Linear Array Synthesis for Wireless Power Transmission Based on Brain Storm Optimization Algorithm

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Introduction
Wireless power transmission (WPT) is a technology that transmits power wirelessly [1,2]. With the deepening of research, this technology has not only been applied to hightech fields such as solar power satellites but also reflected on various fields of daily life, charging mobile phones, computers, or electric cars [3,4]. In recent years, people are studying this technology from more and more aspects, hoping to make a breakthrough in different directions. For a WPT system, its main components include two parts: a transmitting antenna for transmitting energy towards the receiving region and a receiving antenna for converting and receiving energy. erefore, people's research directions can be broadly divided into these two categories. For either of the two parts, the purpose is to achieve higher transmitting/ receiving efficiency.
In order to obtain higher power efficiency, people began to pay more attention to the research of transmitting antenna and proposed a series of methods, including the edge tapering method and stochastic optimization method [5,6]. In 2013, Oliveri et al. took BCE (beam collection efficiency, the ratio of the energy received to the total energy sent by the transmitting antenna) as the objective function and studied the two-dimensional planar antenna array by solving the generalized eigenvalues [7]. en, more and more people have applied optimization algorithms, such as particle swarm optimization (PSO) algorithm and genetic algorithm (GA), to this field [8]. e abovementioned methods all have some disadvantages, such as complex implementation, severe parameter influence, and low precision. erefore, many scholars devote themselves to the study of algorithms to get better performance.
Inspired by the creative problem-solving process of human beings, Professor Shi proposed the brain storm optimization (BSO) algorithm [9] in 2011. is algorithm uses the clustering idea to search the local optimal and obtains the global optimal by comparing the local optimal. e idea of variation increases the multiform of the algorithm, avoids the algorithm falling into the local optimal, and searches for the optimal solution in the process of polymerization and dispersion, which is novel and suitable for solving the problem of multipeak and high-dimensional functions. In [10], Rahmat-Samii used BSO to optimize electromagnetic devices' characteristics. e practicability and effectiveness of BSO algorithm in electromagnetic field are proved. rough reviewing the literature, we find that a lot of researches have been done on linear antenna arrays [11][12][13][14]. But to the best of the author's knowledge, no one has applied BSO to BCE optimizations with multiple constraints on linear antenna arrays, and the characteristics of BSO algorithm are very suitable for application in this field. erefore, the purpose of this paper is to study the optimization of BCE with the BSO algorithm based on multiconstraint conditions.
In this paper, with the constraints of array number, aperture, and minimum array spacing, BCE is optimized by using uniform excitation nonequal spaced linear array and BSO algorithm. e outline of the paper is as follows. e optimized problem is mathematically formulated in Section 2. A brief introduction to the BSO algorithm is given in Section 3. Numerical simulation and analysis are described in Section 4. Finally, some conclusions are drawn in Section 5. Figure 1 shows the geometry of the transmitting part of the linear antenna array. As we can see from Figure 1, the transmitting antenna is composed of 2N + 1 antenna units (the analysis of 2N antenna units is the same as that of 2N + 1); its array factor can be expressed as follows:

Materials and Methods
where the d n and I n (n � −N, . . ., N) represent the element coordinate and current; let d N � L and d −N � −L, thus the aperture size is always 2L; u � cos (θ), θ represents the steering angle from the broadside of the array, 0 ≤ θ ≤ π; k � 2π/λ being the wavenumber, λ represents the wavelength, respectively. In this paper, all the element currents are supposed to be uniform, that is, assume I n � 1, for all n (if transmitting antenna is composed of 2N elements, I 0 does not exist), and the corresponding power pattern can be formulated as (2) e definition mentioned above of BCE is the ratio of the energy received by the receiving antenna to the total energy sent by the transmitting antenna. erefore, BCE can be expressed as where ψ and Ω are the angular region identifying the radiated area and the whole visible range, respectively. Taking BCE as the objective function, the optimization process can be expressed as searching the position vector of antenna elements d � (d 0 , d 1 , . . . , d N ) T within a certain aperture size to achieve the maximum BCE, and where l 0 represents the distance from the first antenna element to the Y axis. e objective function f(d) equals BCE, and the negative sign is used to make it a minimization problem. It is easy to prove that by using the method in [15], all the optimal variables can meet the constraints.

Brain Storm Optimization Algorithm
As mentioned above, BSO uses the clustering idea to search the local optimal and obtains the global optimal by comparing the local optimal. Its pseudocode is given in Figure 2 and BSO algorithm can be described as follows: (1) Population initialization: in this step, a population X is randomly generated. e population's size is N and the problem dimension is D. So, this population can be described as For every x i , it can be randomly generated as where (x min , x max ) are the specified minimum and maximum values for each dimension. e random number function rand returns a number between 0 and 1. (2) Cluster: BSO uses a k-mean clustering method to divide the N ideas into M clusters. (3) Rank: before this step, evaluate each idea with the objective function. Rank each idea according to the evaluation results, and then select the best idea in each group as the cluster center. (4) Disrupting cluster center: to avoid deception and increase population diversity, we occasionally replace a cluster center with a randomly generated idea. e disrupting operator of BSO is given as lines 6-8 in Figure 2. (5) Update: the first step is to determine whether a new idea is based on one cluster or two clusters. is step depends on the parameter p_one. e updating operator of BSO is given as lines 9 to 26 in Figure 2. If the new idea is based on one cluster, it can be described as follows: where x d is the selected idea, N(μ, σ) is the Gaussian random value with mean μ and variance σ, and ξ is a coefficient that weights the contribution of the Gaussian random value and can be described as follows: Figure 1: Geometry of the nonuniformly linear symmetric array. 2 International Journal of Antennas and Propagation where logsig () is a logarithmic sigmoid transfer function, G is the maximum number of iterations, g is the current iteration number, k is for changing logsig () function's slope, and rand () is a random value within (0, 1). If the new idea is based on two clusters, the idea x d is calculated as follows: where R is a random value within (0, 1), while x d1 and x d2 are the two ideas from clusters j 1 and j 2 , respectively. en, use (7) for the update operation.

Results and Discussion
e purpose of this section is two-fold: on the one hand, to test the effectiveness of BSO according to computational efficiency and flexibility when dealing with different aperture sizes and different receiving areas. On the other hand, to compare the results obtained through the BSO method to that through the GA method. Towards this end, a set of simulations have been carried out by assuming. For the BSO algorithm, the parameters we selected are as follows: p_replace � 0.2, p_one � 0.8, p_one_center � 0.4, and p_two_center � 0.5, and it will be used with five clusters by utilizing the k-mean clustering algorithm. e crossover and mutation probabilities for GA are set to P c � 0.7 and P m � 0.01, respectively. Both algorithms will have the same population of N � 30; also, the maximum number of iterations is set to 500. Both algorithms are executed 20 times. Considering the efficiency of the algorithm, we take the minimum adjacent element spacing d c as the fixed population starting position for each run. e first set of experiments was conducted in the case of 2N � 10 elements. In this set of experiments, the parameters involved in (4) are set as d c � 0.4λ and l 0 � 0.5d c . In order to International Journal of Antennas and Propagation make the spacing of all adjacent antenna elements meet d c , the aperture size is set as 2L � 4.5λ, that is, the average element spacing is 0.5λ, and we changed the size of the receiving region ψ by changing u 0 . In this case, the BCE can be formulated as follows: Using the BSO algorithm to solve the optimization problem, the corresponding optimal element distribution is summarized in Table 1 (all the elements are normalized to 0.4λ, that is, 5.625 * 0.4λ � 2.25λ � L). As we can see from Table 1, all antenna elements are within the aperture range, and the spacing of all adjacent antenna elements is less than d c . Table 2 shows the comparison between GA and BSO when u 0 is different, including the comparison between the maximum value and the average value. It is not difficult to find that both the maximum value and the average value of BSO are better than those of GA. With the increase of u 0 , the average characteristic of BSO is more and more prominent than GA. Moreover, the corresponding power pattern is shown in Figure 3. No matter how many antenna elements there are, the power pattern will always exhibit symmetry property.
Additionally, the best and the average convergence rate curves of the fitness function are plotted in Figure 4. It is obvious that although the convergence rate of BSO is slower than that of GA, the BCE optimized by BSO is higher than that optimized by GA. e other set of experiments provided the case of u 0 � 0.2, 2N � 10, 20, and 2N + 1 � 15 elements; the aperture size is constrained by 2L � 4.5λ, 9.5λ, and 7λ, respectively. Also, we changed the size of the receiving region ψ by changing u 0 . In this case, the BCE also can be formulated the same as (10). e conditions are kept the same as the first example, for 2N � 10, 20 elements, l 0 is chosen as l 0 � 0.5d c ,   whereas, for 2N + 1 � 15 elements, l 0 is chosen as l 0 � d c . In the same way, the corresponding optimal element distribution is summarized in Table 3 (all the elements are normalized to 0.4λ). e same result can be concluded.
According to the number of elements, the comparison between BSO and GA is shown in Table 4. Similarly, the comparisons between the maximum value and the average value are included. Moreover, the optimum power pattern and the convergence rate curves of the fitness function are plotted in Figures 5 and 6, respectively, and similar conclusions can be drawn. e abovementioned numerical results demonstrate that the BSO runs well with great efficiency and considerable stability.
All the experiments were simulated on MATLAB 2018a, and the average time of each simulation was about 30 s. e configuration of the computer is Intel Core i7-7700 at 3.6 GHz with 8 GB RAM.

Conclusion
Based on BSO algorithm and taking BCE as objective function, this paper studies the optimization problem under different aperture, different number of antenna elements, and different antenna angles. All experiments show that BSO can better deal with the maximum objective function problem. In this paper, we use BSO algorithm to improve BCE and prove that BSO algorithm is more practical and reliable than GA algorithm. However, this paper also has some shortcomings, such as the algorithm used, which is relatively primitive, and only the one-dimensional antenna array is studied. erefore, in the future, we will study the application of other improved algorithms in this field and try to extend this synthesis technology to more practical antenna structures, such as planar arrays of practical antenna elements and planar antenna power pattern synthesis.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper. International Journal of Antennas and Propagation 7