Design of Linear and Circular Antenna Arrays Using ...

2011 IEEE Jordan Conference on Applied Electrical Engineering and Computing Technologies (AEECT)

Design of Linear and Circular Antenna Arrays Using Biogeography Based Optimization Ashraf Sharaqa and Nihad Dib Electrical Engineering Department Jordan University of Science and Technology P. O. Box 3030, Irbid 22110, Jordan [email protected], [email protected]. Abstract—The design of linear and circular antenna arrays is one of the important electromagnetic optimization problems. In this paper, the problem of designing these arrays for specific radiation properties is dealt with. The biogeography based optimization (BBO) method, which represents a new evolutionary algorithm, is used in the optimization process. BBO is used to minimize the maximum side lobe level (SLL) and null control for isotropic linear antenna arrays by optimizing different array parameters (position, amplitude, and phase). Similarly, for non-uniform circular antenna arrays (CAA), BBO is used to determine an optimum set of weights and positions that provide a radiation pattern with maximum SLL reduction with the constraint of a fixed major lobe beam width. The results obtained show the effectiveness of BBO compared to other optimization methods. Keywords-Antenna arrays; circular arrays; cptimization methods; ciogeography based optimization (BBO)

I.

INTRODUCTION

Antenna arrays play an important role in modern wireless applications, such as radio, TV, mobile and satellite [1]. They are useful in high power transmission, reduced power consumption and enhanced spectral efficiency. The synthesis of antenna arrays with desired pattern has been a subject of very much interest in the literature. Several well-known evolutionary optimization techniques; such as particle swarm optimization (PSO), Taguchi optimization, genetic algorithm (GA), and differential evolution (DE); have been used in the synthesis of antenna arrays [2-11]. Biogeography based optimization (BBO) technique is a new global evolutionary multi-dimensional optimization method [12-14]. It is based on the science of biogeography which is the nature's way of species distribution. The mathematical model of BBO is based on the extinction and migration of species between neighboring habitats which is any island that is geographically isolated from other islands. Islands that are more suitable for habitation than others are said to have a high habitat suitability index (HSI). HSI is considered as a dependent variable. Another interesting variable is called suitability index variable (SIV) which characterizes habitability. It is an independent variable of the habitat. BBO has already proven itself as a valuable optimization technique compared to other already developed techniques [12-14]. The BBO has recently been applied successfully in optimal power flow problems [15, 16]. In the field of electromagnetism, BBO has been applied to the optimal

978-1-4577-1084-1/11/$26.00 ©2011 IEEE

design of Yagi-Uda and linear antenna arrays [17, 18] and to calculate the resonant frequencies of rectangular and circular microstrip patch antennas [19, 20]. Here, the BBO is further applied to the design of linear antenna arrays and nonuniform circular antenna arrays. The goals of linear antenna arrays synthesis are to have an optimal side lobe level and impose nulls in certain directions. The optimized parameters will be the elements excitation amplitude, excitation phase, and location. For the design of circular arrays, one has to choose a sufficient number of antennas in the array, locations along the circle, radius, and feeding currents (amplitudes and phases) of the antenna elements. The rest of this paper is organized as follows: In Section II, the geometry and array factor for both the linear and circular antenna arrays are presented. Then, based on these models, in Section III, the fitness (or cost) function is presented. Moreover, numerical results are given and compared to the results obtained using other optimization methods. Finally, the paper is concluded in Section IV. For brevity purposes, the BBO algorithm will not be presented here; the reader can consult the references cited above for the full details of the BBO algorithm, and [21] to obtain the basic BBO Matlab codes. II.

GEOMETRY AND ARRAY FACTOR

A. Linear antenna array Linear antenna array (LAA) is one of the easiest array antennas in implementation and fabrication. Figure 1 shows a linear antenna array that consists of 2N elements symmetrically distributed along the x-axis.

Figure 1. Geometry of 2N-element symmetric LAA placed along the x-axis.

In general, the array factor of a LAA is given as follows: 2∑

exp

φ

(1)

where k is the wave number, and In , φn, and xn are, respectively, the excitation amplitude, phase, and location of


the array elements. Assuming that the 2N elements are placed symmetrically along the x-axis simplifies the array factor to become as follows: 2∑

cos

φ

(2)

In order to minimize the maximum side lobe level (SLL), it is clear from equation (2) that, there are three parameters controlling the array factor; the amplitudes, the phases, and the positions of the elements. In this paper, BBO method is used to design LAAs by optimizing these parameters individually. B. Non-uniform circular antenna array Figure 2 shows a non-uniform circular antenna array (CAA) of radius a lying in the xy-plane. In the xy-plane (θ=90), the array factor of the circular array is given by [1]: ∑

α

exp

(3)

where N is the number of antenna elements, and In is the excitation amplitude. αn is the phase of the nth element which is set to αn = - k a cos(φ0-φn), where φn is the angular position of the nth element in the xy-plane and φ0 is the direction of the main beam (in our design problems, φ0 is chosen to be zero). dn represents the arc separation (in terms of wavelength) between element n and element n –1.

where [0, φn] is the side lobes region which depends on the number of elements. Here, it is chosen as [0, 76o] and [0, 80o] for 10 and 16 elements LAAs, respectively. Three optimization cases will be considered; optimizing elements amplitudes, optimizing elements positions, optimizing elements phases. 1) Optimizing elements amplitudes (In) To optimize the amplitudes, φn and xn are fixed to be those corresponding to the uniform array. Assuming φn=0 and the spacing between adjacent elements equal to (λ/2), the array factor becomes: 2∑

cos

0.5 π

(5)

The excitation currents amplitudes are assumed to be within the range [0, 1]. Two cases of linear arrays are optimized; 10 and 16 elements using the BBO technique. Example 1: 10 elements LAA Using the fitness function associated with the array factor for 10 elements linear array, Table I shows the optimum amplitudes obtained using the BBO. Figure 3 shows the radiation pattern obtained using these results as compared to that obtained using PSO [8] and Taguchi [9] results. A Laptop with 2 GHz Intel Centrino CPU and 2048 RAM was used for simulating the BBO’s code, and the simulation time was only 7 seconds (population size=50, number of generations=200). The maximum side lobe levels obtained using the BBO, PSO and Taguchi methods are -25.19 dB, -24.62 dB, -24.87 dB, respectively. BBO results are slightly better than other methods. It should be noted that the maximum SLL for the uniform array is -12.97 dB. TABLE I. OPTIMUM AMPLITUDE VALUES FOUND BY BBO METHOD FOR THE 10 ELEMENTS LAA. n

1

2

3

4

5

In

1.0000

0.8999

0.7195

0.5026

0.3860

Figure 2. Geometry of a non-uniform CAA with N isotropic radiators.

0 Uniform BBO PSO Taguchi

-5

FITNESS FUNCTION AND NUMERICAL RESULTS

In all of the following design examples, the following BBO parameters are used: habitat modification probability = 1, mutation probability = 0.01 and elitism parameter = 2. Population size and number of generations are given in each case individually. A. Linear antenna array optimization

-10 Normalized AF (dB)

III.

-15 -20 -25 -30

In order to minimize the maximum SLL, the following fitness function is used:

-35 -40

Minimize Subject to

fitt

max 20 log|AF

∈ 0,

|

0

10

20

30

40

50

60

70

80

90

φ (Degrees)

(4)

Figure 3. Radiation pattern of 10 elements LAA using BBO as compared to the PSO results from [8], Taguchi results from [9] and uniform array.


TABLE III. OPTIMUM POSITIONS VALUES (IN TERMS OF WAVELENGTH) FOUND BY BBO METHOD FOR THE 10 ELEMENTS LAA.

Example 2: 16 elements LAA In this example, a 16-element LAA is optimized using BBO method (population size=60, number of generations=200). The best results are listed in Table II. Figure 4 shows the radiation pattern obtained by BBO compared to other methods. The maximum SLL obtained using BBO method is -33.06 dB, while that obtained using the PSO, Taguchi and the uniform array is -30.7 dB, -31.31 dB, -13.15 dB, respectively.

n

1

2

3

4

5

xn

0.21451

0.60006

1.061

1.587

2.25


-5

n

1

2

3

4

In

1.0000

0.9402

0.8487

0.7104

n

5

6

7

8

In

0.5596

0.4115

0.2697

0.2035

Normalized AF (dB)

-10

TABLE II. OPTIMUM AMPLITUDE VALUES FOUND BY BBO METHOD FOR THE 16-ELEMENT ARRAY.

-15 -20 -25 -30 -35 -40


-5

Normalized AF (dB)

-10

10

20

30

40

50

60

70

80

90

φ (Degrees)

Figure 5. Radiation pattern of 10 elements LAA optimized with respect to positions compared to the PSO results from [8], Taguchi results from [9] and uniform array.

-15

3) Optimizing elements phases (φn)

-20

Here, BBO method will be applied on a 20-element LAA to provide bidirectional null steering at 33.5 and 40 degrees by optimizing the elements phases. As in previous examples, in order to optimize the phases of the element, the other parameters have to be fixed. The amplitudes are set to unity and the spacing between adjacent elements is set to 0.5λ.

-25 -30 -35 -40

0

0

10

20

30

40

50

60

70

80

90

φ (Degrees)

Figure 4. Radiation pattern of 16 elements LAA using BBO as compared to the PSO results from [8], Taguchi results from [9] and uniform array.

2) Optimizing elements positions (xn) In order to optimize the elements positions in the LAA by the BBO, the elements amplitudes and the elements phases are fixed as In=1 and φn=0 for n=1…N. The length of the optimized array should not exceed the uniform one which is 4.5λ for 10 elements array with spacing between elements of 0.5λ. Therefore, the positions of the outermost elements x±5 are fixed at ±2.25λ, and the positions of the other four elements are optimized. This simplifies the array factor to be as follows: 2 ∑

π cos cos 4.5 π

(6)

Table III contains the optimized positions obtained using the BBO (population size=65, number of generations=500), while Figure 5 shows the obtained array factor. The maximum SLL obtained by BBO method (-19.7 dB) is the same as that obtained by PSO [8] and Taguchi [9] methods.

The phases of the element are assumed to be symmetric as φn =φ-n for n=1, 2, ,...N, where φn is the phase of the n-th element. For a symmetric array, the array factor is given as: 2∑

exp φ

cos

0.5 π

(7)

where θ denotes the angle measured from the positive y-axis (i. e., θ = (π/2)-φ). The normalized array factor is given as: 20 log

(8)

where the maximum array factor occurs at θ=0. To impose nulls at specific angles in addition to reducing the maximum SLL by optimizing the phases of the array elements; the fitness function is modified to become:

∑

(9)

where K1 and K2 are weights, fSL(θ) is the fitness function in the feasible region of side lobes (in our design problem, θ is changed from 5.75o to 90o), and fNS(θ) is the fitness function in the direction of the k-th null. Table IV shows the optimum phases (in degrees) for 20 elements array using BBO method without nulls (population size=70, number of generations=350) and with bidirectional null steering at

2011 IEEE Jordan Conference on Applied Electrical Engineering and Computing Technologies (AEECT) o

33.5 and 40o generations=665).

(population

size=170,

number

of

TABLE IV. OPTIMUM PHASE VALUES (DEGREES) FOUND BY BBO METHOD FOR THE 20-ELEMENT ARRAY WITHOUT NULLS AND WITH BIDIRECTIONAL NULL STEERING AT 33.5O AND 40O. The phase values (degrees) BBO (without nulls) BBO (with nulls)

107.819 101.786 100.846 103.407 98.262 100.491 52.127 156.228 115.754 100.038 114.5916 114.5916 135.6821 84.4654 139.8590 -171.8873

97.9128 114.5916 110.5866 135.9400

For the first case, the maximum SLL for the uniform array is -13.19 dB while the maximum SLL for the BBO-optimized array is -16.2165 dB (Taguchi -16.2387 dB). For the second case (with bidirectional nulls), the BBO’s first null value is -99.94 dB (Taguchi -79.51 dB), and the BBO’s second null value is -89.28 dB (Taguchi -77.82 dB), and the maximum SLL for the BBO’s optimized array is -13.62 dB (Taguchi -14.29 dB). Figure 6 shows the radiation pattern obtained by BBO compared to Taguchi method. 0 -10

Normalized AF (dB)

-20 -30 -40 -50 -60

Different examples are optimized using the BBO. The minimum and maximum allowable values for the variables (i.e., the weights and the inter-element arc distances) are set to 0.1 and 1, respectively. The design examples are performed for a specific FNBW, which corresponds to a uniformly-fed circular array with a uniform λ/2 elementspacing and the same number of elements. Moreover, the self-adaptive differential evolution (SADE) with competitive control-parameter setting technique (debr18.m) [22] is used to perform the optimization for the same design problems. For comparison purpose, these methods are compared to genetic algorithm [10] and the PSO [11]. 1) 8 elements CAA The BBO (population size=50, number of generations=300) and SADE codes are run for 10 independent times. Table V shows the best results obtained using BBO and the SADE algorithm. “Best results” are defined as the ones that give the smallest value of the fitness function. The values of the fitness function were 0.0674 and 0.0586 for the BBO and the SADE, respectively. Figure 7 shows the radiation patterns of the optimized array as compared to those obtained using GA [10] and PSO [11] results. TABLE V. THE OPTIMIZED PARAMETERS FOR 8-ELEMENTS NON-UNIFORM CAA USING BBO AND SADE.

-70

N=8

BBO with null Taguchi with null BBO without null Taguchi without null

-80 -90 -100

factor, i.e., its value at φ0. W1 and W2 are weighting factors; both are chosen here to be unity. Thus, the optimization problem is to search for the current amplitudes (In’s) and the arc distances between the elements (dn’s) that minimize the fitness function in equation (10).

0

10

20

30

40

50

60

70

80

φnu2=340

90

BBO

θ (Degrees)

Figure 6. Radiation pattern of 20 elements LAA optimized with respect to phases compared to Taguchi results from [9]. SADE

B. Circular antenna array (CAA) optimization Here, the objective is to design a CAA with deep nulls in the directions φnu1 and φnu2 which define the first null beamwidth (FNBW) while minimizing the side lobes levels. Thus, the following fitness function is used:

| |

| | ,|

[0.3406, 0.7682, 0.2988, 0.5756, 0.6627, 0.8805, 0.6337, 0.4214] ⇒∑=4.58 [0.7637, 0.6075, 0.1090, 1.0000, 0.8722, 0.5396, 0.7177, 0.4858] [0.3438, 0.6668, 0.2059, 0.7951, 0.6272, 0.8437, 0.8295, 0.3383] ⇒∑=4.6505 [0.8749, 0.2302, 0.4633, 0.9542, 1.0000, 0.6442, 0.9099, 0.1844]

|

/| |

[dm1, dm2, dm3, …, dmN] in λ’s [I1,I2,I3,…,IN]

| |

(10)

where the two angles {φnu1, φnu2} define the major lobe, i.e., the first null beam width (FNBW) = φnu2-φnu1=2φnu2. φms1 and φms2 are the angles where the maximum side lobe level is attained during the optimization process in regions (from 180 to φnu1) and (from φnu2 to 180), respectively. An increment of 1o is used in the optimization process. Thus, the function F2 minimizes the maximum SLL around the major lobe. Moreover, AFmax is the maximum value of the array

Figure 7. Radiation pattern for N=8 CAA using the BBO and SADE results as compared to the GA results from [10] and PSO results from [11].


It can be seen that both the BBO and SADE give somewhat better side lobe levels than GA and PSO. Specifically, the maximum SLLs obtained using the BBO, SADE, GA and PSO are -12.18 dB, -12.7 dB, -9.8 dB, -10.8 dB, respectively. It is worth mentioning that a uniform circular array with the same number of elements and λ/2 element-to-element spacing has a maximum side lobe level of -4.17 dB. 2) 12 elements CAA Similar to the previous example, Table VI shows the BBO (population size=150, number of generations=500) and SADE results for 12 elements, while Figure 8 shows a comparison between the array factor obtained using the different optimization methods. It is clear that the BBO results are as good as those obtained using the well-known SADE, GA and PSO techniques. This shows the effectiveness of the BBO in solving antenna array problems. It should be mentioned that for the N=8 case, the PSOoptimized CAA had a circumference of 4.4931λ; while that obtained using the GA had a circumference of 4.4094λ. Both the BBO and SADE-optimized CAAs, shown in Table V, have slightly larger circumference. Moreover, the circumference of the PSO-designed N=12 CAA was 7.1501λ. On the other hand, the GA-designed CAA had a circumference of 7.77λ for N=12.

3) 20 elements CAA It can be noticed that the optimized arrays always have an aperture (i. e., circumference) larger than that of a uniform array with half-wavelength spacing between the elements. Now, to make the comparison between a conventional uniform CAA and an optimized one more fair, one has to force the optimized array to have an aperture as close as possible to that of a uniform array. To accomplish this, the fitness function in equation (10) is modified as follows: | /| ∑ (11) where the desired circumference is that of a uniform CAA, or any other desired circumference. As an example, an N=20 CAA is optimized with the fitness function (11), with a desired circumference of 10λ. The obtained results are shown in Table VII, while Figure 9 shows the corresponding array factor. The conventional uniform CAA has a maximum SLL of -6.08 dB, while the BBO (population size=150, number of generations=500) and SADE optimized CAAs have maximum SLLs of -10.7 dB and -11.3 dB, respectively. Both optimized CAAs have exactly the same aperture as the conventional CAA. TABLE VII. THE OPTIMIZED PARAMETERS FOR 20 ELEMENTS NONUNIFORM CAA USING BBO AND SADE. N=20

TABLE VI. THE OPTIMIZED PARAMETERS FOR 12-ELEMENTS NON-UNIFORM CAA USING BBO AND SADE. N=12 φnu2=230

[dm1, dm2, dm3, …, dmN] in λ’s [I1,I2,I3,…,IN]

BBO

[0.4083, 0.6416, 0.7554, 0.7185, 0.6943, 0.3818, 0.3284, 0.8152, 0.9981, 0.3097, 0.7983, 0.3701] ⇒∑=7.2196

SADE

[0.6567, 0.3879, 0.6960, 0.4596, 0.5627, 0.9600, 0.4168, 0.5890, 0.5368, 0.6230, 0.6910, 1.0000 ] [0.422, 0.6798, 0.6380, 0.6954, 0.9017, 0.5223, 0.7686, 0.6235, 0.2582, 0.5151, 0.8151, 0.3343] ⇒∑=7.1793 [0.3617, 0.3740, 0.3498, 0.6514, 1.0000, 0.8604, 0.4864, 0.3960, 0.3696, 0.3390, 0.5058, 0.8387 ]

Figure 8. Radiation pattern for N=12 CAA using the BBO and SADE results as compared to the GA results from [10] and PSO results from [11].

BBO

SADE

[dm1, dm2, dm3, …, dmN] in λ’s [I1,I2,I3,…,IN] [0.3992 0.4899 0.7261 0.5212

0.9735 0.2235 0.3108 1.0000 0.2110 0.1000 0.3585 0.4811 0.2717 0.5740 1.0000 0.3988 0.2548 0.8606 0.5380 0.3077] ⇒∑=10

[0.5576 0.1000 0.5202 0.6996 [0.2188 0.9706 0.8742 0.4842

0.1000 0.9253 0.1000 0.3295 0.6851 0.6877 1.0000 ] 0.2808 0.7180 0.6226 0.5565 0.1964 0.8028 0.3062] ⇒∑=10

[0.7398 0.7220 0.6459 0.8326

0.4071 0.3937 0.7197 1.0000 0.9281 0.3150 0.8843 0.7853 0.5352 0.3133 0.9790 0.9843 0.7044 0.1857 0.3126 0.6337 ]

0.6877 1.0000 0.3850 0.1000 1.0000 0.8999 1.0000 0.4435 0.2389 0.3800 0.8207 0.2932 0.1364 0.4640 0.5064 0.2789 0.4280 0.6616

Figure 9. Radiation pattern for N=20 CAA using the BBO and SADE.


IV.

CONCLUSIONS

BBO is a new technique in electromagnetics optimization. It was applied on the optimization of linear antenna arrays and circular antenna arrays. Three cases of linear array design have been considered; amplitudes optimization, positions optimization and phases optimization. On the other hand, the positions and the excitations of the antenna elements in a circular array were optimized. BBO results have been compared to well-known optimization techniques (Taguchi, PSO, SADE, GA, and PSO) where it was found that the BBO is a very good algorithm and its results are as good as those obtained using other well-developed methods. REFERENCES [1] [2] [3]

[4]

[5]

[6]

[7]

[8]

[9]

C. A. Balanis, Antenna Theory:Analysis and Design, John Wiley & Sons, New York, 1997. G. Oliveri, and L. Poli, “Synthesis of monopulse sub-arrayed linear and planar array antennas with optimized sidelobes,” Progress In Electromagnetics Research, PIER 99, pp. 109-129, 2009. Perez Lopez, J. R. and J. Basterrechea, “Hybrid particle swarm-based algorithms and their application to linear array synthesis,” Progress In Electromagnetics Research, PIER 90, pp. 63-74, 2009. W. T. Li, X. W. Shi, and Y. Q. Hei, “An improved particle swarm optimization algorithm for pattern synthesis of phased arrays,” Progress In Electromagnetics Research, PIER 82, pp. 319-332, 2008. S. Zainud-Deen, E. Mady, K. Awadalla, and H. Harsher, “Controlled radiation pattern of circular antenna array,” IEEE Antennas and Propagation Symp., 2006, pp. 3399-3402. L. Pesik, D. Paul, C. Railton, G. Hilton and M. Beach, “FDTD technique for modeling eight element circular antenna array,” Electronics Letters, Vol. 42, No. 14, pp. 787-788, July 2006. W. Mahler, and F. Landstorfer, “Design and optimization of an antenna array for WiMAX base stations,” IEEE/ACES International Conference on Wireless Communications and Applied Computational Electromagnetics, 2005, pp. 1006 – 1009. M. Al-Aqil, “Synthesis of antenna arrays using the particle swarm optimization method, ” Master Thesis, Jordan University of Science and Technology, Jan. 2009. Nihad Dib, Sotiris Goudos, and Hani Muhsen, “Application of Taguchi's Optimization Method and Self-Adaptive Differential Evolution to the Synthesis of Linear Antenna Arrays,” Progress In Electromagnetics Research, Vol. 102, pp. 159-180, 2010.

[10] M. Panduro, A. Mendez, R. Dominguez, and G. Romero, “Design of non-uniform circular antenna arrays for side lobe reduction using the method of genetic algorithms,” Int. J. Electron. Commun. (AEU), Vol. 60, pp. 713-717, 2006. [11] M. Shihab, Y. Najjar, N. Dib, and M. Khodier, “Design of NonUniform Circular Antenna Arrays Using Particle Swarm Optimization,” Journal of Electrical Engineering, Vol. 59, No. 4, pp. 216-220, 2008. [12] D. Simon, “Biogeography-based optimization,” IEEE Trans. Evolut. Computat., Vol. 12, No. 6, pp. 702–713, December 2008. [13] D. Simon, M. Ergezer, D. Du, and R. Rarick, “Markov Models for biogeography-based optimization,” IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics, vol. 41, no. 1, pp. 299306, January 2011. [14] D. Simon, R. Rarick, M. Ergezer, and D. Du, “Analytical and numerical comparisons of biogeography-based optimization and genetic algorithms,” Information Sciences, vol. 181, no. 7, pp. 12241248, April 2011. [15] P. Roy, S. Ghoshal, and S. Thakur, “Biogeography based optimization for multi-constraint optimal power flow with emission and nonsmooth cost function,” Expert Systems with Applications, vol. 37, no. 12, pp. 8221-8228, December 2010. [16] A. Bhattacharya and P. Chattopadhyay, “Hybrid differential evolution with biogeography based optimization for solution of economic load dispatch,” IEEE Transactions on Power Systems, vol. 25, no. 4, pp. 1955-1964, November 2010. [17] U. Singh, H. Singla, and T. Kamal, “Design of Yagi-Uda antenna using biogeography based optimization,” IEEE Transactions on Antennas and Propagation, vol. 58, no. 10, pp. 3375-3379, October 2010. [18] U. Singh, H. Kumar, and T. S. Kamal, “Linear array synthesis using biogeography based optimization,” Progress In Electromagnetics Research M, Vol. 11, pp. 25-36, 2010. [19] M. Lohokare, S. Pattnaik, S. Devi, B. Panigrahi, K. Bakwad, and J. Joshi, “Modified BBO and calculation of resonant frequency of circular microstrip antenna,” World Congress on Nature & Biologically Inspired Computing, pp. 487-492, Coimbatore, India, December 2009. [20] M. Lohokare, S. Pattnaik, S. Devi, K. Bakwad, and J. Joshi, “Parameter calculation of rectangular microstrip antenna using biogeography-based optimization,” Applied Electromagnetics Conference, Kolkata, India, 2009. [21] Biogeography-based optimization (BBO). [Online]. Available: http://embeddedlab.csuohio.edu/BBO/. [22] Tvrd´ık, J., V. Pavliska, and H. Habiballa, “Stochastic self-adaptive algorithms for global optimization — MATLAB and C++ library,” http://albert.osu.cz/oukip/optimization/.