Optimal Design of Multimachine Power System Stabilizers using Gbest-guided Artificial Bee Colony Algorithm

Optimal Design of Multimachine Power System Stabilizers using Gbest-guided Artificial Bee Colony Algorithm

Badr M. Alshammari College of Engineering,

University of Hail, Saudi Arabia

Abstract In this paper, the concept of foraging behavior of honey bee is exploited for the optimal design of power system stabilizers. The controller design is formulated as an optimization problem in order to shift the system electromechanical modes. The dynamic performance of the proposed optimization algorithm called gbest-guided artificial bee colony algorithm has been examined at various loading conditions and under different disturbances. The proposed approach is applied to familiar multimachine power system: three machines nine bus system. The nonlinear simulations and eigenvalue analysis show the effectiveness and the robustness of the proposed stabilizers to provide efficient damping compared to famous genetic algorithm. As consequence, the system stability is greatly enhanced and both inter-area and local modes are targeted.

Keywords Power system stabilizer, Optimization problem, Artificial bee colony algorithm.

1. INTRODUCTION

   The improvement of low frequency damping is an important requirement for power system security and maximum power transfer [1-2]. In some cases, these oscillations are very poorly damped and may result a serious consequence such as overload in important transmission line and fatigue at the generators. Kundur et al. [3], demonstrated that Power system stabilizers (PSSs) are very effective for damping electromechanical oscillations in power systems. The main function of PSS is to introduce damping torque to the generator rotor oscillations through the excitation system. Recently, considerable researches have been focused on the designing and using of adequate damping sources [4-8].

   Several methods are developed to guarantee a high performance of PSS. In [9 ] the authors have explained the use of conventional leadlag PSS in power system stability improvement. The conventional PSS (CPSS) which are designed based on linear theory has been reported in [10]. In these methods, a linear dynamic model around an operating point is usually employed. However, the parameters of real power system are time varying causing lack of robustness of the PSS controller. The application of adaptative technique for online controller required real time estimation of power system parameters. The implementation of this kind of controller may not be easy and a fixed controller is more feasible in practice. The design problem of PSS controller can be converted to an optimization problem, and several modern metaheuristic algorithms have been developed [11-13]. These algorithms can be classified into two important groups: evolutionary algorithm (EA) and swarm intelligence.

   Recently, Genetic Algorithm (GA) has become the most popular evolutionary algorithm [14]. GAs simulates the phenomenon of natural evolution such as natural selection, crossover and mutation. The authors, in [15], demonstrated that the performance of PSS tuned with GAs can be significantly enhanced. Unfortunately, recent research demonstrated some limitations of GAs like high computational capacity when the problem to be optimized is complex, premature convergence [16]. The degradation in efficiency also appears, when the optimization problem presented an epistatic objective function and the number of the parameters to be optimized is large.

   A global optimization algorithm, which utilizes the swarm intelligence, has received significant interest from researchers. These algorithms, attempt to simulate the behavior of natural swarms such as colony of ants, flock of birds, colony of bees, etc. In [17], the authors developed several approaches to model the intelligent behaviors of honey bees. In 2005 Karaboga introduced a swarm intelligence based algorithm called Artificial Bee Colony (ABC)[19]. It is based on foraging behavior of bee colony, which is characterized by learning, memorizing and exchanging information. In some publication [20-21], the authors demonstrated that the performance of ABC algorithm is competitive compared with other optimization techniques such as Ant colony optimization (ACO), particle swarm optimization (PSO), differential evolution (DE). Due to, its flexibility and simplicity of implementation, ABC algorithm has been used to solve many optimization problems.

   The solution search equation of ABC is poor at exploitation and good in exploration. In order to achieve good performance in optimization problems, the two aspects should be well balanced. In practice the two abilities contradicts to each other. In [18], the authors proposed a solution of the mentioned drawback based on the modification of the solution search equation .This approach aims to improve the exploitation by applying the global best (gbest) solution in order to guide the search of new candidate solutions. This type of ABC is called Gbest-guided ABC (GABC) algorithm.

   The rest of this paper is structured as follows. In Sect. 2 we introduce the power system model. Section 3 describes the design of the damping controller. The original and the Gbest- guided artificial bee colony algorithm (GABC) are presented in Section 4. Section 5 presents and discusses the results. Finally, the conclusion is drawn in Section 6.

2. SYSTEM MODELING

   1. Synchronous machine model

      In this study, the synchronous machine is modeled as one axis model. The third order nonlinear differential equations for any i-th machine are expressed as follows [1].

      .

      i b i 1

      (1)

      . 1 P

      P D 1

      (2)

      M

      M

      i mi ei i i

      i

      E. ' 1 E

      x x' i E'

      (3)

      Fig. 1. IEEE Type-ST1 excitation system with PSS

      T

      T

      qi '

      d 0i

      fdi di di di qi

      In the previous equation, the washout bloc with time

      Where and are rotor angle and angular speed of the

      constant TW

      is used as high-pass filter to leave the signals in

      i

      machine. b

      i

      is the base frequency in rad/sec.

      Pmi

      and Pei are

      range 0.2-2 Hz associated with rotor oscillation to pass without change. In general, it is in the range of 1-20 s. In this

      the mechanical input and the electrical output powers for the

      E

      E

      qi

      qi

      machine i, respectively. Di and Mi are the damping coefficient and inertia constant, respectively. E fdi and ' are

      study, Tw 5 s .The two first order lead-lag transfer function serve to compensate the phase lag between the PSS output and the control action which is the electrical torque.

      the field and the internal voltages, respectively. id

      armature current .

      is the d-axis

3. DAMPING CONTROLLER DESIGN

   After linearizing the power system model around the

   x

   x

   d

   d

   xd and ' are the d-axis transient reactance and the d-

   operating point, the closed-loop eigenvalues of the system are computed and the desired objective functions can be

   T

   T

   d 0

   d 0

   axis reactance of the generator, respectively. '

   circuit field time constant.

   The electrical torque Te can be expressed by

   Te E' i iqi xqi x' i idi iqi

   is the open

   (4)

   formulated using only the unstable or lightly damped electromechanical modes that need to be shifted.

   In this paper, two eigenvalue objective functions are considered, in order to solve the problem of paameters tuning of the PSS controllers. The first one consists in shifting the

   q d closed-loop eigenvalues in to the left-side of the line defined by

   1. Excitation system with PSS structure

   0 . This function is expressed by

   J1 in equation (7). In

   The PSS acts through the exciter and provides control effect to the power system under study [3]. The IEEE type- ST1 excitation system with PSS shown in Fig. 1 is considered

   in this paper. Where, K Ai and TAi are the regulator gain and

   equation (8), J 2 defines the second objective function. It will place the closed-loop eigenvalues in a wedge-shape sector corresponding to i, j 0 . As consequence, the maximum

   the regulator time constant of the excitation system,

   overshoot is limited.

   respectively.

   Vrefi

   and

   Vti

   are reference and generator np 2

   terminal voltages of the i-th machine, respectively. The field

   J1

   0 i, j

   (7)

   voltage can be modeled by the following equation :

   j 1 i, j 0

   1

   1

   .

   E fdi

   TAi

   Efdi KAi Vrefi Vti Ui

   (5)

   np

   J 2

   0 i, j

   (8)

   2

   2

   As shown the Fig.1, the PSS representation consists of a

   j 1

   gain Ki , a washout block with time constant

   Twi

   and two

   i, j 0

   leadlag blocks. Its input signal is the normalized speed

   Where np is number of operating points and

   and

   deviation, i . While, the output signal is the supplementary

   i, j

   stabilizing signal, Ui . As given in the bloc diagram of Fig. 1,

   i, j

   are respectively, real part and damping ratio of the i-th

   the transfer function of the PSS is given below. eigenvalue corresponding to the j-th operating point.

   U K

   s Twi

   1 sT1i 1 sT3i

   (6)

   Therefore, the design problem is aimed to minimize an

   wi 2i

   wi 2i

   4i

   4i

   i i 1 sT 1 sT

   1 sT i

   eigenvalue based multi-objective function J obtained by combination J1 and J 2 . In order to offset the weight of the

   first and the second objective function a weight factor " a" is

   used. The multi-objective function J is given by the following expression [22].

   np

   max

   max

   min

   min

   J

   2

   Where x j

   and x j are the upper and the lower of a food

   j 1

   np

   i, j

   0

   0

   i, j

   2

   (9)

   source.

   Step2: As mentioned earlier the number of food source sites is equal to the number of employed bees. An employed bee

   a

   0 i, j

   searches a neighborhood of its current solution based on visual

   j 1 i, j 0

   information. The new food source is described by

   The closed-loop eigenvalues will be placed in the D-shape

   vij xij ij ( xij xkj )

   (16)

   sector. In the design process the adjustable parameter bounds given by equations (10)-(14) must be respected.

   The adjustable parameters bounds are formulated as follows:

   It must be noted that ij is a random number in the range

   1, 1 and k has to be different from i. In this work, if the values of the parameters produced by the previous equation

   K min K

   T min T

   K max

   T max

   (10)

   (11)

   exceed their boundaries, the parameters are set to their boundaries values as follows:

   1 1 1

   If x xmax

   x xmax

   T min T

   T max

   (12)

   i i i i

   (17)

   2 2 2

   If x xmin

   x xmin

   T min T T max

   (13)

   i i i i

   3 3 3

   For a minimization problem, the neighborhood solution vi

   T min T T max

   (14)

   4 4 4

   can be evaluated through (18)

   fitness 1/ (1 fi )

   if fi 0

   (18)

4. GBEST-GUIDED ABC (GABC) ALGORITHM

   1 abs( fi )

   if fi 0

   1. Artificial Bee Colony algorithm description

      The artificial Bees colony (ABC) Algorithm is an optimization algorithm which simulates the natural foraging

      Step3: An onlooker bee selects a food source based on the information distributed by the employed bees. The probability of selection of the food source xi is defined by

      behavior of honey bees to find the optimal solution[19]. A colony of honey bees consists of three groups of bees, employed bees, onlookers bees and scouts. The first half of

      pi

      fitnessi

      SN

      fitnessi i1

      (19)

      colony consists of employed bees which are responsible for

      exploring food sources. These bees go to their food sources and come back to hive and dance on this area. The onlooker bees wait in the hive and choose food source to exploit. This decision is depending on employed bees dances. When food sources are exhausted, the employed bees become scouts and start to search in the environment for finding new food sources. The main steps of the ABC algorithm are summarized as follows:

      1. At first, bees explore the search space randomly in order to find a food source.

      2. When the food source is found, the employed bees start to load the nectar. Then they return to the hive to unload the nectar. These bees share information about their source with onlooker bees through a dance in the dance area. The probability of food choice depends on nectar amount information distributed by the employed bees.

      3. The employed bee whose food source has been exhausted, become scouts and starts to search for a new source randomly. The proposed algorithm involves five steps [23]:

         j j j

         j j j

         Step1: In the ABC algorithm a possible solution of the optimization problem is represented by a position of food source. The population size of food source is SN and the number of the parameters to be optimized is D. The initialization phase starts with producing food source randomly within the range of the boundaries via the following expression

         Where is the fitness value of food source.

         Once probabilistic selection is completed, the onlooker bees produce a new food source using eq.(16). After generation a new food source, it will be evaluated and greedy selection will be performed.

         Step 4: The food source whose number of trials exceeds the predetermined limit, are considered to be exhausted and is abandoned. The employed bees associated with exhausted source become a scout. It is assumed that only one employed bee at each cycle can become a scout. The scout generates randomly a new food source for replacing the abandoned source.

         Step 5: The process is stopped, when the termination condition is met and the best food source is memorized. Otherwise repeat the algorithm since the second step.

   2. Gbest-guided ABC

   It is important to point out, that exploitation and exploration are necessary for evolutionary algorithms. The exploitation refers to the ability to use the knowledge of the previous good solutions to generate a better candidate solution. While, the exploration refers to the ability to seek in the various unknown regions of the solutions space to discover the global optimum. in order to achieve good optimization performance the two ability should be well balanced. The search equation of ABC described by Eq. (16) is poor in exploitation, so the new candidate solution is not promising to be better than the previous one.

   xij

   xmin rand (0,1) (xmax xmin )

   (15)

   To improve exploitation, Zhu and Kwong [18] proposed to incorporate the global best solution (gbest) into the solution search equation which described as follows

   vij xij ij xij xkj ij yj xij

   (20)

   table I. The system eigenvalues and damping ratio of mechanical mode for all loading conditions, both with

   p>Where the new added term is called global best (gbest) term,

   y j is the jth element of the global best solution, and ij is a uniformly distributed random number in [0,C], where C is a positive constant.

   GAPSS, GABCPSS and without PSS are given in table II. It is clear, when the PSS is not mounted, the electromechanical modes are poorly damped and both of them are unstable.

   There are 10 parameters to be optimized which are Ki , T1i ,

5. RESULTS AND DISCUSSIONS

   T2i , T3i

   and T4i . The washout time constant and weighting

   1. Test system

      In this study, the 3-machine 9-bus (WSSC) shown in Fig. 2 is considered.. The system data in detail is given in [2].It is assumed that all generators except G1 are equipped with PSS [4].

   2. PSS tuning

      To compute the optimum value of the proposed GABCPSS, three operating conditions are considered. The generators operating conditions and the loads are listed in

      factor a are set to be 5 s and 10 respectively. In designing the objective function to be optimized, the value of 0 and 0 are chosen to be -1.5 and 0.20.

      It is quite clear that the proposed GABCPSS is able to shift all electromechanical mode in the D-shape sector specified by

      0.2 and 1.5 . Hence, compared to GAPSS, the

      proposed GABCPSS greatly enhances the system damping and the dynamic stability is significantly improved.

      2

      2 7 8

      Load C

      3

      9 3

      5 6

      Load A Load B

      4

      1

      1

      Generator	Base Case		Case 1		Case 2		Case 3
      	P [pu]	Q [pu]	P [pu]	Q [pu]	P [pu]	Q [pu]	P [pu]	Q [pu]
      G1	0.72	0.27	2.21	1.09	0.36	0.16	0.33	1.12
      G2	1.63	0.07	1.92	0.56	0.80	-0.11	2.00	0.57
      G3 Load	0.85	-0.11	1.28	0.36	0.45	-0.20	1.50	0.38
      A	1.25	0.50	2.00	0.80	0.65	0.55	1.50	0.90
      B	0.90	0.30	1.80	0.60	0.45	0.35	1.20	0.80
      C	1.00	0.35	1.50	0.60	0.50	0.25	1.00	0.50

      Generator	Base Case		Case 1		Case 2		Case 3
      	P [pu]	Q [pu]	P [pu]	Q [pu]	P [pu]	Q [pu]	P [pu]	Q [pu]
      G1	0.72	0.27	2.21	1.09	0.36	0.16	0.33	1.12
      G2	1.63	0.07	1.92	0.56	0.80	-0.11	2.00	0.57
      G3 Load	0.85	-0.11	1.28	0.36	0.45	-0.20	1.50	0.38
      A	1.25	0.50	2.00	0.80	0.65	0.55	1.50	0.90
      B	0.90	0.30	1.80	0.60	0.45	0.35	1.20	0.80
      C	1.00	0.35	1.50	0.60	0.50	0.25	1.00	0.50

      Fig. 2. Single line diagram for the 3-machine 9-bus Table I. Loading conditions for the system (in p.u).

      TABLE II. ELECTROMECHANICAL MODE AND DAMPING RATIOS OF TEST SYSTEM UNDER DIFFERENT LOADING CONDITIONS

      	Without PSS	GAPSS	GABCPSS
      Base Case	-0.1124±j 7.7400, 0.0145	-1.8096± 4.5504, 0.3695	-2.6760± j 4.6586, 0.4981
      	-1.3346±j 9.1096, 0.1450	-2.4897± 7.5480, 0.3132	-2.6097± j 9.6042, 0.2622
      Case 1	-0.0374±j 7.8347, 0.0048	-1.9889± j 5.6843, 0.3303	-2.2596± j 5.1552, 0.4014
      	-0.7023±j 10.5832, 0.0662	-2.5204± j 7.3736 , 0.3234	-2.6756± j 11.1852, 0.2326
      Case 2	-0.2142±j 6.3226, 0.0339	-1.5189± j 4.1628, 0.3428	-2.2717± j 4.3849, 0.4600
      	-0.8227±j 6.9390, 0.1177	-1.4789± j 6.1042, 0.2355	-1.9109± j 7.6704, 0.2417
      Case 3	0.0181±j 8.0903, – 0.0022	-2.2779± j 5.6069, 0.3764	-2.5524± j 5.1564, 0.4436
      	-0.4515±j 11.3794, 0.0396	-2.2162± j 7.7961, 0.2734	-2.6455± j 12.0431, 0.2146

      The result of the PSS controller parameters set values based on multiobjective function using both the proposed GABC and GA are given in table III. Fig. 3 shows the convergence rate of the objective function with two optimization techniques which are GABC and GA. When the final value of the multiobjective function is J = 0 for two algorithms, the electromechanical modes are restricted in the specified D-shape sector. Also, the parameters of both methods are given in table IV. In order to obtain global optimal solution, the optimization algorithms( GABC and GA ) are run several times and

      Furthermore, the proposed controllers have better performance in term of settling time and overshoots compared to other controllers. With changing operating conditions from the base case to case three, while the performance of GAPSS become poorer, the GABCPSS have robust and stable performances. This confirms the superiority of GABC design approach over GA design approach.

      TABLE IV. GABC AND GA PARAMETERS

      GABC algorithms Genetic algorithms

      then the optimum PSS parameters are selected.

      TABLE III. OPTIMAL PARAMETERS OF THE PROPOSED ABCPSS

      Methods Gen K T1 T2 T3 T4

      GAPSS G2 8.6355 0.7053 0.3223 0.1809 0.3523

      G3 1.6280 0.3325 0.1976 1.3654 0.1687

      GABCPSS G2 10.0000 0.2226 0.3334 0.6000 0.1381

      G3 8.6432 0.0500 0.2575 0.3780 0.2780

      1. Nonlinear time domain simulation

   To demonstrate the robustness of the proposed PSSs tuned over wide range of loading conditions and using the proposed multiobjective function, the following scenarios are considered:

   1. .Scenario 1

      Number of colony size = 100 Food number = 50

      Maximum cycle number = 100 Number of trials "Limit" = 80

      Population size = 100 Number of chromosome = 9 Mutation rate = 0.05

      Cross over rate = 0.5 Number of iterations = 100

      In this scenario, a six cycle three phase fault at bus 7 at the end of line 5-7 is considered. This severe disturbance is cleared without line tripping.The rotor speed deviation of generator for three operating conditions are shown in Figs.47. From these figures, it can be seen that the response with GABCPSS controllers show good damping characteristics to low frequency oscillations and the system is more quickly stabilized than GAPSS.

      Fig.3. Variations of objective function; solid (GABC), dashed (GA)

      Fig 4. Speed response for 6-cycle fault with base case in scenario 1; solid (GABCPSS), dashed (GAPSS) and dotted (without PSS).

      Fig 5. Speed response for 6-cycle fault with case1 in scenario 1; solid (GABCPSS), dashed (GAPSS) and dotted (without PSS)

      Fig 6. Speed response for 6-cycle fault with case 2 in scenario 1; solid (GABCPSS), dashed (GAPSS) and dotted (without PSS)

      Fig 7. Speed response for 6-cycle fault with case 3 in scenario 1; solid (GABCPSS), dashed (GAPSS) and dotted (without PSS).

   2. .Scenario 2

   A more severe disturbance, of a six cycle three phase fault at bus 7 at the end of line 5-7 is considered. The fault is cleared by permanent tripping of the fault line. Fig.8-11 show the speed deviation of the second and the third machines under for operating conditions. The results show that the test power system is adequately

   damped when the GABCPSS is mounted. These responses are consistent with the results of eigenvalue analysis. In addition, it can be seen that GABC base PSSs enhances significantly the first swing stability. This illustrates the potential of the proposed approach to select an optimal set of PSSs parameters.

   Fig. 8. Speed response for 6-cycle fault with base case in scenario 2; solid (GABCPSS), dashed (GAPSS) and dotted (without PSS).

   Fig. 9. Speed response for 6-cycle fault with case1 in scenario 2; solid (GABCPSS), dashed (GAPSS) and dotted (without PSS).

   Fig. 10. Speed response for 6-cycle fault with case 2 in scenario 2; solid (GABCPSS), dashed (GAPSS) and dotted (without PSS)

   Fig. 11. Speed response for 6-cycle fault with case 3 in scenario 2; solid (GABCPSS), dashed (GAPSS) and dotted (without PSS)

   C.3. Performance investigation

   To analyze performance robustness of the proposed stabilizers a performance index: the Integral of the Time multiplied Absolute value of the Error (ITAE) is considered as follows:

   tsim

   ITAE is calculated for two scenarios and at all operating conditions and the results are shown in table V. It can be seen that the values of these system performance index with GABCPSS are much smaller compared to GA tuned controller. This demonstrate that the application of the

   proposed GABCPSS reduce greatly the settling time,

   ITAE 100 0

   t 2 3 dt

   (20)

   overshoot, undershoot and speed deviations. Moreover, these values of the ITAE are smaller than those obtained

   It is to be mentioned that the lower value of this index reflect the better system response characteristics. The

   in [24].

   TABLE V. VALUES OF PERFORMANCE INDEX (ITAE) FOR DIFFERENT CONDITIONS

   	Scenario 1					Scenario 2
   	Base Case	Case 1	Case 2	Case 3		Base Case	Case 1	Case 2	Case 3
   GABC-PSS	0.6286	0.4888	0.4482	0.5784		0.7748	0.8263	0.6459	1.3606
   GA-PSS	1.2357	0.9603	0.5492	3.1383		1.1614	1.5437	1.0990	3.2127

6. CONCLUSION

In this paper, the Gbest-guided artificial bee colony algorithm (GABC) has been successfully applied for power system stabilizer design. The problem of robust PSS is converted to an optimization problem according to eigenvalue based objective function which is solved by GABC algorithm. The effectivness of the proposed design strategy is tested on a multimachine power system subject to severe disturbances for wide range of operating conditions. The nonlinear simulation shows that the proposed GABCPSS controller enhances greatly the dynamic stability over wide range of loading conditions. The eigenvalue analysis reveals that the electromechanical modes are shifted to the left in s-plane. The system performance index demonstrates the superiority of the proposed stabilizers compared to GA based tuned stabilizers.

ACKNOWLEDGMENT

This research work was supported by University of Hail.

REFERENCES

1. P.W. Sauer, and M.A. Pai, Power system dynamics and stability, Englewood Cliffs, NJ: Prentice Hall; 1998.

2. P.M. Anderson PM, and A.A. Fouad, Power system control and stability, Iowa, USA: Iowa State University Press; 1997.

3. P. Kundur, Power System Stability and Control, McGraw-Hill, 1994.

4. M.A. Abido, Robust design of multimachine power system stabilizers using simulated annealing, IEEE transactions on energy conversion, Vol. 15, No. 3, pp. 297-304, 2000.

5. M.A. Abido, and Y.L. Abdel-Magid, Eigenvalue assignments in multimachine power systems using tabu search algorithm, Electrical. Computers and Electrical Engineering, Vol. 28, pp. 527-545, 2002.

6. H.Yassami, A. Darabi, and S.M.R. Rafiei, Power system stabilizer design using Strength Pareto multi-objective optimization approach, Electric Power Systems Research, Vol. 80, pp. 838-846, 2010.

7. M. Kashki, Y.L. Abdel-Magid, and M.A. Abido, Parameter optimization of multimachine power system conventional stabilizers using CDCARLA method, Electrical Power and Energy Systems, Vol. 32, pp. 498-506, 2010.

8. T. Guesmi, A. Farah, H. Hadj Abdallah, and A. Ouali, Adaptative Coordinated Series Capacitor and Power System Stabilizers for Power System Stability Enhancement, International Review of Electrical Engineering (I.R.E.E), Vol. 9, pp. 113-119, 2014.

9. S. Mishra, M. Tripathy, and J. Nanda, Multi-machine power system stabilizer design by rule bacteria foraging, Electric Power Systems Research Vol. 77, pp. 1595-1607, 2007.

10. K.R. Padiyar, Power system dynamics: stability and control, Bangalore: interline Publishing Private Ltd, 1996.

11. M. Mary Linda, N. Kesavan Nair, A new-fangled adaptive mutation breeder genetic optimization of global multi-machine power system stabilizer, Electrical Power and Energy Systems, Vol. 44, pp. 249-258, 2013.

12. H. Alkhatib, and J. Duveau, Dynamic genetic algorithms for robust design of multimachine power system stabilizers, Electrical Power and Energy Systems, Vol. 45, pp. 242-251, 2013.

13. A. Ghasemi, H. Shayeghi, and H. Alkhatib, Robust design of multimachine power system stabilizers using fuzzy gravitational search algorithm, Electrical Power and Energy Systems, Vol. 51, pp. 190-200, 2013.

14. D.E.Goldberg, "Genetic algorithms in search, optimization and machine learning, Reading, MA: Addison-Wiesley, 1989.

15. Y.L. Abdel-Magid, and M.A. Abido, Optimal multiobjective design of robust power system stabilizers using genetic algorithms, IEEE transactions on power systems,Vol. 18. No. 3, pp. 1125-1132, 2003.

16. D.B. Fogel, Evolutionary computation towards a new Philosophy of Machine Intelligence, IEEE, New York, 1995.

17. V. Tereshko, and T. Lee, How information mapping patterns determine foraging behaviour of honey bee colony, Open Syst. Inf. Dyn. Vol. 9, pp.181193, 2002.

18. G. Zhu, and S. Kwong, Gbest-guided artificial bee colony algorithm for numerical function optimization, Applied Mathematics and Computation, Vol. 217, pp.31663173, 2010.

19. D. Karaboga, An Idea Based On Honey Bee Swarm For Numerical Optimization, Technical Report-TR06, Erciyes Universty, 2005.

20. D. Karaboga, and B. Basturk, A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm, J. Global optim. Vol. 39, pp. 459-471, 2007.

21. D. Karaboga, and B. Basturk, On the performance of artificial bee colony (ABC) algorithm, Applied Soft Computing, Vol. 8, pp. 687-697, 2008.

22. H. Shayeghi, H.A. Shayanfar, A. Safari, and R. Aghmasheh, Robust PSSs design using PSO in a multi-machine environment, Energy Conversion and Management, Vol. 51, pp. 696-702, 2010.

23. D. Karaboga, and B. Akay, A comparative study of Artificial Bee Colony algorithm, Applied Mathematics and Computation, Vol. 214, pp. 108-132, 2009.

24. H. Shayeghi, A. Safari, and H.A. Shayanfar, Multimachine power system stabilizers design using chaotic optimization, Energy Conversion and Management, Vol. 5.

BIOGRAPHY

Badr M. Alshammari obtained his B.Sc., M.Sc. and Ph.D. degrees from King Saud University, Riyadh, Saudi Arabia in 1998, 2004 and 2012, respectively. He is currently an Assistant Professor and Vice Dean for Quality & Development, College of Engineering at University of Hail, Hail, Saudi Arabia. He is a chairman of the Research Committee in the college of Engineering and member of the Research Committee at the Hail

University. His main research area of interest is Power System Reliability and Optimized Performance of Electricity Systems.