Comparison of evolutionary algorithms for LPDA antenna optimizationn

Comparison of evolutionary algorithms for LPDA antenna optimizationn

Pavlos I. Lazaridis ⁽¹⁾, Emmanouil N. Tziris ⁽²⁾, Zaharias D. Zaharis, ⁽³⁾, John P. Cosmas ⁽²⁾, Violeta Holmes ⁽¹⁾, and Ian A. Glover ⁽¹⁾

(1) University of Huddersfield, Queensgate, Huddersfield, HD1 3DH, UK.

(2) Brunel University, London, UB8 3PH, UK.

(3) Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece.

Abstract — In this paper a different approach to log-periodic antenna design is carried out where some of the most powerful evolutionary algorithms (EAs) are applied and compared to optimise the design of log-periodic dipole arrays (LPDA). The effort is to achieve optimal antenna design with respect to maximum gain, side lobe levels (SLL) and VSWR. The parameters of the LPDA which are optimized are the dipole lengths, the spacing between the dipoles and dipole wire diameters. The evolutionary algorithms which were compared are the Differential Evolution (DE), Particle Swarm (PSO), Taguchi, Invasive Weed (IWO) and Adaptive Invasive Weed (ADIWO).

Keywords—

I. INTRODUCTION

Broadband log-periodic antenna optimization is a very challenging problem for antenna design. However, up to now, the universal method for log-periodic antenna design is Carrel’s method dating from the 1960s. This paper compares five antenna design optimization algorithms (Differential Evolution, Particle Swarm, Taguchi, Invasive Weed, Adaptive Invasive Weed) as solutions to the broadband antenna design problem. The algorithms compared are evolutionary algorithms which use mechanisms inspired by biological evolution, such as reproduction, mutation, recombination, and selection. The focus of the comparison is given to the algorithm with the best results, nevertheless, it becomes obvious that the algorithm which produces the best fitness values (Invasive Weed Optimization) requires very substantial computational resources due to its random search nature.

Log‐periodic antennas (LPDA: Log‐Periodic Dipole Arrays) are frequently preferred for broadband applications due to their very good directivity characteristics and flat gain curve. The purpose of this study is, in the first place, the accurate modelling of the log‐periodic type of antennas, the detailed calculation of the important characteristics of the antennas under test (gain, vswr, front‐to‐back ratio) and the confrontation with accurate measurements results.

In the second place, various evolutionary optimization algorithms are used, and notably the relatively new (2006) Invasive Weed Optimization (IWO) algorithm of Mehrabian

& Lucas, for optimizing the performance of a log‐periodic antenna with respect to maximum gain, Side‐Lobe‐Levels (SLL) and matching to 50 Ohms, VSWR. The multi‐objective optimization algorithm is minimising a so‐called fitness function including all the above requirements and leads to the

dipole wire diameters. In some optimization cases, a constant dipole wire radius is used in order to simplify the construction of the antenna.

In artificial intelligence, an evolutionary algorithm (EA) is a generic population-based metaheuristic optimisation algorithm. An EA uses mechanisms inspired by biological evolution, such as reproduction, mutation, recombination, and selection. Candidate solutions to the optimisation problem play the role of individuals in a population, and the fitness function determines the environment within which the solutions "live".

Evolution of the population then takes place after the repeated application of the above operators. Artificial evolution (AE) describes a process involving individual evolutionary algorithms. EAs are individual components that participate in an EA.

Antenna arrays play an important role in detecting and processing signals arriving from different directions. The goal in antenna array geometry synthesis is to determine the physical layout of the array that produces a radiation pattern that is closest to the desired pattern. The shape of the desired pattern can vary widely depending on the application.

Before starting to use an EA, setting up the problem is required, which means making sure than an EA is the optimal solution to the problem. Secondly, the parameters that need optimisation must be decided. The parameter which needs to be maximized is the fitness of the population and it is used to generate the next population after being evaluated.

Some basic optimisation concepts for electromagnetic applications will be evaluated for this project and these are the following: 1. Differential Evolution (DE), 2. Particle Swarm Optimisation (PSO), 3. Invasive Weed Optimisation (IWO), 4.

Taguchi's Optimisation Method, 5. Adaptive IWO (ADIWO).

II. C^LASSICAL D^ESIGN A^LGORITHMFOR LPDA^S

The most complete and practical design procedure for a Log-Periodic dipole array (LPDA) is that by Carrel, [13-14].

The configuration of the log-periodic antenna is described in terms of the design parameters: τ, α, and σ, related by:

1 1

tan 4

 



 



 

 

 

(0.1)

Once two of the design parameters are specified, the other can be found. The proportionality factors that relate lengths, diameters, and spacings between dipoles are:

1 1,

n n

n n

L d

L d

  ^  ^

2

n n

s

  L

(0.2) where, Ln and dn are respectively the length and the diameter of the n-th dipole, and sn is the spacing between the n- th and (n+1)-th dipoles. However, for the majority of practical logperiodic array designs, wire dipoles of equal diameters dn are used, or for some more advanced designs, 3 groups of equal diameter dipoles are used to cover the whole frequency range.

In order to reduce some anomalous resonances of the antenna, a short-circuited stub is usually placed at the end of the feeding line at a distance behind the longest dipole. Directivity in dB contour curves as a function of τ for various values of σ are shown in [14], as they have been corrected in [15]. A set of design equations and graphs are used, but in practice it is much easier to use a software incorporating all the necessary design procedure, such as LPCAD, [16-17]. LPCAD also produces a file that can be used for the detailed simulation of the antenna using the NEC software, which employs the Method of Moments for wire antennas, [18-20].

III. EVOLUTIONARY ALGORITHMS

A. Differential Evolution

The general problem that an optimisation algorithm is concerned with, is to determine the vector variable x so as to optimise:

f ( x ) ;x ={x

₁

, x

₂

, …, x

_D

}

f x x

 

; { , ,..., }x x1 2 x_D

(1.1)

(2.1)

Where, D is the dimensionality of the function. The variable domains are defined by their lower and upper bounds:

, , , ; {1,..., }

j low j upp

x x j D .

The population of the original DE algorithm contains NP D-dimensional vectors:

x

_{i ,G}

= { ^x

i ,1,G

, x

_{i ,2,G}

, … , x

_{i , D, G}

} ^{, i=1,2 , … , NP}

, { ,1, , ,2, ,..., , , }, 1, 2,...,

i G i G i G i D G

x  x x x i NP

(1.2)

(2.2)

Where, G is the generation

During one generation for each vector, DE employs mutation and crossover operations to produce a trial vector:

u

_{i ,G}

= { ^u

i ,1,G

, u

_{i ,2,G}

, … , u

_{i , D, G}

} ^{, i=1,2 , … , NP}

, { ,1, , ,2, ,..., , , }, 1, 2,...,

i G i G i G i D G

u  u u u i NP (1.3)

Then, a selection operation is used to choose vectors for the next generation (G+1). The initial population is selected uniform randomly between the lower



xj low,



and upper



xj upp,



bounds defined for each variable

x

^j

. These bounds are specified by the user according to the nature of the problem. After initialization, DE performs several vector transforms (with the above mentioned operations), in a process called evolution.

B. Particle Swarm

In PSO terminology, every individual in the swarm is called “particle” or “agent”. The number S of the particles that compose the swarm is called “population size”. A population size between 10 and 50 is optimal for many problems. All the particles act in the same way like bees do, they move in the search space and update their velocity according to the best positions already found by themselves and by their neighbors, trying to find an even better position. Each particle is treated as point in an N-dimensional space. The position of the i-th particle

(i=1 , …, S)

^{(i1,.... )S} is represented as

1 2

( , ,..., )

i i i iN

x  x x x _{, where} x n_in( 1,..., )N are the position coordinates. Each coordinate

x

_¿ may be limited in the respective (n-th) dimension between an upper boundary

U

_n and a lower boundary

L

_n , so that ( 1,..., )

n n n

L x U n N . The difference RⁿUⁿLⁿ between the two boundaries is called “dynamic range” of the n-th dimension. The performance of each particle is measured according to a predefined mathematical function F called

“fitness function”, which is related to the problem to be solved. The value of the fitness function depends on the position coordinates, i.e, FF x( )_i F x x( ,_i¹ _i²,...,x_iN). Actually, the particle position is considered to be improved as the value of the fitness function is increased/or decreased (maximization or minimization problem). The best previous position (best position) of the i-th particle is recorded and represented as p_i(p p_i¹, _i²,...,p_iN).

The change of

x

_i is:

i i

x u 

   (2.1)

Δt is the time interval, _i



v v_i1, _i2, , v_iN



is the velocity of the i-th particle, and in



n ^{1, ,}N



are the velocity coordinates.

Calculation of velocity:

Considering that  t 1, the position change becomes

i i

x 

  . Thus, the new position of the i-th particle after a time step is given by:



¹

   

¹



i i i

x t x t  t (2.2)

Particle swarms have been studied in two types of neighborhood, called “gbest” and “lbest”.

In the gbest neighborhood, every particle is attracted to the best position found by any particle of the swarm which is called “gbest position”.

In the lbest neighborhood, each (i-th) individual is affected by the best performance of its Ki immediate neighbors which is called “lbest position”.

Equation of Velocity for gbest model:

u

_i

(t +1)= w∗u

_i

(t )+c

₁

rand (t )∗ [ ^p

i

( t )−x

_i

( t ) ] ^+c

2

rand (t )∗[g (t )−x

_i

(t )]

     

         

1 2

1 * *

*

i i

i i i

u t w u t c rand t

p t x t c rand t g t x t

  

  

   

    (2.3)

Where, w = inertia weight (0.0 - 0.1), c1 and c2 are cognitive coefficient, and social coefficient respectively, and rand(t) is a function that generates random numbers from a uniform distribution between 0.0 and 1.0.

Equation of Velocity for lbest model:

u

_i

(t +1)= w∗u

_i

(t )+c

₁

rand (t )∗ [ ^p

i

( t )−x

_i

( t ) ] ^+c

2

rand (t )∗ [ ^l

i

( t )−x

_i

( t ) ]

     

         

1 2

1 * *

*

i i

i i i i

u t w u t c rand t

p t x t c rand t l t x t

  

  

   

    (2.4)

C. Taguchi

The development of Taguchi’s method is based on orthogonal arrays (OAs) that have a profound background in statistics.

Orthogonal arrays were introduced in the 1940s and have been widely used in designing experiments. They provide an efficient and systematic way to determine control parameters so that the optimal result can be found with only a few experimental runs. This section briefly reviews the fundamental concepts of OAs, such as their definition, important properties, and constructions.

The procedure of the Taguchi algorithm consists of five stages. These stages are the following:

1 Problem Initialization: The optimisation procedure starts with the problem initialization, which includes the selection of a proper OA and the design of a suitable fitness function. The selection of an OA (E, P, L, t) mainly depends on the number of optimisation parameters. Where E is the number of Experiments, P is the number of Parameters, L is the number of Levels, and t is the strength.

2 Input Parameters Designation: The input parameters need to be selected to conduct the experiments. When the OA is used, the corresponding numerical values for the levels of each input parameter should be determined. For each i_th iteration and each p_th parameter, the level difference

〖LD〗_pi is calculated by the following formula:

1

1, 1 , ,

i

pi p

LD rr LD^ p  P

LD

_pi

=r r

ⁱ⁻¹

LD

_p1

, p=1 ,… , P

(3.1)

Where,

 

 

1 , 1, ,

1

p p

p

max min A

LD p P

L

   



LD

_p1

= ( ^max

p

− min

_p

A )

( L+ 1) , p=1, … , P

^(3.2)

is the initial level difference and rr is the reduced rate. Also,

max

_p ^maxp _and ^minp

min

_p are respectively the upper and the lower bound of the p_th parameter.

3 Experiments Conduction and Response Table Building: The fitness function

fit

_ei ^fitei for each experiment (e) can be calculated and the fitness value is converted to the signal-to-noise (S/N) ratio (η) in Taguchi’s method using the following formula:

 

20log log Fitness

  

η=−20 log ⁡ log (Fitness)

(3.3) A small fitness value results in a large S/N ratio. After conducting all experiments in the first iteration, the fitness values and corresponding S/N ratios are obtained and listed.

The average fitness values in dB are then extracted for each parameter and each level to build the response table by applying the expression:

η ´

_lpi

= ( E ^L )

_{e ,OA}

^∑

_{(e , p)=l}

^η

^ei

^{, p=1 , … , P ∧l=1 ,… , L( 4.4)}

 

, , 

, 1 , , & 1 , ,

lpi ei

e OA e p l

LE p P l L

 







   

(3.4) 4 Optimal Level Values Identification: Finding the largest S/N ratio in each column of response table can identify the optimal level for that parameter. When the optimal levels are identified, a confirmation experiment is performed using the combination of the optimal levels identified in the response table. This confirmation test is not repetitious because the OA- based experiment is a fractional factorial experiment. The fitness value obtained from the optimal combination is regarded as the fitness value of the current iteration.

5 Optimisation Range Reduction: If the results of the current iteration do not meet the termination criteria, which are discussed in the following subsection, the process is repeated in the next iteration, otherwise, the procedure is terminated.

D. Invasive Weed & Adaptive Invasive Weed

The Invasive Weed Optimisation (IWO) is an optimisation algorithm that is proposed for Electromagnetic applications.

The IWO is a numerical optimisation algorithm inspired from weed colonization and it was first introduced by Mehrabian and Lucas in 2006 [5]. This optimiser can in certain instances outperform other algorithms like the particle swarm optimisation (PSO) and is able to handle new electromagnetic

optimisation problems. The colonization behaviour of weeds follows the steps bellow:

1 First of all, there is a set of variables that are in need of optimising. Once these variables are selected the minimum and maximum values for these variables are set.

2 Once the variables are set, the seeds are randomly positioned in an N-dimensional problem space. Each seed position is considered to be a solution. These positions will contain a value for each variable previously set. That means N values for N variables.

3 Subsequently, each seed will grow into a plant. The fitness function returns a fitness value that represents how good the solution will be for each individual seed. Once each seed is assigned a fitness, it is called a plant.

4 In order for a plant to produce new seeds, and how many, it must meet certain fitness values. Based on the fitness value rank every plant has, it produces a number of seeds between a minimum and maximum possible number. The closer to the set variables a plant is, the more seeds it is allowed to produce.

5 The seeds created in the previous step are spread over the search space. Every new seed is distributed using random numbers for the values of its location but with the numbers whose average value equal to the parent plants location as well as varying standard deviations. The standard deviation (SD) at the present time step can be expressed by:

σ = ( ^I

MAX

−I )

ⁿ

( ^I

MAX

)

ⁿ

( σ

_¿

−σ

_fi

) + σ

_fi

 

^MAX n ⁿ



in fi



fi

MAX

I I

 I    

  

(4.1)

Where, I is the number of iterations and I_MAX the maximum number of iterations.

σ

_¿ and

σ

_fi are defined as the initial and final standard deviations respectively and n is the nonlinear modulation index.

6 Once all seeds have found a position over the search area they become plants and take fitness values and rank along with their parents. In order to keep the maximum number of plants in the colony, plants that are not fit are discarded.

7 The plants that survive produce in turn new seeds and the process is repeated until the maximum number of iterations is reached or the desired fitness achieved.

In the Adaptive IWO, the standard deviation σ of the dispersion of the seeds produced by a weed is a linear function of the fitness value f of this weed. Considering that the goal is the minimization of the fitness function, σ can be estimated according to the following expression:

σ = σ

_MAX

− σ

_min

f

MAX

− f

min

f + σ

_min

f

_MAX

−σ

_MAX

f

_min

f

MAX

− f

min

MAX min min MAX MAX min

MAX min MAX min

f f

f f f f f

   

  

 

  (4.2)

Where,



^MAX σ_{MAX and}



_min σ_min are the standard deviation limits defined in the same way as in the original IWO algorithm, while

f

^MAX f_{MAX and} f_min

f

min represent respectively the maximum and minimum fitness values at a certain iteration. The ADIWO algorithm has the same structure as the original IWO algorithm. The only difference lies in the calculation of σ which is performed by using (4.2). It is easy to realize that the best weed (

f  f

^min₎

f =f

_min disperses its seeds with the minimum σ (

σ =σ

_min

  

min), while the worst weed (

f =f

_MAX

f  f

MAX) disperses its seeds with the maximum σ (

  

MAX

σ =σ

_MAX ). Therefore, the weeds have different behaviour depending on their fitness values. As the fitness value gets closer to

f

_min

f

min, the exploration ability of the weed is reduced and thus the weed can only fine- tune its near-optimal position. On the contrary, as f _gets closer to

f

^MAX

f

_MAX , the exploration ability of the weed increases and thus the weed is capable of exploring the search space to find better positions. In this way, the exploration ability of the weed colony is maintained until the end of the optimisation process. Moreover, the adaptive seed dispersion makes the ADIWO converge faster than the original IWO although less accurate.

IV. R^ESULTS

In the effort to compare the results of each optimisation algorithm, all of the algorithms were applied to a LPDA antenna with 10 dipoles and a rear shorting stub for the frequency range of 450MHz to 800MHz with respect to maximum gain, Side‐Lobe‐Levels (SLL) and matching to 50 Ohms, Standing Wave Ration (VSWR). Therefore, the fitness function has to be a linear combination of the above three parameters, which are calculated by applying the 4NEC2 software, [18], for every weed or seed and for all frequencies which are defined in the above range by steps of 10MHz. The optimised parameters of the antenna therefore, are the dipole’s lengths and diameters as well as the distances between each dipole and the characteristic impedance of the transmission line that feeds the dipoles {1}. Each evolutionary algorithm has been coded in Matlab software and was run to a total of 44000 fitness evaluations. At the end of the execution of every

algorithm the best fitness and the geometry of the optimised antenna were produced and was then extracted to a *.NEC file. The 4NEC2 software was used to run the NEC file produced by Matlab, to derive the VSWR, Gain, F\R Ratio and the Convergence Diagram figures.

On “Figure 1” the comparison of the SWR between all of the evolutionary algorithms which were used to generate the geometries of five different LPDAs shows that the results are very satisfying for all of the algorithms, since the SWR values

are all below 1.8. Nonetheless, as it is expected some algorithms performed better than others, with PSO being the leading algorithm with the lowest values across the frequency range while the Adaptive IWO had the poorest results, being the only method which exceeded the value of 1.5. Differential Evolution, Taguchi and Invasive Weed methods show a standing wave ratio which swings around the 1.25 value which translates to a minimum return loss of 19.1dB and a maximum value of reflection coefficient of 0.111.

Comparing the forward gain of the LPDAs generated by each algorithm gives a better view of the performance of each algorithm than the SWR figure where all the algorithms had a similar average. In the above figure, it is very clear that the best performance comes from IWO and Differential Evolution. IWO is an even better performer since its forward

gain shows a steady value (just below 8dB) and very high compared to the rest of the algorithms across all of the frequency range. Differential Evolution performs similarly but its values are swinging across the frequency range which is not preferred over the steady line that IWO shows. On the other hand, the Taguchi method generated the poorest results with very different and low gain values across the frequency range.

Similarly to the forward gain, the front to rear ratio figure confirms the previous conclusions that the best results are produced by the LPDAs generated from the IWO and Differential Evolution algorithms with F/R ratio values much higher compared to the rest of the algorithms. The PSO method shows average results while the poorest results are again shown by Taguchi (lowest F/R ratio across all the frequency range) and Adaptive IWO (very poor initial F/R ratio values).

At this point, it should be noted, that the performance of the algorithms is a matter of which algorithm can produce the lowest possible fitness value, which as stated before is a linear combination of the VSWR, the forward gain and the F/R Ratio. This means that the algorithm to produce the lowest fitness value is expected to derive the LPDA with the best performance. The figure below depicts the convergence diagram (fitness evaluations/best fitness) of all of the algorithms for 44000 fitness evaluations except for the Taguchi method which automatically selects the amount of fitness evaluations, depending on the fitness function. The amount of the fitness evaluations which was chosen is enough to show which algorithm produces the best fitness value, because after this point, all the algorithms are unable to lower Figure 1 - Standing Wave Ratio

Figure 2 - Forward Gain

Figure 3 – F/R Ratio

much further their fitness value. This is obvious, if the last 10,000 fitness evaluations are observed.

As it is expected, on the figure, the algorithm which produced the lowest fitness value is the IWO algorithm, which also had the best performance shown on the previous figures, while Differential Evolution produces the second best fitness value. Nonetheless, it is very remarkable that PSO manages to have a very fast drop rate on the first 200 fitness evaluations compared to the rest. This means, that in much less time the PSO algorithm will generate the LPDA with the best SWR, forward gain and F/R ratio. Similarly, the adaptive IWO has a better initial drop rate compared to IWO and Differential Evolution, yet not as remarkable as the PSO.

V. CONCLUSIONS

Five different evolutionary algorithms were employed to design Log-Periodic Dipole Arrays to compare them and have the opportunity for the first time to see which one shows the best performance when it comes to LDPA design. All of the algorithms generated LPDA geometries with very satisfying properties (SWR, Forward Gain and F/R Ratio), some algorithms however, were able to produce better properties in less time (PSO and Adaptive IWO), while Invasive Weed and Differential Evolution show the best results, yet the IWO algorithm exhibits the best performance.

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