Particle swarm optimisation applications in FACTS ... - IEEE Xplore

2013 IEEE 7th International Power Engineering and Optimization Conference (PEOCO2013), Langkawi, Malaysia. 3-4 June 2013

Particle Swarm Optimisation Applications in FACTS Optimisation Problem Ahmad Rezaee Jordehi, Jasronita Jasni, Noor Izzri Abdul Wahab, Mohd Zainal Abidin Abd Kadir

University Putra Malaysia Serdang, Selangor, Malaysia heuristic approaches [2]. Among these three groups of approaches, heuristics have shown superior performance in comparison with classical and technical approaches. Among heuristics, particle swarm optimisation introduced by Kennedy and Eberhart in 1995, has shown more promising computational behavior dealing with FACTS optimisation problem and its application on this problem is continually increasing. The following features of PSO make it extremely appropriate for solving FACTS optimisation problem.

Abstract –FACTS optimisation is one of the most important and difficult problems in power systems. For solving this problem, so many different approaches have been proposed in the literature. Among them, particle swarm optimisation (PSO) has exposed so promising behavior. In this paper, applications of PSO in FACTS optimisation problem are explained and analysed from the viewpoint of the objectives, used basic PSO variant, PSO parameter selection, multi-objective handling, constraint handling and discrete variable handling. Eventually, some hints for future research is provided.

¾ ¾

It has few parameters to be tuned by user. Its underlying concepts are so simple. Also its coding is so easy. ¾ It provides fast convergence. ¾ Roughly, initial solutions do not affect its computational behavior. ¾ Its behavior is not highly affected by increase in dimensionality. The rest of the paper is organised as follows; in section II, an overview of PSO is provided. In section III, applications of PSO on FACTS optimisation problem are analysed. Eventually, conclusions and some hints for future research are provided in section IV.

Index Terms—Particle swarm optimisation (PSO), FACTS devices.

T

I.

INTRODUCTION

HE continual increase of power system demand prompts so many problems for electrical transmission systems. The problems include power flow violation in transmission lines, voltage depression in busses, static/dynamic instabilities, voltage collapse and so on. For tackling these problems, there exist twofold solutions. The first solution is to construct new transmission lines. But it is too difficult to be conducted due to economical, environmental or political issues. The second solution, which commonly used, is to install Flexible AC Transmission Systems (FACTS) devices. These devices enhance the existing transmission system such that the above-mentioned problems are removed or mitigated. FACTS devices are categorised into three groups including series devices such as thyristor-controlled series compensator (TCSC) and static synchronous series compensator (SSSC), shunt devices such as static var compensator (SVC) and static synchronous compensator (STATCOM) and combined series-shunt devices such as unified power flow controller (UPFC) [1]. After introducing FACTS devices, the main concern of power system utilities is to use them optimally such that the most potential benefit can be extracted from these expensive devices. Actually for a certain power system, their optimal type, location and setting should be determined. However, from optimisation perspective, this problem is so complex since it is a multi-modal, multi-objective, multidimensional constrained problem with mixed discrete/integer variables. Different approaches have been put forward for solving this complex problem. Those approaches can be classified into three main groups; classical, technical and

II.

AN OVERVIEW OF PSO

A. Primary Version PSO is an optimisation technique introduced in 1995 [3]. PSO starts with the random initialisation of a swarm of particles in the n-dimensional search space (n is the dimension of problem in hand). In PSO, each particle keeps two values in its memory; its own best experience, that is, the one with the best fitness value whose position and objective value are respectively and the best experience of the called and whole swarm, whose position and objective value are called and respectively. The position and velocity of particle i are denoted with the following vectors: 1,

2, … ,

,…,

,…, 1, 2, … , The velocities and positions of particles are updated in each time step according to the following equations:

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2013 IEEE 7th International Power Engineering and Optimization Conference (PEOCO2013), Langkawi, Malaysia. 3-4 June 2013

1

exploration at initial stages and is focused more on exploitation at latter stages of the run.

1 1

1

C. Constricted PSO Constricted PSO, like inertia weight PSO, was invented to enhance the exploration capability of PSO and to hinder explosion of swarm. Velocity update equations in this variant are as follows [7]:

2

and are two positive numbers and and Where are two random numbers with uniform distribution in the interval [0,1]. There are three following terms in velocity update equation: 1) The first term which indicates the tendency of a particle to remain in the same direction it has traversing and is called inertia. 2) The second term, called self-knowledge, is a linear attraction toward the particle’s own best experience scaled by a random weight . 3) The third term, called social knowledge, is a linear attraction toward the best experience of the all particles in the swarm, scaled by a random weight . The procedure for implementation of PSO is as follows: 1) Particles’ velocities and positions are Initialised randomly. 2) Particles’ velocities and positions are updated according to equations (1) and (2). 3) Each particle’s and are updated. and are updated. 4) 5) Steps 2-4 are repeated until stopping criterion is reached.

1

1

and

III.

Or in another form: 5

Although velocity bounding mitigates the above-mentioned problem, but it is not efficient enough, so inertia weight PSO and constricted PSO are introduced.

SVC,

B. Inertia Weight PSO In this variant, which is the most commonly-used PSO variant, the velocities of particles in previous time step is multiplied by a parameter called inertia weight. The corresponding velocity update equations are as follows [5], [6]: V

t

1 1

ωV

t

C r

P 1

X

C r

P

PSO APPLICATIONS ON FACTS OPTIMISATION

In this section, all the PSO-based approaches proposed in literature for tackling FACTS optimisation problem will be analyzed. In [13], under each contingency with high risk index (RI) value, the modal analysis (MA) technique is implemented to determine which buses require static var compensator (SVC) installation. The objective function, formulated as a multiobjective optimization problem, aims to maximize load margin while SVC installation cost is minimized. The PSO particles are constructed as:

4

then

8

D. PSO Variants for Complex Environments The above-mentioned PSO variants are merely appropriate to be applied to uni-modal, single-objective, unconstrained problems with continuous variables, whereas most of realworld optimisation problems are not so. Real-world problems are commonly involved with multi-modalities, multiobjectives, discrete/integer variables and constraints. Consequently, their solution is really hard and efficient strategies should be planned for tackling each of mentioned complexities. For tackling each complexity, so many various strategies exist in literature [8]-[12].

3

|

7

Where

In PSO, since velocity update equations are stochasticbased, the velocities may become too high. So they are bounded to a maximum value as follows [4]:

If |

1

,

SVC,

,,…,

SVC,

,

SVC,

,…,

SVC,

,

SVC,

9

Where SVC, and SVC, represent the location and reactance of th SVC and K indicates the number of SVC’s. The resultant multi-objective problem is solved by using the fitness sharing-based non-dominated multi-objective particle swarm optimization algorithm. In the Pareto front set for each considered contingency, the solution with the biggest performance index value is determined for SVC installation. Finally, an SVC installation scheme derived from the union of the SVC installations for all considered contingencies is recommended for load margin enhancement. However, the

X 6

Commonly, inertia weight is decreased linearly during the course of the run, so that the search effort is mainly focused on

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2013 IEEE 7th International Power Engineering and Optimization Conference (PEOCO2013), Langkawi, Malaysia. 3-4 June 2013

tuning process for PSO parameters has not been implemented and their values are not mentioned in the paper. In [14]-[15], the optimal locations of FACTS devices are determined to minimize the installation cost of FACTS devices and to maximize system loadability. Optimizations are performed on three parameters; namely location of the FACTS devices, their settings, and the installation cost for single-type FACTS devices, while in the case of multi-type FACTS devices, the type of device to be located is also considered as a variable. The particle format for single-type FACTS device is as follows: FACTS,

,

FACTS,

,

FACTS,

,

FACTS,

,…,

FACTS,

,

FACTS,

overvoltages and lines’ overloads) under single line contingencies has been determined. The settings of UPFC to be optimized include series voltage source magnitude, series voltage source phase angle, shunt voltage source magnitude and shunt voltage source phase angle. Thus PSO particles are constructed as: ,

,

FACTS,

,

FACTS,

,…,

FACTS,

,

FACTS,

,

FACTS,

10

11

Where FACTS, represents the type of th FACTS device. According to the results, UPFC results in maximum system loadability but the installation cost is high when compared with all other cases and TCSC possesses minimum installation cost with better improvement in system loadability. SVC leads to the lowest installation cost in IEEE 30 and 118 bus systems but with minimum enhancement in system loadability. In [16]-[17], the problem of finding optimal location and setting of SVC and TCSC is formulated as a mixed-integer multi-objective optimization problem. The PSO particle is constructed in four dimensions as follows: SVC ,

SVC , STATCOM ,

STATCOM

,

,

13

indicates the location of UPFC, and Where represent the magnitude and phase angle of series convertor, and are the magnitude and phase angle of shunt also convertor. Firstly, contingency analysis and contingency ranking is implemented. Then, PSO with linearly-decreasing inertia weight is applied to the problem. The results illustrate PSO’s efficiency especially in terms of stable convergence characteristic, and quickness. Moreover, it is concluded that using UPFC in the optimal location with the optimal parameter settings can significantly improve the security of power systems under single line contingencies. In [21] the same methodology has been used for finding optimal location and setting of TCSC. Also in [22], the installation cost of TCSC is added to the objective function and optimal TCSC allocation is conducted. In [23]-[24], linearly-decreasing inertia weight PSO is applied to determine the optimal location and setting of FACTS devices to enhance the loadability of pool and hybrid models in restructured power systems. Among FACTS devices, TCSC, SVC and thyristor controlled Phase Shifting Transformer (TCPST) have been used. For the hybrid model, firstly, secured bilateral transaction matrix with AC distribution factor considering the impact of slack bus is determined. Then, considering secured bilateral transaction matrix, the loadability is enhanced with optimal location and control of FACTS devices [24]. Moreover, computational behaviour of PSO and differential evolution (DE) on FACTS optimization problem are compared and it is concluded that DE outperforms PSO on this problem although PSO is faster. In [25], optimal location and setting of STATCOMs in power system are determined. The approach is based on the simultaneous application of particle swarm optimization (PSO) and continuation power flow (CPF) to improve voltage profile, minimizing power system total losses, and maximizing system loadability with respect to the size of STATCOM. The resultant multi-objective function is solved via linear weighted sum approach, while setting the weights on different values leads to different outcomes for optimization process. Constricted PSO has been used for optimization while its parameters have not been tuned and presented. Simulation results show the suitability of the applied PSO technique in finding multiple optimal solutions to the problem with reasonable computational burden. In [26], optimal location and setting of distributed FACTS devices (D-FACTS) is determined by linearly-decreasing inertia weight PSO in order to minimize line overloads considering the cost of D-FACTS devices. Among D-FACTS devices, distributed series capacitor and distributed series

Where FACTS, and FACTS, represent the location and setting of th FACTS device and K indicates the total number of FACTS devices to be installed. In the case of multi-type devices, the type of devices should be optimized together with their location and setting. So, the particle is formed as: FACTS,

,

12

Where SVC and STATCOM represent the location of SVC and STATCOM respectively while SVC and STATCOM represents their settings. Once, two competing objectives namely static voltage stability margin and power losses are optimized. Thereafter, the voltage deviation is added to the two proceeding objectives and the new multi-objective problem is solved. In each case, the optimal location and setting of FACTS devices is determined. A crowding distance-based non-dominated sorting MOPSO technique is used to solve multi-objective problem. Furthermore, a fuzzy-based mechanism is employed to extract the best compromise solution from the Pareto-front. The results indicate that crowding-distance-based nondominated sorting MOPSO provides well-distributed nondominated solutions while desirable exploration of the search space is implemented as well. In [18], the same problem is solved with MOPSO invoked by -based approach for leader selection. In [19]-[20], optimal location and settings of UPFC for enhancing the security of power system (minimizing buses’

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2013 IEEE 7th International Power Engineering and Optimization Conference (PEOCO2013), Langkawi, Malaysia. 3-4 June 2013

objective function to maintain transient stability is formulated as: |Max ∆δ 16 ∆δ |

reactor are used. The problem is formulated as a bi-objective optimisation problem and linear weighted sum approach is invoked to solve it. Like most of other researches, PSO parameters have not been tuned for this particular problem but are adopted from PSO literature. In [27]-[28], linearly-decreasing inertia weight PSO is used to find optimal location and setting of STATCOMs in power system. The objective is to minimize both voltage deviation and STATCOM size. Here, in contrast to most other researches done on this area, all PSO parameters are tuned according to this particular problem. Even a distinct value for maximum velocity in each dimension of particles has been set. In [29]-[30] a new and efficient constraint-handling strategy is introduced which is particularly appropriate for highly-constrained optimization problems such as power system problems. In this approach which has been applied to determine optimal location and setting of STATCOMs, different flight equations are applied to the particles according to the feasibility or infeasibility of ’s and as follows. 1. If the particle’s and swarm’s are both feasible, regular PSO flight equations are applied. is infeasible while swarm’s is 2. If the particle’s feasible, it is concluded that the particle should not be attracted toward its , but should merely be attracted by social leader, that is: 1 3.

Where ∆δ and ∆δ are the rotor angle deviation following a disturbance of generators in areas 1 and 2, respectively, and ∆δ represents the absolute value of maximum Max ∆δ rotor angle deviation difference of two areas. The results illustrate that optimally located STATCOM extends the critical fault clearing time and hence enhances transient stability. In [33], PSO is invoked for optimally tuning FACTS devices and power system stabilizers (PSSs) in order to damp power system oscillations. A fuzzy adaptive inertia weight is used in which inertia weight is updated at each iteration according to the swarm best and based on some fuzzy rules. 17 is weight and is the damping ratio of mode Where and n is the number of system modes. The results show the capability of FACTS devices in damping inter-area modes and their adverse effect on local modes. In [34] the STATCOM’s settings are determined by a hybrid PSO so that low frequency oscillations in power system are minimized. PSO is hybridised with genetic algorithm to benefit from its mutation and crossover operators for avoiding premature convergence. The objective function is formulated in linear-weighted sum form as:

14

If the particle’s and swarm’s are both infeasible, it is concluded that neither nor are able to guide the particle toward promising feasible regions, so it is re-initialised via: 1

,

18

15

19

represents the th dimension of maximum Where , velocity. In [31], a modified PSO is invoked to find optimal location, type and setting of FACTS devices in power system. TCSC and SVC are used as FACTS devices. In the proposed modified PSO, when a new gbest is encountered, a small percentage of randomly selected particles are re-randomized around the new gbest. This improves convergence rate while does not affect the global search properties. In [32], linearly-deceasing inertia weight PSO is used to determine optimal location and controller parameters of STATCOM. First, a systematic procedure to find the optimal location of STATCOM for transient stability enhancement following a severe disturbance is proposed. The proposed algorithm is applied to find the optimal location of STATCOM in a two-area system employing PSO. Further, a STATCOM-based damping controller is proposed and the parameter of the controller are determined via PSO. The

20 and are the real part of the eigenvalue and Where damping ratio related to th mechanical mode and W is the weight is linear weighted sum approach. The relative stability is determined by . The obtained results testify that power system low frequency oscillations have been damped efficiently by such optimization process.

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2013 IEEE 7th International Power Engineering and Optimization Conference (PEOCO2013), Langkawi, Malaysia. 3-4 June 2013

Table 1.1: Characteristics of PSO applications in FACTS optimization problem Reference Number

Applied PSO variant and parameter selection

Objective Function

MO Approach

[13]

Basic PSO with inertia weight, No tuning for PSO parameters has been done and they are not mentioned in paper. Linearly-decreasing inertia weight PSO. Tuning has been merely done for swarm size. Other PSO parameters have not been tuned and their values have not been mentioned in the paper. Linearly-decreasing inertia weight PSO. Tuning has not been done for PSO parameters, but they are adopted from other papers directly.

Maximizing loading margin, Minimizing SVC installation cost. Minimizing voltage violation, power flow violation and installation cost Static voltage stability margin, real power losses, voltage deviation

Fitness-sharing-based non-dominated MOPSO Linear weighted sum

Not mentioned

Discrete Variable handling Approach Not mentioned

Penalty function Approach

Round off for integer variables.

Discarding infeasible solutions

Round off for integer variables.

Linearly-decreasing inertia weight PSO with 0.9, 0.4, 1.5, 30, parameters have not been tuned for this particular problem but are adopted from PSO literature. Linearly-decreasing inertia weight PSO with 0.9, 0.4, 1.5, 30. Parameters have not been tuned for this particular problem. Linearly-decreasing inertia weight PSO with 0.9, 0.4, 100, have been tuned on try and error basis. Constricted PSO

Minimization overloads overvoltages

Crowding distance aided non-dominated sorting Approach, and -based approach Linear weighted sum

Not mentioned

Round off

Minimization of line overloads and bus overvoltages and TCSC installation cost Maximising loadability

Linear weighted sum

Not mentioned

Round off

Single-objective

Not mentioned

Round off

Minimizing voltage violation, power losses, STATCOM size, maximizing load factor Minimizing line overloads and compensator cost

Linear weighted sum

Discarding infeasible particles

Round off

Linear weighted sum

Discarding infeasible particles Enhanced constrainthandling strategy

Round off

[14], [15]

[16]-[18]

[19]-[21]

[22]

[23]-[24]

[25]

[26]

[27]-[28]

[29]-[30]

[31]

[32]

[33]

[34]

Linearly-decreasing inertia weight PSO with 0.9, 0.4, 2, 10, parameters have not been tuned for this problem. Linearly-decreasing inertia weight PSO is used. All PSO parameters are tuned according to this particular problem. Even a distinct value for maximum velocity in each dimension of particle has been set. Linearly-decreasing inertia weight PSO is used. All PSO parameters are tuned according to this particular problem. Even a distinct value for maximum velocity in each dimension of particle has been set. Linearly-decreasing inertia weight PSO is used. Tuning has merely been done for swarm size. Other PSO parameters are not tuned and their values are not mentioned in the paper. Linearly-decreasing inertia weight PSO is used. Tuning has not been done for PSO parameters. A fuzzy adaptive inertia weight is used in which inertia weighted is updated at each iteration according to the swarm best and based on some fuzzy rules. Other PSO parameters have not been tuned for this particular problem. PSO hybridized with GA (with mutation and crossover)

of and

line bus

Constraint handling Approach

Round off

Voltage STATCOM size

deviation,

Linear weighted sum

Voltage STATCOM size

deviation,

Linear weighted sum

Enhanced constrainthandling strategy

Round off

Linear weighted sum

Not mentioned

Round off

Single-objective

Not mentioned

Round off

system

Linear weighted sum

Not mentioned

Not mentioned

frequency

Linear weighted sum

Discarding infeasible particles

No discrete variable

Minimization overloads overvoltages

of and

Transient enhancement

line bus stability

Damping power oscillations

Damping low oscillations

IV.

CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS In this paper, applications of PSO in FACTS optimisation problem have been explained and analysed from the viewpoint of the objectives, used basic PSO, PSO parameter selection, 5

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2013 IEEE 7th International Power Engineering and Optimization Conference (PEOCO2013), Langkawi, Malaysia. 3-4 June 2013

[17] R. Benabid, M. Boudour, M.A. Abido, P. Venkatesh, and J. P. S. Abraham, “Optimal placement of FACTS devices for multi-objective voltage stability Problem,” in proc. IEEE Int. Conf. on Power Systems, 2009, pp. 111. [18] A. Laifa, and M. Boudour, “FACTS allocation for power systems voltage stability enhancement using MOPSO,” in proc. IEEE Int. Conf. on Systems, Signals and Devices, 2008, pp. 1-6. [19] H. I. Shaheen, G. I. Rashed, and S. J. Cheng, “Application and comparison of computational intelligence techniques for optimal location and parameter setting of UPFC,” Engineering Applications of Artificial Intelligence, vol. 23, pp. 203-216, 2010. [20] H. I. Shaheen, G. I. Rashed, and S. J. Cheng, “Optimal location and parameters setting of unified power flow controller based on evolutionary optimization techniques,” in proc. IEEE Power Engineering Society Meeting, 2007, pp. 1-8. [21] G. I. Rashed, H. I. Shaheen, and S. J. Cheng, “Evolutionary optimization techniques for optimal location and parameters setting of TCSC under single line contingency,” in proc. IEEE Power Engineering Meeting, 2008, pp. 1-9. [22] G. I. Rashed, H. I. Shaheen, and S. J. Cheng, “Optimal location and parameter settings of multiple TCSCs for increasing power system loadability based on GA and PSO techniques,” in proc. IEEE Int. Conf. on Natural Computation, 2007, pp. 335-334. [23] S. Nagalakshmi, and N. Kamaraj, “Comparison of computational intelligence algorithms for loadability enhancement of restructured power system with FACTS devices,” Swarm and Evolutionary Computation, vol. 23, pp. 1-11, 2012. [24] S. Nagalakshmi, and N. Kamaraj, “Loadability Enhancement for Pool Model with FACTS devices in transmission system using differential evolution and particle swarm optimization,” in proc. IEEE Int. Conf. on Power Electronics, 2011, pp. 1-8. [25] S. N. Azadani, S. H. Hosseinian, and P. Hasanpor, “Optimal placement of multiple STATCOM for voltage stability margin enhancement using particle swarm optimization,” Electrical Engineering, vol. 90, pp. 503-510, 2008. [26] D. Das, A. Prasai, R. G. Harley, and D. Divan, “Optimal placement of distributed FACTS devices in power networks using particle swarm optimization,” in proc. IEEE Int. Congr. on Energy Conversion, 2009, pp. 527-534. [27] Y. Del Valle, J. C. Hernandez, , J. K. Venayagamoorthy, and R. G. Harley, “Optimal STATCOM sizing and placement using particle swarm optimization,” in proc. IEEE Int. Conf. on Transmission and Distribution, 2006, pp. 1-6. [28] Y. Del Valle, J. C. Hernandez, J. K. Venayagamoorthy, and R. G. Harley, “Multiple STATCOM allocation and sizing using Particle swarm optimization,” in proc. IEEE Int. Conf. on Power Systems, 2006, pp. 18841891. [29] Y. Del Valle, M. Digman, A. Gray, J. Perkel. J. K. Venayagamoorthy, and R. G. Harley “Enhanced Particle Swarm optimizer for power system applications,” in proc. IEEE Int. Symp. on Swarm Intelligence, 2008, pp. 1-7. [30] Y. Del Valle, R. G. Harley, and J. K. Venayagamoorthy, “Comparison of enhanced-PSO and classical optimization methods: a case study for STATCOM placement,” in proc. IEEE Int. Conf on Intelligent System Applications to Power Systems, 2009, pp. 1-7. [31] A. Parastar, A. Pirayesh, and J. Nikoukar, “Optimal location of FACTS devices in a power system using modified particle swarm optimization,” in proc. IEEE Int. Univ. Power Engineering Conf , 2007, pp. 1122-1128. [32] S. Panda, and N. P. Padhy, “Optimal location and controller design of STATCOM for power system stability improvement using PSO,” Journal of Franklin Institute. vol. 345, pp. 166-181, 2008. [33] G. Cheng, and L. Q. Zhan, “Simultaneous coordinated tuning of PSS and FACTS damping controllers using improved particle swarm optimization,” in proc. IEEE Int. Conf on Power and Energy Engineering, 2009, pp. 1-4. [34] M. Zarringhalami, S. M. Hakimi, and M. Javadi, “Optimal regulation of STATCOM controllers and PSS parameters using hybrid particle swarm optimization,” in proc. IEEE Int. Conf on Harmonics and Quality of Power, 2010, pp. 1-7.

multi-objective handling, constraint handling and discrete variable handling. Followings are some hints and proposals for future research in this area. • Since round off approach exposes some drawbacks, attempting to remove its drawbacks or applying other strategies for tackling discrete variables is of high value. • Application of other constraint-handling approaches like Deb’s approach may lead to superior results. • Application of other MOPSO approaches like subpopulation-based approaches, lexicographic-based ordering approaches, vector-evaluated MOPSO may lead to more desirable results. • Tuning all PSO parameters for each particular problem is highly recommended. • Using other basic PSO variants may lead to better results. • Attempting in order to decreasing PSO’s runtime is so valuable. V. REFERENCES [1] N. G. Hingorani, and L. Gyugyi, Understanding FACTS: Concepts and technology of flexible AC transmission systems.: IEEE Press, 2000. [2] A. Rezaee Jordehi and J. Jasni, “A comprehensive review on methods for solving FACTS optimisation problem in power systems,” International Review of Electrical Engineering., vol. 6, no. 4, pp. 1916-1926, August. 2011. [3] J. Kennedy, and R. Eberhart, “Particle swarm optimisation,” in Proc. IEEE Int. Conf. on Neural Networks, November. 1995, pp. 1942-1948. [4] R. Eberhart, Y. Shi, and J. Kennedy, Swarm Intelligence, San Mateo, CA, Morgan Kaufmann, 2001 [5] Y. Shi, and R. Eberhart, “A modified Particle swarm optimiser,” in Proc. IEEE Int. Conf. on Computational Intelligence, May. 1998, pp. 69-73. [6] Y. Shi, and R. Eberhart, “Empirical study of particle swarm optimisation,” in Proc. IEEE Int. Conf. on Computational Intelligence, July. 1999, pp. 19451950. [7] M. Clerc, and J. Kennedy, “The particle swarm explosion, stability and convergence in a multi-dimensional complex space,” IEEE Trans. Evol. Compt., vol. 6, no. 1, pp. 58-73, February. 2002. [8] A. Rezaee Jordehi and J. Jasni, “particle swarm optimization for discrete optimisation problems: a review,” Artificial Intelligence Review, pp.1-16, 2012. [9] A. Rezaee Jordehi and J. Jasni, “Parameter selection in particle swarm optimisation: a survey,” Journal of Experimental and Theoretical Artificial Intelligence, pp. 1-12, 2013. [10] A. Rezaee Jordehi, and M. Joorabian, “Optimal placement of Multi-type FACTS devices in power systems using evolution strategies,” in proc. IEEE Int. Power Engineering and Optimization Conference (PEOCO),2011, pp. 352-357. [11] A. Rezaee Jordehi, and J. Jasni, “Approaches for FACTS optimization problem in power systems,” in proc. IEEE Int. Power Engineering and Optimization Conference (PEOCO), 2012, pp. 355-360. [12] A. Rezaee Jordehi, and J. Jasni, “Heuristic methods for solution of FACTS optimization problem in power systems,” in proc. IEEE Int. Research and Development (SCOReD), 2011, pp. 30-35. [13] Y. C. Chang, “Multi-objective optimal SVC installation for power system loading margin improvement,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 19, April. 2012. [14] M. Saravanan, S. M. R. Slochanal, P. Venkatesh, and J. P. S. Abraham, “Application of particle swarm optimization technique for optimal location of FACTS devices considering cost of installation and system loadability,” Electric Power Systems Research, vol. 77, pp. 276-283, 2007. [15] M. Saravanan, S. M. R. Slochanal, P. Venkatesh, and J. P. S. Abraham, “Application of PSO technique for optimal location of FACTS devices considering system loadability and cost of installation,” in proc. IEEE Int. Conf. on Power Engineering, 2005, pp. 716-721. [16] R. Benabid, M. Boudour, M.Abido, “Optimal location and setting of SVC and TCSC devices using non-dominated sorting particle swarm optimization,” Electric Power Systems Research, vol.79, 2009.

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