Grasshopper optimization algorithm

GOA is a kind of natural heuristic algorithm, whose design inspiration comes from the social behavior of grasshopper in nature. The grasshopper swarm moves slowly and slowly in its infancy but has a wide range of activities in its adulthood. These two characteristics make two trends in the search process of the natural heuristic algorithm: exploration and exploitation, which, as well as target search, are naturally accomplished by grasshoppers. In order to solve the optimization problem for avoiding grasshoppers reaching comfort zone quickly and the population not converge to a specific point, the mathematical model for simulating grasshopper swarm behavior is revised as follows [Reference Yu, Jiang, Hui and He22, Reference Saremi, Mirjalili and Lewis23]:

(1)$$X_i^d = c\left({\sum\limits_{\scriptstyle j = 1 \hfill \atop \scriptstyle j\ne i \hfill}^N {c\displaystyle{{ub_d-lb_d} \over 2}s( {\vert {x_j^d -x_i^d } \vert } ) \displaystyle{{x_j-x_i} \over {d_{ij}}}} } \right) + T_d, \;$$

(2)$$s( r ) = fe^{{{-r} \over l}}-e^{{-}r}, \;$$

(3)$$c = c_{max}-\displaystyle{{l( {c_{max}-c_{min}} ) } \over L}, \;$$

where $X_i^d$is defined as the location of i-th grasshoppers in the d-th dimension solution space, ub _d, lb _dare the upper and lower bound of the d dimension solution space; d _ij is the distance between i-th grasshopper and j-th grasshopper, i, j = 1, 2, …, N (N is the population size); $s( r)$ is the force function of social activities, f, l are the attraction intensity and the attraction length between grasshoppers; T _d is the target value of the d-th dimension (best solution found so far). c is used to reduce the decline coefficients of the comfort zone, exclusion zone, and attraction zone, c _max, c _min are the maximum and minimum values of c, l is the current iteration times and L is the maximum iteration times respectively.

Equation (1) cleverly simulates the interaction between grasshoppers in a swarm. The next position of grasshoppers is determined by their current position, target position, and the position of all other grasshoppers, which is different from PSO whose next position is based on the current position, personal best, and global best, GOA requires all search agents to participate in defining the next location of each search agent. The first component of the equation takes into account the position of grasshoppers relative to other grasshoppers, the second component simulates the trend of grasshoppers transferring to food sources, the parameter c simulates the deceleration of grasshoppers approaching food sources and slowing down to eat.

The mathematical model of the GOA requires grasshoppers to move towards the target gradually during the iteration process, avoiding convergence to the target too quickly, so as to fall into local optimum. GOA saves the most promising target in the search space in each iteration and requires grasshoppers to move towards it gradually. This is to find a better and more accurate target as the best approximation of the real global optimum in the search space. Like other evolutionary algorithms, GOA uses fitness function to guide grasshoppers to search their optimal location in d-dimensional space in order to meet the requirements of the objective function. At each stage of the algorithm, the position vector corresponding to the optimal fitness value is taken as the global optimal position vector, and this information is transmitted to other grasshoppers around so that grasshoppers can adjust their steps and position vectors accordingly until they reach the target position of food. The main optimization process of GOA is shown in Fig. 3. The advantages of applying GOA to broadband antenna loading and matching network optimization have been verified in paper [Reference Wang, Liu, Wu and Xie25, Reference Wang, Liu, Wu, Li and Xie26], and this paper will not be repeated.

Fig. 3.

Flow chart of the GOA.

Algorithm design

For wire antenna, the solution object of moment method is generally electric field integral equation, When the length of the loading area is much smaller than that of the antenna, Dirac function can be used to replace the distribution of loading impedance in the loading area and its neighborhood. Taking the feed point at the root of the antenna as the coordinate origin and the antenna axis as the z-axis to establish the column coordinate system, the load impedance distribution of the antenna can be expressed as follows:

(4)$$Z( z ) = Z_i\delta ( {z-z_i} ) , \;$$

where Z _i is the impedance of the lumped element at the loading position, z _i is the position of the loading element.

In the simulation calculation, considering that the current on the surface of the long straight wire antenna is axisymmetric with respect to the z-axis, the surface current can be equivalent to the line current on the antenna axis, that is, $I( {z^{\rm^{\prime}}} ) {\rm d}z$ is used instead of $J_s( {r^{\rm^{\prime}}} ) {\rm d}A^{\rm ^{\prime}}$. The antenna body will be divided into several sections, and the current on each section can be expressed by a polynomial. The current function is described as follows:

(5)$$I_i( {z^{\rm^{\prime}}} ) = \sum\limits_{k = 1}^{N_i} {I_{i, k}\left\vert {\displaystyle{{z^{\rm^{\prime}}} \over {z_{i + 1}}}} \right\vert } ^{k-1}, \;$$

where $z_i \le z^{\rm ^{\prime}} \le z_{i + 1}, \;i = 1, \;2, \;\ldots , \;N + 1$ is segment number, k = 1, 2, …, N _i;N _i is the term number of the current polynomial of the i-th segment (i.e. the segment between z _i point and z _i+1 point); N _i−1 is the order of the current approximate polynomial of the i-th segment.

The current expansion function is substituted into the integral equation, and the area along the antenna surface is replaced by the line integral along the antenna axis. The weight function is δ function, and a matrix equation can be obtained. Solving the matrix equation, the current distribution on the antenna can be obtained, and then the input impedance, VSWR, gain and pattern of the antenna can be calculated.

When studying antenna loading and broadband matching network optimization, VSWR and gain are generally used as performance indicators to measure the degree of optimization.

(6)$$\Gamma ( {\omega_i} ) = \displaystyle{{Z_q( {\omega_i} ) -Z_0} \over {Z_q( {\omega_i} ) + Z_0}}, \;$$

(7)$$VSWR( {\omega_i} ) = \displaystyle{{1 + \vert {\Gamma ( {\omega_i} ) } \vert } \over {1-\vert {\Gamma ( {\omega_i} ) } \vert }}, \;$$

where Z _q(ω _i) is input impedance, Γ(ω _i) is the reflection coefficient at ω _i, Z ₀ is the feeder characteristic impedance, which is generally taken as 50 Ω.

GOA is used to improve the gain of the antenna on the premise that VSWR is <3 as much as possible by optimizing the loading and broadband matching network of the antenna. For the loading network, the objective function of the optimization is to minimize the maximum VSWR and maximize the minimum gain of each sampling frequency point [Reference Wang, Liu, Wu and Xie27, Reference Wang, Liu, Wu and Xie28].

(8)$$F = min\left\{{\sum\limits_i^n {[ {W_i{( {VSWR( {\omega_i} ) -1} ) }^2 + A_i( {G_0-G( {\omega_i} ) } ) } ] } } \right\}, \;$$

where G (ω _i) is the gain of antenna at i-th frequency point ω _i(i = 1, 2, …, n), G ₀ is rated gain, which is an adjusting parameter to weigh the broadband impedance characteristics and gain characteristics of antenna. W _i is a weighted value of VSWR at i-th frequency point, whose value depends on the relative importance of VSWR(i). On the one hand, it retains a good VSWR, on the other hand, it discards a bad VSWR. Obviously, the smaller the value of the objective function, the better the optimization effect will be.

In order to reduce the power loss as much as possible, the matching network designed in this paper is low or even lossless, so the gain of the antenna is basically determined after loading. The matching network is only used to further reduce the VSWR for matching the antenna with the 50 Ω transmission line. For matching network optimization, the objective function is to minimize the average value of VSWR.

(9)$$F = min\left[{\sum\limits_{i = 1}^n {VSWR( {\omega_i} ) } } \right]. $$

Sub-band optimization

Both the loading network optimization of the upper part of the antenna and the adjusting inductance and matching network optimization of the bottom part are based on the FEKO model of the antenna. The editable program of FEKO is called by MATLAB software, and the loading and matching elements are both optimized by GOA, seeing Fig. 4 for details.

Fig. 4.

Antenna calculation and optimization process.

An antenna model is established with a RL loading network at 7.5 m, and the loading network parameters are optimized by equation (8). Since both the adjusting inductance and matching network are composed of lossless elements without resistance, once the loading network is determined, the gain and efficiency are basically determined. Therefore, the antenna gain should be given priority here, and the VSWR can be left to the matching network for further optimization. The optimized values of the loading element are shown in Table 1.

Table 1.

Optimal values of loading

For band I–IV, connecting the upper and lower antennas at 5 m, for band V, disconnecting them, and the antenna could obtain better impedance characteristics in each sub-band by optimizing the adjusting inductance L ₀ and matching network using GOA; the optimized values of the matching network elements of each sub-band are shown in Table 2. After adding the optimized loading and matching network of each sub-band, the input impedance and VSWR are shown in Figs 5 and 6.

Fig. 5.

Impedance curves of the proposed antenna in each sub-band.

Fig. 6.

VSWR curves of the proposed antenna in each sub-band.

Table 2.

Optimal values of L ₀ and matching network for each sub-band

It can be seen from Fig. 5 that when the loading network is added to the antenna, the input resistance of the low-frequency band increases highly (from 4 to 22.4 Ω), the capacitive reactance decreases to some extent, the change degree of input impedance in the medium and high frequency is obviously flattened, the input resistance is closer to 200 Ω, and the reactance fluctuates within the range of ±130 Ω; Moreover, the change of the input impedance of the antenna is further flattened by network matching for each sub-band, the input resistance is close to 50 Ω, and the input reactance basically fluctuates around ±40 Ω, which can better realize the impedance matching between the antenna and the feeder. As shown in Fig. 6, The VSWR is greatly reduced whether it is loaded or network matched, and the VSWR of the proposed antenna through adding load and matching network is basically <3.

Test and analysis of scaled antenna

In order to better verify the effectiveness of the proposed reconfigurable antenna, and considering the possibility of implementation, a 10:1 simple scale model is manufactured, and the electrical parameters of the scale antenna are measured.

As shown in Fig. 7, a scale-up model of 1-m reconfigurable whip antenna is established with 5 sub-bands (I–V). The frequency range is 30–300 MHz, a loading point is set at the upper part of the antenna with a parallel form of RL elements. A RF switch is added in the cavity at the middle of the antenna to control the connection of upper and lower whips in band I–IV and the disconnection of upper and lower whips in band V, and the DC control circuit of the switch enters from the bottom of the antenna and reaches the radiator control switch in the middle of the antenna through the inner cavity of the antenna. Corresponding inductance L ₀ and matching network designed for five sub-bands are integrated on a 1.6 mm thick printed circuit board to simplify the circuit structure, which is placed in the antenna base. In Fig. 7, L ₀ is the adjusting inductance, and the labels “I–V” respectively, represent the matching network of sub-band I–V, the transformer is connected parallel to a “г”-shaped LC network, and the power is supplied by a 50 Ω feeder. Here, a finite metal plate (iron is used in this paper) of 1 m × 1 m is selected as the ground medium.

Fig. 7.

Scale test antenna of the proposed antenna.

The actual antenna is placed in the microwave laboratory to measure the performance parameters, seeing Fig. 8. In this paper, the comparison method is used to measure the antenna gain. Because the space of the microwave laboratory is relatively small, it cannot fully meet the distance requirements of the whole VHF antenna far-field test, especially the far-field measurement in the lower frequency band. In order to verify the effectiveness of the proposed antenna, only the far-field of the antenna at the higher frequency point are measured.

Fig. 8.

Actual measurement environment of the proposed antenna.

Figure 9 shows the characteristic curves of the proposed frequency reconfigurable antenna, including VSWR, gain, and efficiency. Through the simulation and measurement of the VSWR, it can be seen that 98% of the frequency points are <3, with an average value of 2.34, the measured value of all the frequency points is <3, with an average value of 2.14, and the measured value is consistent with the simulated one, but there is a certain fluctuation due to the impact of the environment, the VSWR is greatly improved in the lower-frequency band, mainly because the nonlinearity and loss of the core of the transmission line transformer can smooth the high capacitive reactance of the input end in the lower-frequency band. The gain of the proposed antenna is all >−2.5 dB, with an average of 3.90 dB, the efficiency is >18.2%, with an average of 71.59%, both are much higher than that of the traditional antenna. Under the existing measurement conditions, the gain of the proposed antenna in the range of 150~300 MHz is measured, and the measured value is basically consistent with the simulation value, which is slightly affected by the manufacturing accuracy.

Fig. 9.

Electrical performance parameters (a) VSWR, (b) gain, and (c) efficiency of each sub-band of the proposed antenna.

It can be seen from Fig. 9(b) and (c) that the gain and efficiency of the frequency reconfigurable antenna increase greatly at 150 MHz, because after 150 MHz (i.e. band V), the RF switch in the middle part of the antenna is disconnected, and only the lower part of the antenna is involved in the radiation. At this time, the radiation height of the antenna is close to 1/4 wavelength, so the antenna can get a better match. Moreover, the failure of the upper part of the antenna (contains loss resistance) to access the radiation greatly reduces the power loss of the antenna, seeing Fig. 10, so the gain and efficiency of the antenna are greatly improved in band V.

Fig. 10.

Loading loss power of the proposed antenna.

Due to the limitation of size and frequency, it is impossible to measure the far-field radiation pattern of the antenna in the microwave laboratory. Figure 11 shows the gain pattern of the proposed antenna calculated by the moment method. In the whole band, the horizontal patterns are all omnidirectional, the vertical patterns all keep the half shape of “∞”, and there is no such phenomenon of up-warping as the traditional antenna, which is also consistent with the gain curve of Fig. 9(b). Therefore, the frequency reconfigurable antenna designed in this paper can achieve omnidirectional radiation with high gain and high efficiency to better realize long-range, medium-range, and short-range communication in the whole band.

Fig. 11.

(a) Horizontal pattern, and (b) vertical pattern of the proposed antenna.