Reliability Maximization of Power System Using Firefly Algorithm

International Electrical Engineering Journal (IEEJ) Vol. 7 (2016) No.1, pp. 2116-2123 ISSN 2078-2365 http://www.ieejournal.com/

Reliability Maximization of Power System Using Firefly Algorithm Khaled Guerraiche1, Mostefa Rahli2, Abd elkader Zeblah3, Latifa Dekhici4 1 Department of Electrical, University of Sciences and the Technology, Oran, Algeria 2 Department of Electrical, University of Sciences and the Technology, Oran, Algeria 3 Department of Electrical, University of Sidi Bel Abbes, Sidi Jillali, Algeria. 4 Department of Computers sciences, University of Sciences and the Technology, Oran, Algeria 1

[email protected], [email protected], [email protected], [email protected]

 Abstract— In electrical systems, maximizing reliability is considered as an important issue. Generally, the aim of reliability optimization is to maximize product reliability under budgetary constraints, or to minimize cost subject to reliability constraints. This paper uses a firefly Algorithm (FA) that is a bio-inspired meta-heuristic to solve the problem of electrical structure optimization of series-parallel production systems. We proposed a procedure that determines the maximal reliability of series-parallel electrical topology of power system. The electrical system components are selected from a list of available components on the market. Electrical components are characterized by their reliability, cost and performance (capacity). The proposed approach could be useful for managers and engineers to better understand their system reliability and performance, and also achieve better configuration. To estimate the series -parallel system reliability, we suggest a rapid method based on Universal Moment Generating Function (UMGF). The results obtained using the proposed technique was compared with those of the ant colony algorithm (ACA). Index Terms— Firefly Algorithm (FA), Reliability Optimization (ROP), Multi-State System (MSS), Universal Generating Moment Function (UMGF).

I. INTRODUCTION The redundancy problem is a complex combinatorial optimization problem, which is used in many industrial applications. These applications include electrical power systems, electronic systems, telecommunications systems and manufacturing production systems [1]. The purpose of the present paper is to develop an efficient firefly algorithm (FA) to solve the Redundancy optimization problem (ROP) for series-parallel Multi-state system (MSS). The resolution involves the selection of power components and the appropriate levels of redundancy in order to maximize system reliability of series-parallel power systems, under constrained cost. In past researches about reliability, equipments had only two states; zero and one (one is entirely

safe and zero is entirely fail). Today, in complex systems, it is needed to investigate the reliability of system in different states with different efficiencies. So new systems are not in binary states and they experience different states between zero and one. These systems are called multi state systems. Redundancy allocation problem (RAP) was first introduced by Ushakov [2-3]. The problem is to determine which topology to select and what kind of devices to use in order to achieve the maximum reliability. This expression of the redundancy allocation problem leads to the maximization of electrical system reliability of the series–parallel structure. II. FORMULATION OF THE RELIABILITY OPTIMIZATION PROBLEM FOR MSS A. General description of the problem The series-parallel multi state system (MSS) consists of a number of components link in series, such that each component can contain various elements connected in parallel Fig.1. Failed elements are detected and the elements availabilities are known. For each component there are various element versions, which are proposed by the suppliers on the market [1]. These elements are characterized according to their version by their, Reliability , cost and performance . Redundancy allows system availability improvement, but increases the total cost. The objective is to conception the series parallel system so that the reliability maximized, subject total cost constraint. The composition of subsystem i can be defined by the numbers of parallel components for , where represents the number of versions available for component of type i. The entire system can therefore be defined by the set of triplets { , , } ( , ). K represents the initial system structure. In fact, for given K, the total cost of the electrical structure can be calculated as

2116 Guerraiche et. al.,

Reliability Maximization of Power System Using Firefly Algorithm

International Electrical Engineering Journal (IEEJ) Vol. 7 (2016) No.1, pp. 2116-2123 ISSN 2078-2365 http://www.ieejournal.com/ n

Vi

 k ivCiv  C0

(1)

i 1 j1

We denote by and the vectors and , respectively. As the reliability A is a function of , and . In the case of electrical power system, the vectors and indicate the cumulative load curve (consumer demand). In general, this curve is known for every power system [6]. C. Optimal design problem formulation Dual problem: the multi stage system Reliability heterogeneous Redundancy Optimization Problem of electrical energy system can be formulated as follows: find the maximal reliability system configuration , such that the corresponding to the cost less than or equal the specified cost . That is, Maximize

Fig. 1 Series–Parallel Power System Topology



n  Vi  K AK1 , K 2 , K3 ,......K n ,W , t     uiv z  iv i 1  v 1

B. Availability of reparable MSS In electric systems, reliability is considered as a measure of the system ability to respond needs the load demand ., i.e., to give an adequate supply of electrical energy . This definition of the reliability clue is widely used for power systems: see e.g., [1-4]. The Loss of Load Probability index (LOLP) is usually utilized to estimate the reliability clue in [5]. This clue is the overall probability that the load demand can not be achieved. Thus, we can write and the . This reliability index depends on consumer demand W. For reparable MSS, a multi-state stable-state availability E is used as , the distribution of states probabilities is given by expression (2), while the MSS stationary reliability is formulated by equation (3):

p1  lim[ proba((t )   l )] t 

(2)

E

P

(3)

j

 j W

(2)

M

1 M

T l 1

 P ( l 1

s

 Wl ) Tl

l

(4)

Vi

 k i 1 v 1

iv

(5)



Civ  C0

(6)

D. MSS reliability estimation method In this work the procedure used is based on the universal z transform (UMGF), which is a modern mathematical technique introduced in Ushakov [7]. This method, convenient for numerical implementation, is revealed to be very effective for high dimension combinatorial problems. In the literature, the universal z-transform (Laplace transform) is also called UMGF or simply u transform. The UMGF extends the widely known universal moment generating function [8-9]. II.4.1 Definitions and properties of the U function The UMGF of a discrete random variable  is determined as a polynomial: J

If the functioning period T is divided into M intervals (3) (with durations … ) and each interval has a required demand level ( …, respectively), then the generalized MSS reliability index A is:

A

n

Subject to



u ( z )   Pj z

j

(7)

j 1

Where the variable (performance) has possible values and is the probability that is equable to . The probabilistic characteristics of the random of variable can be found employing the function u (z). In particular, if the discrete random variable is the multi state system (MSS) stationary outlet performance, the availability is 2117

Guerraiche et. al.,

Reliability Maximization of Power System Using Firefly Algorithm

International Electrical Engineering Journal (IEEJ) Vol. 7 (2016) No.1, pp. 2116-2123 ISSN 2078-2365 http://www.ieejournal.com/ given by the probability as follows:

which can be defined



Proba(  W)   u ( z ) z  W



(8)

Where a distributive operator is defined by expressions (9) and (10):

  P, if   W ( Pz  W )    0, if   W

(performances) [11]. The u-function of MSS device n containing parallel devices can be calculated by using the  operator: u p ( z )  (u1 ( z ), u2 ( z), ..., un ( z)) , where (1 ,  2 , ...,  n ) 

(10)

It can be easily represents that equations (9)–(10) meet j

m

  Pi Q j z



,by using the operator Ф,

the coefficients of polynomial u (z) are added for every term with , and the probability that is not less than some arbitrary value is systematically got. Consider single devices with total failures and each device i has reliability Ai and nominal performance. The UMGF of such a device has only two terms can be defined as:

ui ( z )  (1  Ai ) z 0  Ai z i  (1  Ai )  Ai z i

(11)

To assess the multi state heterogeneous system (MSS) availability of a series-parallel system, two fundamental composition operators are introduced. These operators define the polynomial u (z) for a group of devices. E. Series-parallel MSS availability evaluation using U functions To assess the (MSS) heterogeneous availability of a series-parallel system, two fundamental composition operators are introduced. These operators define the polynomial u (z) for a group of elements.

Q j 1

b

j

z j)

ai  b j

i 1 j 1

The parameters and are physically interpreted as the performances of the two devices. and m are numbers of feasible performance levels for these devices. and are steady-state probabilities of feasible performance levels for apparatus. One can see that the  operator is quite simply a product of the individual u-functions. Thus, the formula Jn

UMGF is:

u p ( z )   u j ( z ) . Determines the individual j 1

UMGF of devices defined in equation (11). G. Series devices When the devices are linked in series, the element with the least performance becomes the bottleneck of the system. Therefore, this element may give the entire system productivity. To calculate the U-function for system containing  elements connected in series, the operator ´ should be used:

u s ( z )   (u1 ( z), u2 ( z ), ..., um ( z )) ,





where  (1 ,  2 , ...,  m )  min 1 ,  2 , ...,  m so that



n

 (u1 ( z ), u 2 ( z ))     Pi z a , n

F. Parallel devices Let consider a system device containing devices link in parallel. The total performance of the parallel system is the addition of performances of all its equipments. In energy systems, the term capacity is usually used to denote the quantitative performance measure of a device [10] Examples: generating capacity for a production unit, carrying capacity for an energy transmission line, etc. Thus, the total performance of the parallel unit is the sum of capacities

.

i

m

(u1 ( z ), u 2 ( z ))  ( Pi z ai , i 1



P

i 1

n

(9)

J  J  W   W   Pj z j     Pj z j  j 1  j 1

 j W



Therefore for a pair of devices connected in parallel:

n

condition

n

m

  Pi Q j z

 i 1 min ai , b j 

i

m

Q z j 1

j

bj

   

i 1 j 1

Applying composition operators of parallel devices () and Series devices ( ) consecutively, one can obtain the UMGF of the entire system. To do this we must first define the individual UMGF of each device.

2118 Guerraiche et. al.,

Reliability Maximization of Power System Using Firefly Algorithm

International Electrical Engineering Journal (IEEJ) Vol. 7 (2016) No.1, pp. 2116-2123 ISSN 2078-2365 http://www.ieejournal.com/ H. Devices with total failures Let consider the frequent case where only total failures are tested and each subsystem of type and version has nominal performance and reliability . In this case, we have: Proba(   ij )  Aij and Proba(  W)  1  Aij . The UMGF of such a device has just two terms can be defined as in equation (9) by

u *i ( z )  (1  Aij ) z 0  Aij z

ij

 1  Aij  Aij z

ij

Using the  operator, we can get the UMGF of the i-th system device containing Parallel devices



ui ( z )  u * i ( z )

  A z ki

ij

ij



ki

 (1  Aij ) .

The UMGF of the all system containing n system devices connected in series composed with identical elements is:

  

  

 A z 1 j  (1  A ) k1 ,  1j  1j    k 2  u s ( z )    A2 j z 2 j  (1  A2 j ) , ...,    kn   Anj z nj  (1  Anj )   

(12)

To evaluate the probability Pr oba(  W ) for the entire system, the operator  is applied to equation (12):



Pr oba(  W )   u s ( z ) z  W



(13)

The next study presents firefly meta-heuristic optimization method to solve the heterogeneous redundancy optimization problem for multi-state systems.

III. THE FIREFLY OPTIMIZATION APPROACH The problem formulated in this work is a complicated combinatorial optimization problem. The total number of various solutions to be tested is very large, even for rather small problem. An exhaustive examination of the wide number of solution is not feasible given reasonable period limitations. Thus, because of the search space size of the ROP for MSS, we applied an relatively a new meta-heuristic in this area. This meta-heuristic consists in adaptation of the firefly optimization method. It was reported in [12-13-14].

A. Fireflies in nature Fireflies, belong to family of Lampyris, are small winged beetles capable of producing a cold light flashes in order to seduce mates. They are consider to have a capacitor-like mechanism, that slowly charges until the certain threshold is attained, at which they emit the energy in the form of light, after which the cycle begins again [15][16]. The basic principle and its implementation in our problems are briefly presented in the following paragraphs. B. Basic principles of firefly method Firefly method inspired by the flashing behavior of fireflies was recently put forth by. The fundamental functions of flashing light of fireflies are to transmit (like seducing mating partners) and to attract potential prey. Inspired by this nature, the firefly method was developed by idealizing some of the flashing characteristics of fireflies and representing every individual solution of optimization problem as a firefly in population. Three major idealized rules are [12-13-17-14-18]. All fireflies in the population are unisex so that any individual firefly will be attracted at other fireflies;  For any pair of fireflies, the fewer bright one will move towards the brighter one. The attractiveness of a firefly is proportionally linked to the brightness which diminishes with increasing distance between two fireflies;  The brightness of a firefly is proportionally associated to the value of objective function in the similar way to the fitness in (GA) genetic algorithm. The procedure of apply the firefly algorithm for a maximum optimization problem is summarized by the pseudo code shown in Fig. 2 [12-13-15-14-18].

C. Parameters In the firefly algorithm, there are five important items: Light Intensity. In the simplest case for minimum optimization problems solution, the brightness I of a firefly at a particular place x can be chosen as

,

Attractiveness. In the firefly algorithm, the leading form of attractiveness function can be any monotonically decreasing functions such as the following generalized form: (14) Where r is the distance separating two fireflies is the attractiveness at r = 0 and γ is a fixed light absorption coefficient.

2119 Guerraiche et. al.,

Reliability Maximization of Power System Using Firefly Algorithm

International Electrical Engineering Journal (IEEJ) Vol. 7 (2016) No.1, pp. 2116-2123 ISSN 2078-2365 http://www.ieejournal.com/ with total failures. The maximum numbers of components put in parallel are set to (7, 5, 4, 9 and 4). The number of firefly used to find the best solution is 10. The simulation results depend broadly some values of the coefficients , and . In this experiment the values firefly algorithm parameters are set to: =0.01, = 20, = 0.01 and =0.5. The stopping criteria are the number of generation which must be 1000. Three different values of are considered. The algorithm was implemented in C++. The numerical tests were executed on an AMD-A4-3300M APU 1.9GHz CPU and 4GB RAM.

Procedure FF Meta-heuristic (Nbr_gen: the maximal number of generations) Begin γ: the light absorption coefficient Define the objective function of f(x), where x=(x1,........,xd) in domain d Generate the initial population of fireflies or xi (i=1, 2 ,..., nb) Determine the light intensity Ii at xi via f(xi) While (t Ii) Attractiveness βi,j varies with distance ri,j move firefly i towards j with attractiveness βi,j else move firefly i randomly end if Evaluate new solutions update light intensity I End for j End for i Rank the fireflies and find the current best t++ End while End procedure

A. Test example The electrical energy station system, which provides electrical energy for consumers, is designed with five basic subsystems as. The electrical feeding system can be described as follows: the electrical energy is generated from the production units (subsystem 1). Then, transformed for high voltage (HT) by the HT transformers (subsystem 2) and transported by the high voltage lines (subsystem 3). A second transformation takes place in HT/MT transformers (subsystem 4), which furnishes the MT load by means of the MT lines (subsystem 5). Each equipment of the system is considered as unit with total breakdown. The characteristics of the products available in the industry for each type of device are shown in Table 1. This table shows for each device Reliability A, nominal performance Ξ, and cost C.

Fig.2: The pseudo-code of the FA [13]

 Distance

Subsystems

The distance between any two fireflies and at the Cartesian distance as follows:

and can be Power Units

(15) Where,

is the

component of the

firefly.

HT Transform er

 Movement The movement of a firefly i lured by another more attractive (brighter) firefly j, is determined by

HT lines

(16) where the first and second term is owing to the attraction while the third term is randomization with α being the “rand” and randomization parameter is a random number generator evenly distributed in [0, 1].

HT/MT Transforme rs

IV. EXPERIMENTS AND COMPUTATIONAL RESULTS In order to illustrate the proposed firefly algorithm, a numerical example is solved by use of the data provided in table 1. Each element of the subsystem is considered as a unit

MT Lines

Table 1. Data examples Versions Reliability Cost A (%) C(mln $) 1 0.798 0.590 2 0.977 0.535 3 0.982 0.470 4 0.978 0.420 5 0.983 0.400 6 0.990 0.180 7 0.980 0.220 1 0.695 0.205 2 0.896 0.189 3 0.797 0.091 4 0.997 0.056 5 0.898 0.042 1 0.871 7.525 2 0.973 4.720 3 0.971 3.590 4 0.676 2.420 1 0.977 0.180 2 0.678 0.160 3 0.978 0.150 4 0.983 0.121 5 0.881 0.102 6 0.971 0.096 7 0.783 0.071 8 0.982 0.049 9 0.877 0.044 1 0.984 0.986 2 0.883 0.825 3 0.987 0.490 4 0.981 0. 475

Capacity Ξ (MW) 100 100 85 85 48 31 26 100 92 53 28 21 100 60 40 20 100 100 91 72 72 72 55 25 25 100 100 60 51

2120 Guerraiche et. al.,

Reliability Maximization of Power System Using Firefly Algorithm

International Electrical Engineering Journal (IEEJ) Vol. 7 (2016) No.1, pp. 2116-2123 ISSN 2078-2365 http://www.ieejournal.com/ Table 2 present the power demand levels and their corresponding times. The energy system structure should be planned from available components and be able to satisfy the demand requirements at all load levels.

Optimal reliability obtained by Firefly Algorithm (presented by symbol A in table 3) were compared to reliability given by Ant algorithm (presented by symbol A1 in table 3) in the reference [19], For this genre of problem, we define the maximal reliability heterogeneous system configuration under budget constraint , (where =17 is given in [19] taken as reference). We notice clearly the betterment of the reliability of the system compared to the Ant method (see fig 3). We granted more importance to the reliability of the system compared to its cost what warranting the increased in the cost compared to the reference. The maximization reliability was treated successfully in this work. The objective is to choose the optimal combination of elements utilized in series-parallel structure of power system. This has to satisfy the maximal system reliability level with the cost constraint. The Firefly algorithm allows each subsystem to contain elements with several versions technologies. The FA proved very efficient in solving the ROP and best quality results in terms of reliability levels was achieved compared to ACA [19].

Table 2. Parameters of the cumulative demand curve

Wm (%)

100

80

50

20

Tm (h)

4203

788

1228

2536

B. Discussion and numerical results The object is to find the maximal reliability feasible MSS series-parallel structure under constrained cost C0. Three different solutions (for = 12, = 15 and = 17) are presented in Table. 3 (unit cost is given in mln $).

Table 3. Optimal solutions for reliability maximization problem Cost Constraint (mln $)

Subsystems of configuration

FIREFLY

ANT COLONY

Best configuration

A (%)

C (mln $)

(%)

(mln $)

12

Subsystem 1 Subsystem 2 Subsystem 3 Subsystem 4 Subsystem 5

1*(6) - 5*(7) (1) - (2) – (3) - (4) (2) - (3) 7*(9) (1) – (3)

0.936

11.91

not available

not available

15

Subsystem 1 Subsystem 2 Subsystem 3 Subsystem 4 Subsystem 5

7*(7) (2) - 4*(4) – (5) 2*(2) 4*(5)- (7) - 2*(9) 6*(4)

0.968

14.85

not available

not available

17

Subsystem 1 Subsystem 2 Subsystem 3 Subsystem 4 Subsystem 5

4*(6) – 2*(7) 4*(3) - 2*(4) (2) - 2*(3) 2*(7) – 8 – 3*(9) (1) - 3*(3)

0.984

16.49

0.87

13.98

2121 Guerraiche et. al.,

Reliability Maximization of Power System Using Firefly Algorithm

International Electrical Engineering Journal (IEEJ) Vol. 7 (2016) No.1, pp. 2116-2123 ISSN 2078-2365 http://www.ieejournal.com/ Table 4. Comparison of optimal solution obtained by Firefly and Ant Colony algorithms

(mln $)

17

Reliability FA (A)

Reliability ACA (A1)

Percentage (%)

0.984

0.87

11.4

Table 4 also shows reliability Comparison with two of the most popular meta-heuristic algorithms, the firefly algorithm (FA) and the Ant Colony algorithm (AC). Firefly achieved better quality results in terms of structure reliability Table 4. We remark that ACA performed better by achieving a less expensive configuration; however FA algorithm achieved a near optimal configuration with a higher reliability level Table 4. We take, for example, for reference cost ( = 17), It is noticed, according to table 4, that FA tends, to increase the reliability of the system with 11.4 %.

problem, namely the redundancy optimization problem of series-parallel multi-state systems. That why we applied a universal generating function technique to evaluate system availability. Results on an illustrative example proved the efficiency and adaptability of our firefly algorithm combined to UGFT to reach the best solutions in few seconds. It shows that Firefly Algorithm (FA) provides better results than ACA

ACKNOWLEDGMENT I would like to thanks Mr.Rahli Mostefa, and Mr.Zeblah Abdelkader for the assistance they provided to understand the redundancy optimization problem (ROP). I would also like to thanks the other PhD Scholars of my University. Very special thanks goes out to Dr. Dekhici Latifa, without whose motivation and encouragement, I confess that it would be difficult for me to move forward in my PhD Program. REFERENCES [1]

[2]

[3]

[4]

[5]

Fig 3. Reliability for the cost constraint C0 = 17

[6] [7] [8] [9] [10]

[11] Fig 4. Convergence curves for FA [12]

V. CONCLUSION In this paper, we discussed a reliability optimization problem of a series-parallel system with multiple choices of budget constraints, where the objective is to maximize reliability of the whole system under budget constraint. We proposed the firefly meta-heuristic to solve the reliability optimization

[13]

[14]

M. Ouzineb, M. Nourelfath and M. Gendreau, “Tabu search for the redundancy allocation problem of homogenous series–parallel multi-state“, Reliability Engineering and System Safety, pp. 1257–1272, 2008. I. Ushakov, “Optimal standby problems and a universal generating Function“, Journal computer system science, Vol. 25, pp. 61-73, 1987. V. Ebrahimipour, A. Tohidi and F. Rahmanniya, “Simultaneously Optimizing of Reliability Cost and Completion Time in Series-parallel Systems Considering Redundant Allocation Using a Genetic Algorithm“, IEEE, pp. 385-390, 2011. M. S. Fang, S. Xuemin and H. Pin-H, “Reliability optimization of distributed access networks with constrained total cost“, IEEE transactions on reliability,Vol. 54, No. 3, pp. 421-430, 2005. M. Hiroyuki and Y. Takafumi, “Probabilistic distribution network expansion planning with multi-objective memtic algorithm“, IEEE Electrical Power & Energy Conference, pp. 1-6, 2008. O. Bendjeghaba, D. Ohahdi and A. Zeblah, “Hybrid approach for redundancy optimization of multi state power system“, Leonardo Journal of Sciences, pp. 115-130, 2007. I. Ushakov, “Universal generating function“, journal Compt. Syst. Sci, Vol. 24, No. 5, pp. 118-129, 1986. S. M. Ross, “Introduction to probability model“, Academic Press, 1993. F. Harrou, A. Tassadit, B. Bouyeddou and A. Zeblah, “Efficient optimization algorithm for preventive maintenance in transmission sytems“, Journal of Modelling and Simulation of Systems, Vol. 1, pp. 59-67, 2010. W. Coit and A.E. Smith, “Optimization approaches to the redundancy allocation problem for series-parallel systems“, Proceeding of the fourth industrial engineering research conference (IERC), May 1995. M. Nourelfath, N. Nahas and A. Zeblah, “An ant colony approach to redundancy optimization for multi-State system“, International Conference on industrial engineering and production management , (IEPM’2003), Porto-Portugal, 2003. Y. Liu, HZ. Huang, Z.Wang, YF. Li and XL. Zhang. “Joint optimization of redundancy and maintenance staff allocation for multi-state series-parallel systems“, Maintenance and Reliability, Eksploatacja i Niezawodnosc, Vol. 14, No. 4, pp. 312–318, 2012. X. S. Yang, “Nature-Inspired Metaheuristic Algorithm“, Luniver Press, Beckington, 2008.

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Reliability Maximization of Power System Using Firefly Algorithm

International Electrical Engineering Journal (IEEJ) Vol. 7 (2016) No.1, pp. 2116-2123 ISSN 2078-2365 http://www.ieejournal.com/ [15] X. S. Yang, “Engineering Optimization An Introduction with Metaheuristic Applications“, New York, Wiley, 2010. [16] L. Dekhici and K. Belkadi, “Metaheuristic fireflies Discrete for the Flow Shop Hybrid“, Cari, 2012. [17] K. Durkota, “Implementation of a discrete firefly algorithm for the QAP problem within the sage framework“, BSc thesis, Czech Technical University, 2011. [18] L. S. Coelho, D. L. A. Bernert and V. C. Mariani, “A chaotic firefly algorithm applied to reliability- redundancy optimization“, Proceedings of IEEE, Congress on Evolutionary Computation, New Orleans, LA, USA, pp. 517-521, 2011. [19] A. H. Gandomi, X. S. Yang and A. H. Alavi, “Mixed variable structural optimization using firefly algorithm“, Computers and Structures, Vol. 89, pp. 2325-2336, 2011. [20] A. Zeblah, “Reliability maximiszation of power system using ant colony approach“, Inernational journal of electrical and power engineering, pp. 192-201, 2008.

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Reliability Maximization of Power System Using Firefly Algorithm