems, Evolutionary Particle Swarm Optimization (EPSO), Swarm Intelligence, Transmission Expansion Planning, Unified Particle Swarm Optimization (UPSO).

I. INTRODUCTION Meeting the power demand at minimum cost is a major requirement in the electric power systems industry. For this end, it is indispensable to carry out multiple hierarchical optimization tasks. These tasks range from the power system expansion planning to scheduling real-time operation. The Transmission Expansion Planning (TEP) entails to determine all the changes needed in the transmission system infrastructure, i.e. additions, modifications and/or replacements of obsolete transmission facilities, in order to allow the balance between projected demand and the power supply, at minimum investment and operational costs. In addition, the TEP must meet technical, economic and environmental constraints in a long-term planning horizon. It is well-known that the TEP is an integer-mixed, nonlinear, and nonconvex optimization problem. The solution to the TEP problem consists of determining the quantity and type of transmission facilities, their location in the power system and

This work was supported in part by Fundação de Amparo a Pesquisa do

Estado de São Paulo / FAPESP) S. Torres and C. Castro are with the University of Campinas (UNICAMP),

Brazil, Campinas (email: santiago.ieee@gmail.com; ccastro@ieee.org). R. Pringles and W. Guaman are with the University of San Juan (UNSJ), San Juan, Argentina.

the scheduled year of these throughout the planning horizon. The size of today’s electric power systems, the discrete nature of the investment decisions and the uncertainties in the demand growth rate and future generation sites, make the TEP problem a large scale, stochastic and hard combinatorial problem. As a consequence, the TEP optimization problem has been simplified with the use of static and multi-stage models to reduce the scope, complexity and the calculation time [1]. Classical optimization techniques were employed in seeking solutions to the TEP problem. These techniques demand large computing time, due to the dimensionality curse, in this kind of problem. Heuristic methods emerged as an alternative to classical optimization methods; their use has been very attractive since they were able to find good feasible solutions, demanding a small computational effort. However, they cannot guarantee the global optimal solution to the TEP problem. Meta-heuristic methods emerged as an alternative to the two previous approaches, producing high quality solutions with moderate computation time. Nowadays, novel meta-heuristic techniques like Particle Swarm Optimization (PSO) have been successful in tackling power systems related problems, and constitute a serious option when one has to solve complex optimization problems [2-6]. Also, some PSO variants such as Unified Particle Swarm Optimization (UPSO) and Evolutionary Particle Swarm Optimization (EPSO) have been developed in order to improve the overall PSO performance [7-9]. In this paper, the main objective is to provide the Power System community with a comparative evaluation of the two above-mentioned novel, state of art, PSO variants applied to the Static Transmission Expansion Planning (STEP). Although simpler than the multistage model, STEP is a formidable large-scale, mixed integer programming problem that frequently presents many local sub-optimal solutions, which grows exponentially with the network size. The performances of both algorithms are tested using small and medium-sized power systems, and the results are compared among them. In the following sections the PSO techniques are described and the STEP problem is formulated. Subsequently, the proposed solution algorithms are presented and the results of their application are reported.

II. MATHEMATICAL TOOLS In this section, the Particle Swarm based optimization tools used in this work are described, mainly based on references

Comparison of Particle Swarm Based Meta-heuristics for the Electric Transmission Network

Expansion Planning Problem Santiago P. Torres, Senior Member, IEEE, Carlos A. Castro, Senior Member, IEEE, Rolando M.

Pringles, Member IEEE, Wilson Guaman, Member IEEE

978-1-4577-1002-5/11/$26.00 ©2011 IEEE

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[7-11]. Also, the mathematical model used to represent the STEP problem is presented.

A. Particle Swarm Optimization Basics Swarm intelligence is a branch of artificial intelligence that studies the collective behavior of complex, self-organized, decentralized systems with social structure. Such systems consist of simple interacting agents organized in small societies (swarms). The aggregated behavior of the whole swarm exhibits traits of intelligence. In the algorithms based on swarm intelligence, there are five basic principles:

1) Proximity: ability to perform space and time computations.

2) Quality: population ability to respond to environmental quality factors.

3) Diverse response: ability to produce a wide set of different responses.

4) Stability: ability to retain robust behaviors under mild environmental changes.

5) Adaptability: ability to change behavior when it is dictated by external factors.

PSO, a swarm intelligence technique developed by Kennedy and Eberhart [11], is a stochastic optimization algorithm based on social simulation models. The development of PSO was based on concepts that govern socially organized populations in nature, such as bird flocks, fish schools, and animal herds. It employs a population of search points that moves stochastically in the search space. The best experience or position achieved by each individual is retained, and then communicated to part or the whole population. The communication scheme is determined by a fixed or adaptive social network that plays a crucial role on the convergence properties of the algorithm. In a mathematical framework, let, nA ⊂ R , be the search space, and, : nf A Y→ ⊆ R , be the objective function, the population is called the swarm and its individuals are called the particles. The swarm is defined as a set

1 2( , ,..., )= Ti i imx x xS of N particles, defined as

1 2( , , ..., ) , 1, 2,...., , = ∈ =Ti i i imx x x x A i N where N is a user-

defined parameter dependent on the problem. Each particle can be an m-component vector, which also defines the dimension of the problem. The objective function, f(x), is assumed to be available for all points in A. Therefore, each particle has a unique function value ( )= ∈i if f x Y . The particles are assumed to move within the search space A iteratively, adjusting their position shift, called velocity, and denoted as 1 2( , , ..., ) , 1, 2,...., . = =T

i i i imv v v v i N Velocity is updated based on information obtained in previous steps of the algorithm. This is implemented in terms of a memory, where each particle can store the best position it has ever visited during its search. Also, let { }1 2, ,....., ,= mp p pP be the memory set which contains the best positions

1 2( , , ..., ) , 1, 2, ...., , = ∈ =Ti i i imp p p p A i N ever visited by

each particle. Assuming a minimization problem, let g be the index of the best position with the lowest function value in P at a given iteration t, then the first developed version of PSO was defined by

( ) ( )1 1 2 2( 1) ( ) ( ) ( ) ( ) ( ) ,ij ij ij ij gj ijv t v t c R p t x t c R p t x t+ = + − + − (1)

( 1) ( ) ( 1), + = + +ij ij ijx t x t v t (2) where i=1, 2,…,N j=1, 2,...,m, t represents the iteration

counter; R1 and R2 are random variables distributed uniformly within [0, 1], c1 and c2 are weighting factors called the cognitive and social parameters respectively. At each iteration, after the update and evaluation of particles, the best positions (memory) are also updated. The new determination of index g for the updated best positions completes an iteration of PSO.

B. Unified Particle Swarm Optimization Two variants of PSO were developed to take advantage of properties of exploration/exploitation, namely, the global (gbest), where the entire swarm is considered as the neighborhood of each particle, and the local (lbest), where neighborhoods are strictly smaller. The concept of neighborhood comes into play in order to reduce the global information exchange scheme to a local one, where information is diffused only in small parts of the swarm at each iteration. Each particle assumes a set of other particles to be its neighbors and, at each iteration, it communicates its best position only to these particles, instead of to the whole swarm. Thus, information regarding the overall best position is initially communicated only to the neighborhood of the best particle, and successively to the rest through their neighbors. In general, a neighborhood of a particle xi belonging to a swarm 1 2{ , ,..., },= Nx x xS

is defined as a set

1 2{ , ,..., },=

Si n n nNB x x x where, 1 2{ , ,..., } {1,2,..., },⊆Sn n n N is the set of indices of its neighbors. The global variant converges faster towards the overall best position than the local variant. Therefore, the former stands out for its exploitation ability. The local variant has better exploration abilities, since information regarding the best position of each particle is gradually communicated to the other particles through their neighbors. The precise neighbor’s definition adopted here is as following

1 1 1 1{ , ,..., , , ,..., , },− − + − + + − +=i i r i r i i i i r i rNB x x x x x x x where parameter r determines the neighborhood size, and is known as neighborhood radius (NR). The Unified Particle Swarm Optimization (UPSO), developed by Parsapoulus and Vrahatis [9], tries to combine the two variants, in terms of their exploitation/exploration properties, in a generalized manner. It can be described by the following equations.

( ) ( )1 1 2 2( 1) ( ) ( ) ( ) ( ) ( ) , ij ij ij ij gj ijG t v t c R p t x t c R p t x tχ ⎡ ⎤+ = + − + −⎣ ⎦ (3)

( ) ( )1 1 2 2( 1) ( ) ( ) ( ) ( ) ( ) ,ij ij ij ij lj ijL t v t c R p t x t c R p t x tχ ⎡ ⎤+ = + − + −⎣ ⎦ (4)

( 1) ( 1) (1 ) ( 1),+ = + + − +ij ij ijv t uG t u L t (5)

( 1) ( ) ( 1),+ = + +ij ij ijx t x t v t (6) where Gij(t+1) is the velocity update of the particle xi for each component j, for the global PSO variant, g denotes the index of the overall best position Pg. Lij(t+1) is the corresponding part of the local PSO variant, l represents the index of the best position Pl in the neighborhood of xi, χ is a parameter called

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the constriction factor introduced by Clerc and Kennedy [10] in order to enhance the PSO performance, u∈[0, 1] is the new parameter called unification factor, which controls the influence of the global and local velocity update.

C. Evolutionary particle swarm optimization Evolutionary Particle Swarm Optimization (EPSO) is a novel optimization meta-heuristic algorithm. It combines the concepts of Evolutionary Strategies (ES) and Particle Swarm Optimization. Under the name of Evolutionary Strategies an important number of models have been development. Evolutionary algorithms have been inspired in the biological evolution of species; these rely on Darwinist selection to promote progress toward the optimal. The algorithm EPSO relies on a set of particles that evolve in the search space trying to find the optimal point in this space. Unlike PSO, the evolution not only looks at the behavior of particles but also in the weights that affect movement of these when they move forward in the search space. One of the most important features of EPSO is that it is a self-adaptive method, which automatically tunes its parameters or behaviors in order to produce an adequate rate of progress towards the optimum [7]. EPSO has two mechanisms (evolutionary and self-adaptive) acting in sequence each one with its own probability of producing not only better individuals, but also a better average group. At a given iteration, each particle is defined by a position in the search space xi(t) and a velocity vi(t). At a given moment in time t, there is at least one particle that has the best position in the search space. The population of particles recognizes such position bg, then the particles tend to move in that direction, furthermore each particle is also attracted to his previous best position bi. In the course of each generation (iteration), the particles will reproduce and evolve according to the following steps: Replication: each particle is replicated r times, giving rise to identical new particles. Mutation: strategic parameters (wi) which affect the particles movement are mutated. Reproduction: each mutated particle generates an offspring according to the particle movement rule. Evaluation: each offspring are evaluated with an objective function. Selection: by stochastic tournament or other selection procedure, the best particles survive to form a new generation. The movement or reproduction rule of the particles is the same as represented by (6). In EPSO, the velocity of the particle (5) is replaced by

( ) ( )* * * *1 2 3( 1) ( ) ( ) ( )ij i ij i i ij i g ijv t w v t w b x t w b x t P+ = + − + − (7)

where bi is the best point found by particle i in its past life up to the current generation; bg is the best global point found by the warm of particles in their past life up to the current generation, xij(t) is the location of particle i, dimension j, at generation t; vij(t) is the velocity of particle i at generation k, wi1is the weight of inertia term, wi2 is the weight of memory term, wi3 is the weight of cooperation term, P is the communication factor.

The particle velocity vij(t) is made up by three terms: the first term of the summation represents inertia, because the particle keeps moving in the direction it had previously moved; the second term represents memory, the particle is attracted to the best point in its past trajectory, finally the third term represents the cooperation between particles of the swarm, i.e. the particle is attracted to the best point found by all particles. The particle movement rule is illustrated in Figure 1.

Fig. 1 - Movement of particles in EPSO algorithm.

In the movement rule, symbol * indicates that these parameters present an evolution as a result of the mutation process. This is an important difference with PSO where the weights are fixed in the optimization process. The mutation rule is

[ ]* log (0,1)ik ikw w N τ= (8)

where logN(0,1) is a random variable with lognormal distribution obtained from the Gaussian distribution with mean 0 and variance 1, i.e. N(0,1); τ is a learning parameter, fixed exogenously, which controls the amplitude of the mutation. Commonly, another form used to mutate the weight is

( )* 1 0,1ik ikw w Nτ= +⎡ ⎤⎣ ⎦ (9)

where N(0,1) is a random variable with Gaussian distribution of mean 0 and variance 1. The previous two equations are equivalent provided that τ is small, so that the negative weights are rejected. Moreover, the global best solution bg is randomly disturbed, which is expressed by

( )* *4 0,1g g ib b w N= + (10)

where, wi4 is the fourth strategic parameter (weight) associated with particle i. This parameter controls the size of the neighborhood of bg where it is more likely to find the global best solution or, at least, a better solution than the current bg. This weight wi4 is mutated according to the general mutation rule shown above. The communication factor P introduces a stochastic draw for the communication between particles. P is a diagonal matrix that affects all particles, contains binary variables of value 1 with probability p and value 0 with probability (1-p). The value of p is an exogenous parameter and it controls the communication of information within the particle swarm. In the classical formulation, p is considered equal to 1. This

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parameter allows a more individual search by each particle, avoiding premature convergence to local optima.

D. Mathematical Modeling – DC Model The mathematical modeling used in this work corresponds to the DC model [12], as shown below.

( , )min

∈Ω

= ∑ kl klk l

v c n (11)

Subject to ,+ =Sf g d (12)

( )( ) 0,− + + =okl kl kl kl k lf n nγ θ θ (13)

( ) ,≤ +okl kl kl klf n n f (14) 0 ,≤ ≤g g (15)

0 ,≤ ≤kl kln n (16)

( )integer; and unbounded, ∈ Ω

kl kl ln fk l

θ

where , , , , and okl kl kl kl kl klc n n f fγ correspond respectively to the

cost of a circuit that can be added to the right of way k-l, the susceptance of that circuit k-l, the number of circuits in the right of way k-l, the number of circuits in the base case, the total power flow and the corresponding maximum power flow by circuit in the right of way k-l. Variable v is the investment, S is the branch-node incidence transposed matrix of the power system, f is a vector with fkl elements, g is a vector with elements gr (generation in bus r) whose maximum value is g , d is the demand vector, kln is the maximum number of circuits that can be added to the right of way k-l, lθ is the phase angle of bus l, and Ω is the set of all rights of way. The objective function is given by (7), the constraints (8) represent the power conservation for each node. This constraint models Kirchhoff’s current law in the equivalent DC network. Constraint (9) corresponds to applying Ohm’s law for the equivalent DC network. Thus, Kirchhoff’s voltage law is implicitly taken into account, and these constraints are nonlinear.

III. IMPLEMENTATION ISSUES This section describes the pseudo codes of UPSO and EPSO implementation in the TEP problem, and the comparison settings to test the performance of both algorithms.

A. Pseudo code of UPSO 1) Prepare electric data network. 2) Set parameters of UPSO (i.e. swarm size, number of

neighbors, maximum number of iterations, c1, c2, χ, initial iteration, unification parameter, etc).

3) Initialize particles positions and velocities randomly. 4) Evaluate the objective function by using the DC model

presented in D. 5) Update the best overall, individual, and local particle

positions. 6) While stop criteria is not met, do the following:

6.1) Increase iterations counter.

6.2) For each particle update velocities by using equations (3, 4, 5, 6, 19). 6.3) Check velocity limits. 6.4) Update swarm. 6.5) Check swarm limits. 6.6) Evaluate the objective function by using DC model presented in D. 6.7) Update the best overall, individual, and local particle positions.

7) End

B. Pseudo code of EPSO 1) Prepare electric data network. 2) Set parameters of UPSO (i.e. swarm size, maximum

number of iterations, initial iteration, etc.). 3) Initialize particles positions and velocities randomly. 4) Evaluate the objective function by using the DC model

presented in D. 5) Update the best overall and individual particle

positions. 6) Replicate particles r times (clones). 7) While stop criteria is not met, do the following:

7.1) Increase iterations counter. 7.2) Mutate clones using strategic parameters (8-10) 7.3) Reproduce swarm and clones using (7, 19). 7.4) Check velocity limits. 7.5) Update swarm. 7.6) Check swarm limits. 7.7) Evaluate the objective function by using DC model presented in D. 7.8) Update the best overall and individual particle positions. 7.9) Select the best particles.

8) End

C. Network Data Normally, the data used for the STEP implementation is related to the analyzed electric networks. The dimension of the problem is given by the number of right of ways, where it is possible to add circuits to the system according to the planned power generation and load patterns.

D. Parameter settings Some parameters are very important in order to assure the convergence of PSO algorithms. The number of particles N is a problem dependent parameter, which should be chosen according to the dimension of the problem. There is no formal procedure to select such parameter, which is mainly a trial and error process. In the case of UPSO, constants c1 and c2 are set to 2.05; therefore, factor χ is 0.729 [10]. In general, the neighborhood radius NR can take values less than the number of particles, in this work NR was chosen to be 1. For EPSO, τ was chosen equal to 0.1.

E. Swarm and velocities initialization The technique of random uniform initialization is used in this work to generate the initial swarm. It is the most popular in evolutionary computation since it allows an equal exploration of the search space. In practice, it only consists of a random

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vector defined within [0, 1] with uniform probability distribution. The produced value is then scaled in the corresponding search space, so that the particles and their corresponding velocities lie strictly within their bounds.

F. Swarm and velocities bounds In order to limit the search space, the minimum and maximum bounds of each particle x has to be defined. In this problem, it corresponds to the number of circuits in each right of way in the original electric network (xmin) and the maximum number of circuits (xmax) allowed in each right of way of the future network, respectively. In addition, the velocity components are also checked in order to clamp it within its limits, [-vmax, vmax], where vmax was defined as xmax/2.

G. Function evaluation For the purposes of this paper, the objective function (11) was slightly modified in order to manage the constraints, in the following way. ( ) ( ), = +FO f x P x (17)

( , )

( ) ,∈Ω

= ∑ kl klk l

f x c n (18)

1 2( , , ..., ), = ∈ ⊂ ni i i imx n n n x A R

where x is the vector that contains the number of circuits nkl, for each particle i of dimension m (in this case, it is the number of candidate rights of way to add circuits), added to each right of way k-l. P(x) corresponds to a penalty function defined as: i) 0 if x is a feasible point, ii) P1 if x violates constraints (8), (9), and (11), and iii) (P2*nl) if x violates constraint (10), where nl is the number of lines where the power flows are over their limit; P1 and P2 can be set to much higher values than the cost of each addition. Constraint (12) is implicitly taken into account when the swarm bounds are defined. With this way of dealing with the constraints, it is possible for the algorithm to avoid isolated nodes in the power network and to converge to a feasible solution. Moreover, the addition of any kind of constraints to the problem is straightforward. G. Update velocities and swarm The PSO techniques presented in Section II are suited for continuous variables. The STEP problem formulated in this paper works with integer variables. Therefore, (6) has been slightly modified, as mentioned in [9], in order to cope with integer numbers in the following way

( 1) [ ( ) ( 1)],ij ij ijx t round x t v t+ = + + (19) So each time xij is updated with the movement equation, then it is rounded to the nearest integer.

H. Stop criteria In this work, two criteria have been used to stop the algorithm and get a solution. The first one is the maximum allowed number of iterations, which also limits the number of function evaluations. Since test systems with optimal known solutions were used, the second criterion is related to the convergence towards function known values. In practice, search stagnation could also have been used as an alternative stop criterion. In addition, the number of function evaluations was used as measure of performance.

IV. TEST RESULTS In this work, it has been used three test systems: 1) Garver 6- bus, 2) IEEE 24-bus, and 3) Southern Brazilian 46-bus. These systems are well known in the transmission planning literature as benchmark systems. The complete data can be found in [12-13]. The algorithms were implemented in MATLAB, running in an Intel i5, 2.27 GHz, 4GB RAM, hardware platform. The open source tool MATPOWER [14], a well-known tool for power systems analysis, was also used, and can manage the function evaluation process by using the DC model presented in Section II. The tests were performed without allowing generation rescheduling.

1) Garver’s 6-bus system This system has 6 buses, 15 candidate branches, with a total load of 760MW. The maximum allowed lines per right-of- way is four. The two PSO variants succeeded to obtain the optimal known value for this system, US$ 200,000, without generation rescheduling. The following circuits were added in this solution: n2-6=4, n4-6=2 and n3-5=1. This topology corresponds exactly to the one shown in the literature [12-13]. The solution for this system can be seen in Fig. 2. In order to compare the performance of the two PSO variants applied to the transmission expansion planning, it follows below (Table 2 a, b) an analysis of the results for the variation of the swarm size. In general, 200 maximum iterations were allowed for the tests. Table 2 (a) Performance of UPSO and EPSO using SS=60 (Max. Iter. = 200) UPSO EPSO u=0 u=0.5 u=1 Test times 100 100 100 100 Success rate (%) 100 88 50 98 Average iterations 29 11 10 20 Standard deviation iterations 8 2 5 5 Average function evaluations 1753 656 1215 2425

Table 2 (b) Performance of UPSO and EPSO using SS=80 (Max. Iter. = 200) UPSO EPSO u=0 u=0.5 u=1 Test times 100 100 100 100 Success rate (%) 100 89 60 100 Average iterations 28 11 12 17 Standard deviation iterations 7 4 4 4 Average function evaluations 2244 883 1922 2784

In both tables, it can be seen that UPSO (u=0) outperforms EPSO. In Table 2 (a), it is shown that UPSO (u=0) uses less function evaluations than EPSO to get full success. In this case, EPSO got 98% of success in obtaining the optimal solution. Table 2 (b) shows that both algorithms reach 100% of success, however UPSO (u=0) demanded a smaller number of function evaluations. It must be mentioned that EPSO outperforms UPSO when the latter uses other u values different from 0. One of the strengths of EPSO is that most of its parameters are tuned automatically; however, the most critical parameter in this work is the swarm size, which is common to all PSO techniques, and its choice is a mix of experience with a trial and error process.

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Fig. 2. Garver System Optimal Plan without Generation Rescheduling

2) IEEE 24-bus system The system consists of 24 buses, 41 rights of way, and 8550 MW. The original topology is shown in Fig.3. The maximum allowed number of circuits per right of way is four. The solution reached by the algorithm matches that optimal solution presented in [12-13], with a cost of US$ 218,000,000, and the following circuits were added: n6-10=1, n7-8=2, n10-12=1, n14-16=1, n16-17=1, n20-23=1. For both algorithms, ten tests have been performed using 600 particles with 500 maximum iterations allowed; in the case of UPSO, different values of unification parameter (0, 0.5, 1) were used. The simulations using u=0 achieved 100% of success, being much better than the results achieved by EPSO. For this system, the results obtained by EPSO were not as good as those obtained for the Garver 6-bus system. Table 3 summarizes the results for this test system. Table 3 Performance of UPSO and EPSO using SS=600 (Max. Iter. = 500) UPSO EPSO u=0 u=0.5 u=1 Test times 10 10 10 10 Success rate (%) 100 20 30 20 Average iterations 212 29 37 35 Standard deviation iterations 41 4 9 11 Average function evaluations 105800 17100 22000 41400

3) Brazilian 46-bus system This is a realistic system that represents the southern part of the Brazilian interconnected system, with 46 buses and 79 rights of way, where it is possible to add new circuits. The total demand for this system is 6880MW. The maximum number of lines allowed per right of way is five in this case. The optimal solution obtained for this system was US$154,420,000, and matches results obtained in previous works using other approaches [12-13]. The circuits added were: n20-21=1, n42-43=2, n31-32=1, n28-30=1, n26-29=3, n29-30=2, n24-25=2, n19-25=1, n46-6=1, n6-5=2. The test was performed by using 1000 particles for both algorithms. In the case of UPSO, it was used u=0 and there were necessary 282000 function

evaluations with 282 iterations to get the optimal solution. EPSO got the optimal solution in 273 iterations by using 546000 function evaluations. In this test system, both algorithms got the optimal solution; however, EPSO used more function evaluations than UPSO. Figure 4 shows the convergence process for this system by using the UPSO algorithm.

V. CONCLUSION This work shows the use of two state of art PSO variations, called Unified Particle Swarm Optimization and Evolutionary Particle Swarm Optimization, applied to the Static Transmission Expansion Planning in electrical networks. Extensive simulations were performed for small and medium-size systems. The results obtained show that the local PSO feature (UPSO u=0) outperformed EPSO not only in robustness but also in computing efficiency. In addition, due to the flexibility of this approach, it is easy to use any mathematical model (DC, AC, etc) and incorporate any kind of constraints to the problem. One drawback, which is common for this kind of techniques, is that parameters such as the size of the swarm must be determined for each system by experience and a trial and error process. The simulation performed for a realistic system showed that Particle Swarm Optimization tools can be applied by planners. The large calculation time spent in the case of large systems could be overcome by using parallel processing or some previous fast heuristic technique in order to obtain good quality initial solutions, which will be treated in a forthcoming work. Ongoing work is being carried out in order to apply other PSO variants to the same problem using more complex mathematical models and larger test systems.

Fig. 3. IEEE 24-bus system

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Fig. 4. Convergence process for the South Brazilian 46-bus system achieved by UPSO.

VI. REFERENCES [1] G. Latorre, R. Cruz, J. Areiza, A. Villegas, “Classification of

Publications and Models on Transmission Expansion Planning”, IEEE Transactions on Power Systems, Vol. 18, No. 2, May 2003.

[2] Y. Del Valle, et.al, “Particle Swarm Optimization: Basic Concepts, Variants and Applications in Power Systems”, IEEE Transactions on Evolutionary Computation, Vol. 12, No. 2, April 2008.

[3] V. Miranda, et. al., “Improving Power System Reliability Calculation Efficiency With EPSO Variants”, IEEE Transactions on Power Systems, vol. 24, no. 4, November 2009.

[4] J. Yi-Xiong, G. Hao-Zhong, Y. Jian-yong, Z. Li, “New discrete method for particle swarm optimization and its application in transmission network expansion planning”, Electric Power Systems Research, pp. 227-233, Elsevier, 2006.

[5] R. Pringles, V. Miranda, F. Garcés. “Expansión optima del sistema de transmisión utilizando EPSO”, VII Latin American Congress on Electricity Generation & Transmission, October 24-27 2007.

[6] H. Shayegui, M. Mahdavi, A. Bagheri, “Discrete PSO algorithm based optimization of transmission lines loading in TNEP problem”, Energy Conversion and Management, pp. 112-121, Elsevier, October 2009.

[7] V. Miranda, “Evolutionary Algorithms with Particle Swarm Movements”, Proceedings of the 13th International Conference on Intelligent Systems Application to Power Systems, pp 6-21, 6-10 Nov. 2005.

[8] V. Miranda, Hrvoje Keko, Alvaro Jaramillo, “EPSO: Evolutionary Particle Swarms,” en Advances in Evolutionary Computing for System Design, Serie: Studies in Computational Intelligence, vol 66, L. Jain, V. Palade, D. Srinivasan, Eds. Springer, 2007, pp 139-168.

[9] K. Parsopoulus, M. Vrahatis, “Particle Swarm Optimization and Intelligence: Advances and Applications”, ISBN 978-1-61520-666-7, Information Science Reference, USA, 2010.

[10] M. Clerc, “Particle Swarm Optimization”, ISBN 978-1-905209-04-0, ISTE, Great Britain, 2006.

[11] J. Kennedy, R. Eberhart, “Swarm Intelligence”, ISBN 1-55860-595-9, Academic Press, USA, 2001.

[12] R. Romero, A. Monticelli, A. García, S. Haffner, “Test systems and mathematical models for transmission network expansion planning”, IEE Proc.-Gener. Transm. Distrib. Vol. 149, No. 1, pp. 27-36, January 2002.

[13] R. Romero, C. Rocha, J.R.S. Mantovani, I.G. Sanchez, “Constructive heuristic algorithm for the DC model in network transmission expansion planning”, IEE Proc.-Gener. Transm. Distrib., Vol.152, No. 2, pp. 277-282, March 2005.

[14] http://www.pserc.cornell.edu/matpower/

VII. BIOGRAPHIES Santiago P. Torres (S’05, M’07, SM’10) received the B.S. from the University of Cuenca, in Ecuador, in 1998, and the Ph.D. degree from the Institute of Electrical Energy of the National University of San Juan in Argentina, in 2007. He is a Post-Doctoral Fellow at the Power Systems Department, University of Campinas, Brazil. His research interests are operation and planning of electric power systems, computational intelligence and optimization applications in power systems.

Carlos A. Castro (S’90, M’94, SM’00) received the B.S. and M.S. degrees from the University of Campinas (UNICAMP), Campinas, Brazil, in 1982 and 1985, respectively, and the Ph.D. degree from Arizona State University, Tempe, in 1993. He has been with UNICAMP since 1983, where he is currently an Associate Professor. Rolando M. Pringles (M’08) obtained the B.E.E. degree from National University of San Juan (UNSJ), San Juan, Argentina in 2003 Currently he is pursuing the Ph.D. degree at Institute of Electrical Energy (IEE) from National University of San Juan (UNSJ). During 2006 he was a visiting researcher at the Institute of Engineering in Systems and Computers of Porto, Portugal (INESC-Porto). His research interests are power systems expansion, methodologies for economical evaluation, power investment under uncertainty, and reliability and risk management. Wilson Guamán (M’10) obtained the B.E.E. from Escuela Politécnica Nacional in Quito, Ecuador in 2003. From 2003-2005, he worked as Maintenance Engineer in a private maintenance service company for the Ecuador Transmission System. Currently, he is a candidate to the Ph.D. degree in Electric Engineering from the Instituto de Energía Eléctrica from Universidad Nacional de San Juan, in San Juan, Argentina. Between January and May of 2010, he was a visiting researcher at the Elektrische Anlagen und Netze Institute from Duisburg-Essen Universität in Duisburg, Germany. His special fields of interest include transmission expansion planning under a framework of uncertainties, optimization methods and the application of artificial intelligence techniques in power systems.