Determination of voltage stability boundary values in electrical power systems by using the Chaotic Particle Swarm Optimization algorithm

Electrical Power and Energy Systems 64 (2015) 873–879

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Determination of voltage stability boundary values in electrical power systems by using the Chaotic Particle Swarm Optimization algorithm L. Kuru a,⇑, A. Ozturk a, E. Kuru b, O. Kandara c a

Department of Electrical and Electronics Eng., Duzce Uni., 81100 Konuralp, Duzce, Turkey Department of Mechatronics Eng., Duzce Uni., 81100 Konuralp, Duzce, Turkey c Dept. of Computer Sci., Southern Univ., Baton Rouge, LA 70813, USA b

a r t i c l e

i n f o

Article history: Received 23 July 2013 Received in revised form 7 August 2014 Accepted 19 August 2014

Keywords: Power system Voltage stability Chaotic Particle Swarm Optimization Logistic map Henon map

a b s t r a c t A power system has critical values which are the limit values of voltage stability. These values are the highest active power taken by the load busses, voltage amplitude and the angle of the busses. In this research the critical values in electric power systems are defined with use of Chaotic Particle Swarm Optimization (CPSO) algorithm. In this study CPSO has been aimed to use logistic map and Henon map as chaotic maps to control the values of the parameters in velocity update formulation. Initially, critical values of voltage stability have been found by simple Particle Swarm Optimization (PSO). Then the same values have been found with CPSO. Accordingly, the results have been evaluated and observed that the stability critical values found by CPSO can be used to produce good potential solutions. Simulation results are promising and show the effectiveness of the applied approach. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction The highest active power values transferring to the load bus, the voltage amplitude and the phase values are determined by the critical values in electrical power systems. Voltage stability is analyzed statically with various methods. Many chaotic maps used in the literature possess certainty, ergodicity and the stochastic property. Recently, chaotic sequences have been adopted instead of random sequences and are used in many applications with good results. They have also been used together with some heuristic optimization algorithms to represent optimization variables [32–34]. In this study, the chaotic maps such as logistic map and Henon map are used against the random number generator which is an important control parameter that affect the PSO convergence. The theory of PSO was shortly presented and how critical values are obtained with PSO are explained. The theory of CPSO algorithm based on logistic map and Henon map are described. CPSO is a direct mechanism without a continuous power flow to compute the critical values as in the NR method. The critical limit values in the IEEE-nine bus sample power system have been found by PSO and CPSO. The results are compared to each other. The simulation results show that the application of deterministic chaotic signals

⇑ Corresponding author. E-mail address: [email protected] (L. Kuru). http://dx.doi.org/10.1016/j.ijepes.2014.08.012 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

instead of random sequences is a possible strategy to prove the performances of PSO.

Literature review Voltage stability is directly related to the maximum load capacity of power system transmission line and it is defined as the ability of keeping load busses in determined limit values [1,2]. Voltage stability is analyzed statically with various methods. One of these methods for determining the limits of the critical values in power system is Newton Raphson method (NR). In this method, load value of the load bus is increased step by step until power flow analysis produces no solution where the Jacobian matrix is singular. Thomas et al. have developed a global voltage indicator based on singular value of Jacobian matrix [3]. The critical values calculation have been generally made on Jacobian matrix obtained with load flowing [4–6]. Similar studies, various techniques were reported in the literature to identify and estimate the maximum loadability to indicate its importance in power system studies [7–14]. One of the popular techniques and fast search techniques is by using the Artificial Intelligence (AI) search techniques. Musirin and Rahman developed a new algorithm to execute to Evolutionary Programming (EP) base optimization technique for estimating maximum loadability or critical loading condition in power system for one load bus [15]. Begovic et al. have studied on a voltage

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stability which Thevenin equivalent impedance is based on qualification and rate of load equivalent impedance [16]. Other optimization techniques which can also perform similar task are linear programming, genetic algorithm, quadratic programming, Ant Colony Optimization and Artificial Immune System [17]. PSO is widely used in various applications to optimize the system parameters in the last years [17–20]. The critical limit values in the IEEE-nine bus sample power system have been found by using CPSO. Logistic map and Henon map as chaotic maps are used to control the values of the parameters in velocity update formulation. The results are compared to each other. The results show that using chaotic maps in PSO is an effective approach against the random number generator.

where Z is the control parameter, x is a vector and F is a nonlinear transformation [27]. In this study Logistic map and Henon map are used. Logistic map The logistic map is a one-dimensional discrete-time non-linear system exhibiting quadratic non-linearity. The logistic map is given by the function

f : ½0; 1 ! R defined by f ðxÞ ¼ l  ð1  xÞ

ð4Þ

which is expressed in state equation form as Materials and methods

xiþ1 ¼ f ðxi Þ ¼ lxi ð1  xi Þ i ¼ 0; 1; 2; . . . ;

Particle Swarm Optimization

where xi 2 (0, 1), x0 is not equal to {0.25, 0.5, 0.75} and l is set to 4 for ergodicity. l is known as the control parameter or bifurcation parameter. Here xi is the state of the system at time i. Xi+1 denotes the next state and i denotes the discrete time. The logistic map is frequently used by PSO [28–30]. Repeated iteration of f gives rise to a sequence of points {xi}, known as an orbit. The bifurcation diagram of the logistic map is shown in Fig. 1. The chaotic numbers generated by logistic map are shown in Fig. 2.

PSO is a population-based optimization algorithm developed first time by Kennedy and Eberhart [21]. It was inspired by the moving school of fish and swarm inspects. PSO is used to solve nonlinear optimization problems in power systems [22–25]. Basically, PSO is an algorithm based on swarm intelligence. It is similar to the computational techniques based on development such as genetic algorithms (GA) based on the fact while PSO is searching for the optimum in the search space, it uses the population which holds the possible solutions for the function to be optimized [26]. Each individual is called a particle. Particles make up the population called a swarm. In PSO, however, each member of the swarm has a variable speed which changes based on the conditions and determines its movement in the search space. In addition, each member has a memory holding the best spot previously visited. Each particle adjusts its position toward the best position in the swarm by taking the advantage of its previous experience. PSO is basically based on the fact that the positions of the particles in the swarm are brought closer to the best position in the swarm. The speed of ‘‘bringing closer’’ happens in a random fashion and in most cases the particles in the swarm find better positions in their new movements. This process continues until the goal. Assuming the search space with D dimensions, the position of the ith particle in the swarm can be expressed in a D dimensional vector as in (Xi = (xi1, xi2, . . ., xiD)T) and its speed can be expressed in a D dimensional vector as in (Vi = (vi1, vi2, . . ., viD)T). In addition, the best position of the particle has ever visited can be expressed in a D dimensional vector as in (Pi = (pi1, pi2, . . ., piD)T). In the following swarm expressions numbered 1 and 2, g denotes the index number of the best particle, and the upper scripted numbers denote the iteration numbers.

v nþ1 ¼ wv nid þ c1 r nid ðpnid  xnid Þ þ c2 Rnid ðpngd  xnid Þ id

ð1Þ

xnþ1 ¼ xnid þ mnþ1 id id

ð2Þ

Henon map Henon map, is a two dimensional dynamical system that is the simplified version of the Lorenz system [31]. The Henon equations are given by

xiþ1 ¼ 1 þ yi  ax2i

ð6Þ

yiþ1 ¼ bxi

ð7Þ

The Henon map has a strange attractor for a = 1.4 and b = 0.3 values. The Henon map used in this study is shown in Fig. 3. The chaotic numbers generated by logistic map are shown in Fig. 4. Chaotic Particle Swarm Optimization Chaotic behavior refers to one type of complex dynamical behavior that possess some very special features such as being extremely sensitive to tiny variations of initial conditions. Chaotic systems exhibit irregular, unpredictable behavior. The boundary between linear and chaotic behavior is often characterized by period doubling, followed by quadrupling, etc., although other routes to chaos are also possible.

Here, d = 1, 2, . . ., D, and i = 1, 2, . . ., PS, where PS = the size of the population in the swarm. r1 and r2 are randomly selected values between 0 and 1. n denotes the iteration number. xid and vid denote the position and the speed values, respectively. w is for the value of inertia weight and c1 and c2 are for the scaling factors. Chaotic maps A chaotic map is a discrete-time dynamical system in the iteration form of:

X iþ1 ¼ Fðxi ; ZÞ

ð3Þ

ð5Þ

Fig. 1. The bifurcation diagram of the logistic map.

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Chaos is applied to PSO in various ways. In this study, the chaotic maps such as logistic map and Henon map are used against the random number generator. Chaos is used; to control the values of the parameters in velocity update formulation, to generate the c⁄rand() coefficients [33–34]. The parameters rnid and Rnid are important control parameters that affect the PSO convergence. Chaotic Particle Swarm Optimization algorithm (CPSO) based on logistic map and Henon map is described as follows:

1 0.9 0.8

numbers

0.7 0.6 0.5 0.4

-CPSO1-

0.3

Parameter rnid of Eq. (1) is modified with the following equation.

0.2

n v nþ1 ¼ w  v nid þ c1  hid  id

0.1 0

0

20

40

60

80

100

120

140

160

180

200

sample Fig. 2. The chaotic numbers generated by logistic map.

 
Pnid  xnid þ c2  Rnid  Pngd  xnid

ð8Þ

where hnid is given first by logistic map and secondly by Henon map scaled with values between 0 and 1. -CPSO2Parameter Rnid of Eq. (1) is modified with the following equation

v nþ1 ¼ w  v nid þ c1  rnid  id

 
Pnid  xnid þ c2  Hnid  Pngd  xnid

ð9Þ

where Hnid is given first by logistic map and secondly by Henon map scaled with values between 0 and 1. -CPSO3Parameters rnid and Rnid of Eq. (1) are modified with the following equation n v nþ1 ¼ w  v nid þ c1  hid  id

 
Pnid  xnid þ c2  Hnid  Pngd  xnid

ð10Þ

where hnid and Hnid are given first by logistic map and secondly by Henon map scaled with values between 0 and 1. Simulations and results In Fig. 5, IEEE nine bus power system of which critical values will be searched is seen [35]. In this system bus 1 refers slack bus, bus 2 and 3 refer generator busses, bus 5, 7, 9 refer load busses. The line data of the nine busses system and generator limit

Fig. 3. Henon map for a = 1.4 and b = 0.3.

1

1 0.9 0.8

G

Pg=0.85PU Qg=0.0PU

numbers

0.7 0.6 0.5

4

0.4

1.03 PU 5

0.3

1.04 PU

G

1.02 PU

6

1.04 PU

7

1.03 PU

PL=0.9PU QL=0.3PU

0.2 0.1 0

3

1.04 PU

9 0

10

20

30

40

50

60

70

80

90

1.00 PU

8

1.03 PU

100

sample Fig. 4. The chaotic numbers generated by Henon map.

Typically chaotic motions result when the system operating at a stable periodic orbit undergoes a series of bifurcations leading to the birth of a strange attractor under some parametric variations. The chaotic map can be helpful to escape from a local minimum [32], and it can also improve the global/local searching capabilities.

2

PL=1.25PU QL=0.5PU co

Pg=1.63PU Qg=0.0PU

1.03PU

G

Fig. 5. IEEE 9 bus system.

PL=1.00PU QL=0.35PU

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values given in Fig. 5 are shown in Tables 1 and 2. For each load bus in the IEEE-9 bus system, the actual critical values are obtained with the well known load flow program by increasing the active power gradually until the convergence condition is not achieved, are shown in Table 3 [36]. In this system the loads on all the busses work under the stable power number. Also the study is executed through releasing reactive power borders of PV generators. Under these circumstances the critical values of 5, 7, 9 busses are estimated these critical values refer the maximum active power values will be able to be removed from the busses, voltage amplitude and angle values. PSO, which is a heuristic method is used initially to obtain these critical values (see Table 4). Then those indicated critical values are obtained using CPSO algorithm which is based on chaotic maps. The power system in Fig. 5, if critical values are required to find using PSO algorithm, the non-linear equations referring the active power value in busses 5, 7, 9 are considered as the objective function. The maximum values are found optimization with PSO algorithm with these equations. The circumstances required to be provided searching maximum value are considered together, appropriately function shown in Eq. (2). The active and reactive power equations (Pr, Qr) shown in Eqs. (11) and (12) are obtained according to exemplary system shown in Fig. 5 [37].

Pr ¼

Qr ¼

V s  V r  ðb1  cos d þ b2  sin dÞ  V 2r  ða1 b1 þ a2 b2 Þ 2

2

b1 þ b2 V s  V r ðb1  sin d þ b2  cos dÞ þ ða1 b2 þ a2 b1 Þ  jV r j2 2

2

b1 þ b2

Bus N.

1 2 3

Active power limits

Reactive power limits

Pmax (MW)

Pmin (MW)

Qmax (MVAr)

Qmin (MVAr)

0 163 85

0 0 0

100 40 40

100 40 40

Table 3 IEEE 9 bus system stability critical values obtained by power flow. Bus

V (pu)

d (deg)

P (pu)

5 7 9

0.74 0.78 0.74

26.96 22.71 24.31

3.02 2.60 3.18

Table 4 Stability critical values with using PSO. Bus

V (pu)

d (deg)

P (pu)

5 7 9

0.69 0.83 0.68

31.26 19.96 30.99

3.04 2.68 3.17

ð11Þ

ð12Þ

In Eqs. (11) and (12), Vs is production bus voltage, Vr is load bus voltage, d is load bus angle, a1 and a2 are real and imaginary parts of constant A for long transmission line, b1 and b2 are real and imaginary parts of constant B for known long transmission line. A and B constant of transmission line are given as A = a1 + j ⁄ a2 and B = b1 + j ⁄ b2 respectively. Since it has been searched the maximum power values with PSO, Eq. (11) has been accepted as the goal function (GF). If there are constraints, constraints are added to the fitness function as penalty function. With this way fitness function values are constrained. In this study, since solutions are evaluated for constant power values, power angle u, and so tan u = Qr/Pr must be constant. This situation has been taken as constraint. Under light of these explanations, CF will be as in Eq. (13):

CF ¼ Q r  tan u  Pr

Table 2 IEEE 9 bus system generator limit values.

ð13Þ

CF is ideally equal to zero. While PSO is running, CF value decreases and lastly takes zero value. If CF is greater than zero it is given a penalty by multiplying with a proper coefficient (r). Eq. (14) is described as the penalty function (PF). In this equation r is taken as a proper coefficient and in order to avoid negative results the square of CF has been taken [38]. Since the aim in this study

Table 1 IEEE 9 bus system line datas. Line N.

R (pu)

X (pu)

B (pu)

1.-4 4.-5 6.-5 3.-6 6.-7 8.-7 2.-8 8.-9 4.-9

0.0000 0.0170 0.0390 0.0000 0.0119 0.0085 0.0000 0.0320 0.0100

0.0576 0.0920 0.1700 0.1008 0.0720 0.0720 0.0625 0.1610 0.0850

0.0000 0.1580 0.3580 0.0000 0.1490 0.1490 0.0000 0.3060 0.1760

was maximization the FF, PF is added to the goal function (GF) to obtain the best result. According to these definitions: PF will be as in Eq. (12) and FF will be as in Eq. (15):

PF ¼ r  ðCFÞ2

ð14Þ

FF ¼ GF  PF

ð15Þ

The biggest possible ranges of Vr and d are given in Eqs. (16) and (17) respectively.

0:4 6 V r 6 1:1 pu

ð16Þ

1:5 6 d 6 þ1:5 rad

ð17Þ

This problem solving with PSO, was taken by some PSO values as population size = 50, c1 = 2, c2 = 2.2, inertia weight factor = 0.73, iteration size = 48. These values are obtained by 50 times running the algorithm in the matlab program. Every time the best solutions are reached before 30 iterations. Therefore the iteration size is selected greater than 30 to see on which iteration each bus reaches the best fitness value. Solutions of the PSO algorithm to obtain the maximum values of active power load bus voltage magnitude and angle values are given in Table 5. To control the values of the parameters in velocity update formulation and generate the c⁄rand() coefficients chaos is used. The parameters rnid and Rnid are the important control parameters that affect the PSO convergence. Eq. (5) is used to obtain the parameters generated by the Logistic map. Eqs. (6) and (7) are used to obtain the parameters generated by the Henon map. After obtaining these parameters, CPSO-1 method based on logistic map and Henon map described in Eq. (8), CPSO-2 method based on logistic map and Henon map described in Eq. (9), CPSO-3 method based on logistic map and Henon map described in Eq. (10) are used to obtain the maximum value of active power, load bus voltage magnitude and angle. The results found by CPSO are shown in Table 3. It can be seen that the best solutions are found by CPSO-3- Logistic and Henon map. Figs. 6 and 7 illustrate the changes in active power values based on the generation numbers for the 5th numbered load

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L. Kuru et al. / Electrical Power and Energy Systems 64 (2015) 873–879 Table 5 Stability critical values with using CPSO. BUS-5

BUS-7

3

BUS-9

CPSO-1 Logistic map V (pu) 0.684695113113558 d (rad) 0.544471174392134 P (pu) 3.000364429956300

0.838521601281897 0.348182070654064 2.677462232665930

0.677728663937260 0.543192129421035 3.177674231436110

CPSO-2 Logistic map V (pu) 0.684250837025725 d (rad) 0.544922468555660 P (pu) 3.000368350991450

0.838664440673861 0.348017686725989 2.677462360432380

0.677879742532725 0.543040875604862 3.177675405151280

CPSO-3 Logistic map V (pu) 0.684145065386543 d (rad) 0.545041829036886 P (pu) 3.000369011504720

0.838650386747121 0.348040666363368 2.677462409178120

0.677863405436707 0.543099336568527 3.177675765268820

2.95 2.9 2.85 2.8 2.75 2.7 2.65 2.6 2.55

0.838607062273872 0.348121579155258 2.677462223649360

0.677542877012470 0.543455694327456 3.177674244895290

CPSO-2 Henon map V (pu) 0.683298030222442 d (rad) 0.545872946736051 P (pu) 3.000363439816950

0.838271471590708 0.348471973562935 2.677461010978930

0.677653126597084 0.543363241277351 3.177672810777040

CPSO-3 Henon map V (pu) 0.684025435880896 d (rad) 0.545168229367486 P (pu) 3.000369176430990

0.838604255107994 0.348102662193627 2.677462450115870

0.677785581280436 0.543166364939317 3.177675923012710

BEST FITNESS VALUES NUMBER OF FIVE

3.01 3.005 3 2.995

0

5

10

15

20

25

30

35

40

45

50

GENERATION Fig. 7. 5th Numbered load bus fitness functions values with CPSO-3 using Henon map.

BEST FITNESS VALUES NUMBER OF SEVEN 2.75 2.7

ACTIVE POWER X 100 MW

CPSO-1 Henon map V (pu) 0.684107982658997 d (rad) 0.545097282445681 P (pu) 3.000368949786140

ACTIVE POWER X 100 MW

ACTIVE POWER X 100 MW

Critical values

BEST FITNESS VALUES NUMBER OF FIVE

3.05

2.65 2.6 2.55 2.5 2.45 2.4 2.35 2.3

2.99

2.25 2.985

0

5

10

15

20

25

30

35

40

45

50

GENERATION 2.98 Fig. 8. 7th Numbered load bus fitness functions values with CPSO-3 using Logistic map.

2.975 2.97

0

5

10

15

20

25

30

35

40

45

50

GENERATION

BEST FITNESS VALUES NUMBER OF SEVEN 2.68

Fig. 6. 5th Numbered load bus fitness functions values with CPSO-3 using Logistic map.

ACTIVE POWER X 100 MW

buses, Figs. 8 and 9 illustrate the changes in active power values based on the generation numbers for the 7th numbered load buses, Figs. 10 and 11 illustrate the changes in active power values based on the generation numbers for the 9th numbered load buses obtained by CPSO-3 method using Logistic and Henon map respectively. The 5th numbered Load Bus reaches the best fitness value at the 30th generation by using logistic map where as it reaches the best fitness value at the 25th generation by using Henon map. The 7th numbered Load Bus reaches the best fitness value at the 6th generation by using logistic map where as it reaches the best fitness value at the 29th generation by using Henon map. The 9th numbered Load Bus reaches the best fitness value at the 15th generation by using logistic map where as it reaches the best fitness value at the 17th generation by using Henon map.

2.675 2.67 2.665 2.66 2.655 2.65 2.645 2.64 2.635 2.63

0

5

10

15

20

25

30

35

40

45

50

GENERATION Fig. 9. 7th Numbered load bus fitness functions values with CPSO-3 using Henon map.

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L. Kuru et al. / Electrical Power and Energy Systems 64 (2015) 873–879

References

BEST FITNESS VALUES NUMBER OF NINE

3.2

ACTIVE POWER X 100 MW

3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 0

5

10

15

20

25

30

35

40

45

50

GENERATION Fig. 10. 9th Numbered load bus fitness functions values with CPSO-3 using Logistic map.

BEST FITNESS VALUES NUMBER OF NINE 3.2

ACTIVE POWER X 100 MW

3.15 3.1 3.05 3 2.95 2.9 2.85 2.8

0

5

10

15

20

25

30

35

40

45

50

GENERATION Fig. 11. 9th Numbered load bus fitness functions values with CPSO-3 using Henon map.

Conclusion To find the critical boundary voltage stability values with classical methods require a continual power flow by increasing load stage by stage until it cannot provide the system’s power flow. This situation leads to the prolongation of the solution. Instead of continual power by increasing load stage by stage, PSO and CPSO algorithms can obtain the critical values directly. It is quite important knowing these critical values before to plan and process of a power system. CPSO algorithm based on Logistic map and Henon map have been successfully applied to IEEE nine bus power system. The results compared with the classical PSO show that CPSO algorithm based on Logistic map and Henon map can also reach directly the voltage stability limit values and are alternative solution methods. Simulation results are promising and show the effectiveness of the applied approach. Future work will include a different approach such as chaos is used in order to interact with the PSO algorithm for searching the solution space that is called chaos search.

[1] Taylor CW. Power system voltage stability. McGraw-Hill Inc.; 1994. ISBN 0-07113708-4. [2] Anderson PM, Fouad AA. Power system control and stability. 4th printing. The Iowa State University Press; 1986. [3] Thomas RJ, Tiranuchit A. A posturing strategy against voltage instabilities in electric power systems. IEEE Trans Power Syst 1988;3(1):87–93. [4] Löf PA, Andersson GA, Hill DJ. Voltage Stability Indices for Stressed Power Systems. IEEE Trans Power Syst 1993;8(1):326–35. [5] Löf PA, Andersson GA, Hill DJ. Voltage dependent reactive power limits for voltage stability. IEEE Trans Power Syst 1995;10(1):220–8. [6] Kundur P. Power system stability and control. McGraw-Hill Inc.; 1994. ISBN 007-5958-X. [7] Kundur P, Paserba J, Ajjarapu V. Definition and classification of power system stability. IEEE Trans Power Syst 2004;19(2):1387–401. [8] Morison GK, Gao B, Kundur P. Voltage stability analysis using static and dynamic approaches. IEEE Trans Power Syst 1993;8:1159–71. [9] Rahman TKA, Jasmon GB. A new voltage stability index and load flow technique for power system analysis. Int J Power Energy Syst 1997;17(1):28–37. [10] Yome S, Mithulananthan N, Lee KY. A maximum loading margin method for static voltage stability in power systems. IEEE Trans Power Syst 2006;21(2):799–808. [11] Musirin I, Rahman TKA. Novel fast voltage stability index (FVSI) for voltage stability analysis in power transmission system. In: IEEE international student conference of research and development SCOReD; 2002. p. 265–8. [12] Ozsag˘lam MY, Cunkasß M. Particle swarm optimization algorithm for optimization problems. J Poly Tech, Tech Educ Fac 2008;11(4):299–305. [13] Kumar Sanjeev, Chaturvedi DK. Optimal power flow solution using fuzzy evolutionary and swarm optimization. Int J Electr Power Energy Syst 2013;47:416–23. [14] Gnamabali K, Babulal CK. Maximum loadability limit of power system using hybrid differential evolution with particle swarm optimization. Int J Electr Power Energy Syst 2012;43(1):150–5. [15] Kalil MR, Musirin I, Othman MM. Ant colony based optimization technique for voltage stability control. In: Proceeding of the WSEAS international conference on power system, Lisbon, Portugal; 2002. p. 149–54. [16] Begovic VUK, Novosel M, Saha D. Use of local measurements to estimate voltage-stability margin. IEEE IEEE Trans Power Syst 1999;14(3):1029–35. [17] Musirin I, Rahman MH, Rahman TKA. Optimization of reactive power dispatch via artificial immune system for voltage stability enhancement. In: Proceeding of colloquium of signal processing and its applications, Pangkor Perak, Malaysia; 2005. p. 28–30. [18] Badar Altaf QH, Umre BS, Junghare AS. Reactive power control using dynamic particle swarm optimization for real power loss minimization. Int J Electr Power Energy Syst 2012;41(1):133–6. [19] Zhang Wei, Ma Di, Wei Jin-Jun, Liang Hai-Feng. Parameter selection strategy for particle swarm optimization based on particle positions. Expert Syst Appl 2014;41:3576–84. [20] Sun Shichang, Liu Hongbo. 6 – Particle swarm algorithm: convergence and applications. Swarm Intell Bio-Inspired Comput 2013:137–68. [21] Kennedy J, Eberhart RC. Particle swarm optimization. In: Proc of the IEEE int conference on neural networks, vol. 4; 1995. p. 1942–8. [22] Khamsawang S, Jiriwibhakorn S. Solving the economic dispatch problem using novel particle swarm optimization. Int J Electr Comput Syst Eng 2009;3(1):41–6. [23] Park JB, Jeong YW, Kim HH, Shin JR. An improved particle swarm optimization for economic dispatch with valve-point effect. Int J Innovations Energy Syst Power 2006;1(1):1–7. [24] Gaing ZL. Constrained dynamic economic dispatch solution using particle swarm optimization. In: Power engineering society general meeting; 2004. p. 153–8. [25] Sriyanyong P. Solving economic dispatch using particle swarm optimization combined with Gaussian mutation. In: Proceedings of ECTI-CON 2008; 2008. p. 885–8. [26] Aslantasß V, Dog˘an A, Kurban R. Parçacık Sürü Optimizasyonu ile DWT-SVD Tabanlı Resim Damgalama. Uluslar arası Katılımlı Bilgi Güvenlig˘i ve Kriptoloji Konferansı, Ankara; 13–14 Aralık 2007. p. 213–8. [27] Thompson JMT, Stewart HB. Nonlinear dynamics and chaos. 2nd ed. John Wiley & Sons; 2002. [28] Liu B, Wang L, Jin YH, Huang CX. Designing neural networks using hybrid particle swarm optimization. Lect Notes Comput Sci 2005;3496:391–7. [29] Meng HJ, Zheng P, Wu RY, Hao XJ, Xie Z. A hybrid particle swarm algorithm with em-bedded chaotic search. In: Proc IEEE conference on cybernetics and intelligence systems; 2004. p. 367–71. [30] Chen AL, Wu ZM, Yang GK. LS-SVM based on chaotic particle swarm optimization with simulated annealing. Lect Notes Comput Sci 2006;3959:99–107. [31] He´non M. A two-dimensional mapping with a strange attractor. Commun Math Phys 1976;50:69–77. [32] Coelho LS, Mariani VC. A novel chaotic particle swarm optimization approach using Henon map and implicit filtering local search for economic load dispatch. Chaos Solutions Fractals 2009;39:510–8.

L. Kuru et al. / Electrical Power and Energy Systems 64 (2015) 873–879 [33] Alatas B, Akin E, Ozer AB. Chaos embedded particle swarm optimization algorithms. Chaos Solutions Fractals 2009. [34] Caponetto R, Fortuna L, Fazzino S, Xibilia MG. Chaotic sequences to improve the performance of evolutionary algorithms. IEEE Trans Evol Comput 2003;7(3):289–304. [35] Huang YH, Yeh SN. Cogeneration pricing based on avoided costs of power generation and transmission. J Chin Inst Eng 2004;27(2):211–21.

879

[36] POWERWORLD Power System Simulation Program. . _ [37] Yalçın MA. Enerji Iletim Sistemlerinde Gerilim Kararlılıg˘ının Yeni Bir Yaklasßım _ PhD thesis. I.T.U., F.B.E.; 1995. ile Incelenmesi. [38] Saruhan H, Uygur I. Design optimization of mechanical systems using genetic algorithms. J Inst Sci Technol Sakarya University 2003;7(2).