# Multiobjective Particle Swarm Optimization to Design the Top Load of a Very Low Frequency Thirteen-Tower Umbrella Antenna

Authors: Hui Xie; Xiangyu Xu; Huaning Wu; Ning Wang
Journal: International Journal of RF and Microwave Computer-Aided Engineering
 (2023)

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## Abstract

In this paper, an innovative method for designing the top load of a very low frequency (VLF) thirteen-tower umbrella antenna based on multiobjective particle swarm optimization (MOPSO) is proposed. The design of the thirteen-tower umbrella antenna’s top load mainly involves the sag and position distribution of the cables. On the basis of analyzing the influence of the cables’ sag on the antenna’s radiation efficiency, a MOPSO based on the Pareto optimality is used to optimize the location distribution of the antenna top load catenaries. The radiation resistance and static capacitance of the antenna can be both slightly improved when each diamond-shaped top load maintains 8 top load cables, while the radiation resistance and static capacitance of the antenna can be improved by more than 6% when each diamond-shaped top load is increased to 10 top load cables.


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## Body

## 1. Introduction

Very low frequency (VLF) communication is a long-range communication method operating in the frequency range of 3~30 kHz. Due to its long propagation distance, low propagation loss, and strong seawater penetration capability, VLF communication is widely used in the fields of ocean communication, submarine communication, and timing [1–3]. Generally, a VLF transmitting antenna is an electrically small monopole antenna placed on the ground, and top loading is the most important method to improve the radiation efficiency of the monopole antenna [4]. Compared with the VLF transmitting antennas with other top load forms, the thirteen-tower umbrella antenna is the most efficient shore-based VLF transmitting antenna in practice [5]. The Cutler antenna and the NWC antenna are the best known thirteen-tower umbrella antennas in use and have been working for more than fifty years now [6, 7]. Therefore, until new communication methods such as blue-green laser communication [8] and neutrino communication [9] are put into practical use, VLF communication is still the most reliable way of submarine communication at present.Many studies have been carried out to analyze the electrical performance of the umbrella top load antennas. Trainotti used the transmission-line technique to study the top-loaded monopoles exhaustively, and the antenna’s radiation resistance has been obtained under different top-loading conditions [10]. On this basis, the electrical performance of the VLF transmitting antenna is optimized by changing the size, shape, and electrical connection of the top load of the monopole antenna [11–13]. However, these optimization designs mainly use the traditional empirical formula analytical calculation method, rather than the current popular numerical calculation method combined with intelligent optimization algorithm; the calculation is complex and cannot adequately solve the optimal value during the antenna optimization.Intelligent optimization algorithms have been widely used in the field of antenna optimization, such as spider monkey optimization (SMO) [14], multiobjective particle swarm optimization (MOPSO) [15], whale optimization algorithm (WOA) [16], butterfly mating optimization (BMO) [17], equilibrium optimization algorithm (EOA) [18], and sparrow search algorithm (SSA) [19]. Among them, particle swarm optimization has been rapidly developed and widely used since it was proposed in 1995 [20], such as designing artificial magnetic conductors [21] and designing antennas for environmental sensing [22]. However, these antenna optimization works are mainly focused on the phase control and beam-pattern synthesis of array antennas, and there are few studies on the structural optimization of VLF transmitting antennas.Due to the irreplaceability of VLF communication in the field of long-range submarine communication at present, it is still very valuable to study and optimize the structure of VLF transmitting antennas. In this paper, a MOPSO based on the Pareto optimality is proposed to optimize the top load of the thirteen-tower umbrella antenna. Different from the traditional analytical calculation, the numerical calculation method combined with intelligent optimization algorithm can quickly traverse the entire search space to solve the optimal value of antenna design. The structure is organized as follows: Section2 presents the effect of antenna top load cables’ sag on the antenna’s radiation efficiency. Section 3 establishes the optimization of the antenna’s top load by using the MOPSO with the Pareto optimality. Finally, Section 4 concludes the paper.


## 2. Model and Electrical Performance of the Thirteen-Tower Umbrella Antenna Varying with Top Load Catenaries’ Sag

### 2.1. The Model of the Thirteen-Tower Umbrella Antenna

The VLF thirteen-tower umbrella antenna is an electrically short top-loaded monopole antenna. The top load of the antenna is made up of six diamond-shaped panels. These panels are formed by eight wires that run out from the antenna center and one catenary for support [5]. The NWC antenna, as the highest radiation efficiency shore-based VLF transmitting antenna, has a radiation efficiency of over 80% when operating at 19.8 kHz [7, 23]. In this study, the NWC VLF transmitting antenna is used as the research object, and the modeling and electrical performance simulation calculation of the antenna are performed by FEKO and MATLAB software. In contrast to other antennas operating in a higher frequency range, the VLF antenna is an electrically small antenna with a usually omnidirectional directivity and an antenna gain of roughly 3 [24]. Since the power capacity and radiation capability of a VLF antenna are its primary concerns, this study focuses on calculating the antenna’s static capacitance and radiation resistance in order to assess the performance of the optimization algorithm.Due to the effect of gravity, the top load of the NWC antenna is composed of several catenaries with a certain sag. As shown in Figure1, A and B are the two support points of the catenary, respectively, l is the distance between the two support points A and B, ζm is the maximum sag, and h is the vertical distance between the two support points A and B and, in general, defines the sag as
(1)ζ=ζml.Figure 1
The catenary. A and B are the two support points of the catenary,l and h are the horizontal distance and vertical distance between A and B, respectively, ζx is the sag of each point on the catenary, and ζm is the maximum sag.As the exact catenary equation is very complicated to calculate, the error of applying the flat parabolic equation to approximate the catenary at small distances and small sag (h/l<0.1 and ζ=ζm/l<0.1) is less than 2% [25], so the flat parabolic equation is used to make an approximate simulation. The equation of the flat parabola is
(2)y=gx22τ0,where g/τ0 can be derived from the flat parabola’s sag formula ζm=gl2/8τ0.By establishing parabolas with a sag of 5%, a simulation model of the NWC antenna is constructed in this study (see Figure2), and the simulation results of the electrical performance of the antenna model are shown in Table 1; the antenna operates at the frequency of 19.8 kHz.Figure 2
Model of the NWC antenna.Table 1
The measured and simulated results of the antenna’s electrical performance.
Radiation resistance (Ω)Static capacitance (nF)Self-resonance (kHz)Measured result [7]0.2400162.634.2Simulated result0.2389152.332.3Error-0.46%-6.33%-5.56%The result in Table1 shows that the top load of the NWC antenna model constructed by parabolas with a certain sag is valid and accurate. In the following study, this model is used as a basis for investigating the effect of the sag of the top load catenaries on the antenna’s electrical performance.


### 2.2. Calculation of the Antenna Radiation Efficiency

Because the earth is neither an ideal conductor nor a perfect medium, losses are unavoidable when electromagnetic forces interact with it. Two mechanisms are thought to be responsible for ground loss in antenna systems. The first isE-field loss, which is the loss of displacement current from the antenna via the ground into the ground system. The second process is the H-field loss, which is caused by the current induced in the ground by the tangential magnetic field of the vertical down-lead current flowing toward the antenna bottom [24, 26].In this study, theH-field loss and E-field loss of the antenna are calculated, respectively. Firstly, according to the current distribution of the antenna, the tangential component of the earth surface magnetic field Ht near the antenna can be obtained. Then, the surface current density caused on the earth surface is
(3)Jt=ez×Ht,where ez is the normal vector of the interface. Ignoring the displacement current, all the current flows back to the antenna through the earth, and the tangential electric field on the surface is
(4)Et=Zeez×Ht,where Ze is the surface impedance of the earth. Then, the loss power of the earth is
(5)PH=Re∫AJt⋅EtdA=∫ARe⋅Ht2dA,where Re is the surface resistance of the earth and A is the area of the near field of the antenna.The earth with ground screen can be equivalent to the earth without ground screen with equivalent surface impedanceZg. When the distance between the conductors of the ground screen is less than the skin depth, the equivalent surface impedance Zg is equal to the parallel connection of the individual earth surface impedance Ze and the impedance per unit area of the ideal conductor ground screen in free space Zs. And the real part RH′ of Zg is the equivalent surface resistance of the ground screen area. If there is a dense ground screen on the ground with low conductivity, the equivalent surface resistance is [24]
(6)RH′≈2.1×10−9⋅s2f3/2σ1/2lgs/πd2,where s is the distance between ground screen conductors, f is the operating frequency, σ is the earth conductivity, and d is the diameter of ground screen conductors. Then, the H-field loss resistance is
(7)RH=1Io2∫ARH′⋅Ht2dA,where I0 is the input current of the antenna system.Then, theE-field loss is calculated. Ez is the vertical electric field component at the ground surface, and the displacement current perpendicular to the ground is Jd=ωε0Ez; the E-field loss power is
(8)PE=∫AJd2⋅RE′dA=∫ARE′⋅ωε02Ez2dASo theE-field loss resistance is
(9)PE=1Io2∫ARE′⋅ωε02Ez2dAwhere RE′ is the effective series resistance per unit area of the ground network area, ω is the operating angular frequency, ε0 is the vacuum permittivity, and Io is the input current of the antenna. And the total ground loss resistance is
(10)Rg=RH+REThen, the radiation efficiency of the antenna is(11)η=RrRr+Rg+Rcwhere Rr is the radiation resistance and Rc is the loss resistance of conductors.


### 2.3. The Influence of Top Load Catenaries’ Sag on the Antenna Electrical Performance

By modifying the sag of the NWC antenna top load catenaries, the models of the thirteen-tower umbrella antenna with sag from 4% to 8% are constructed. In order to reduce the ground loss resistance and improve the electrical performance of the antenna, a radiant ground screen with a radius of 1600 meters is laid, the number of conductors is 360, and the total length of the ground screen is5.76∗105 meters. The parameters of the antenna system are shown in Table 2. In addition, the operating frequency changes between 16 and 28 kHz.Table 2
The parameters of the antenna system model.
ParametersValueCenter tower height (m)387.4Inner ring tower height (m)358.1Inner ring radius (m)647.4Outer ring tower height (m)303.6Outer ring radius (m)1257.6Sag of top load wires4%-8%Conductor conductivity (S∗m−1)0.001Earth conductivity (S∗m−1)5.8×107Input power (kW)2000Figures3 and 4 describe the radiation resistance, radiation reactance, and radiation efficiency of the NWC antenna model with different top load cable sags, respectively. The simulation results show that the radiation resistance of the antenna falls dramatically when the top load catenaries’ sag increases from 4% to 8%, with an average reduction of 17% in the frequency range of 16-28 kHz. The antenna’s radiation efficiency is also reduced by around 2% as a result of this. The variation of the antenna radiation reactance with the antenna top load catenaries’ sag is minimal, which is compatible with the variation of the antenna static capacitance under various top load sags shown in Table 3.Figure 3
The radiation resistance and reactance of the antenna with different sags. (a) Radiation resistance. (b) Radiation reactance.
(a)(b)Figure 4
The radiation efficiency of the antenna with different sags.Table 3
The simulated results of the antenna’s static capacitance with different sags.
Sag0.040.050.060.070.08Static capacitance (nF)150.9152.3153.6155.2156.8


## 2.1. The Model of the Thirteen-Tower Umbrella Antenna

The VLF thirteen-tower umbrella antenna is an electrically short top-loaded monopole antenna. The top load of the antenna is made up of six diamond-shaped panels. These panels are formed by eight wires that run out from the antenna center and one catenary for support [5]. The NWC antenna, as the highest radiation efficiency shore-based VLF transmitting antenna, has a radiation efficiency of over 80% when operating at 19.8 kHz [7, 23]. In this study, the NWC VLF transmitting antenna is used as the research object, and the modeling and electrical performance simulation calculation of the antenna are performed by FEKO and MATLAB software. In contrast to other antennas operating in a higher frequency range, the VLF antenna is an electrically small antenna with a usually omnidirectional directivity and an antenna gain of roughly 3 [24]. Since the power capacity and radiation capability of a VLF antenna are its primary concerns, this study focuses on calculating the antenna’s static capacitance and radiation resistance in order to assess the performance of the optimization algorithm.Due to the effect of gravity, the top load of the NWC antenna is composed of several catenaries with a certain sag. As shown in Figure1, A and B are the two support points of the catenary, respectively, l is the distance between the two support points A and B, ζm is the maximum sag, and h is the vertical distance between the two support points A and B and, in general, defines the sag as
(1)ζ=ζml.Figure 1
The catenary. A and B are the two support points of the catenary,l and h are the horizontal distance and vertical distance between A and B, respectively, ζx is the sag of each point on the catenary, and ζm is the maximum sag.As the exact catenary equation is very complicated to calculate, the error of applying the flat parabolic equation to approximate the catenary at small distances and small sag (h/l<0.1 and ζ=ζm/l<0.1) is less than 2% [25], so the flat parabolic equation is used to make an approximate simulation. The equation of the flat parabola is
(2)y=gx22τ0,where g/τ0 can be derived from the flat parabola’s sag formula ζm=gl2/8τ0.By establishing parabolas with a sag of 5%, a simulation model of the NWC antenna is constructed in this study (see Figure2), and the simulation results of the electrical performance of the antenna model are shown in Table 1; the antenna operates at the frequency of 19.8 kHz.Figure 2
Model of the NWC antenna.Table 1
The measured and simulated results of the antenna’s electrical performance.
Radiation resistance (Ω)Static capacitance (nF)Self-resonance (kHz)Measured result [7]0.2400162.634.2Simulated result0.2389152.332.3Error-0.46%-6.33%-5.56%The result in Table1 shows that the top load of the NWC antenna model constructed by parabolas with a certain sag is valid and accurate. In the following study, this model is used as a basis for investigating the effect of the sag of the top load catenaries on the antenna’s electrical performance.


## 2.2. Calculation of the Antenna Radiation Efficiency

Because the earth is neither an ideal conductor nor a perfect medium, losses are unavoidable when electromagnetic forces interact with it. Two mechanisms are thought to be responsible for ground loss in antenna systems. The first isE-field loss, which is the loss of displacement current from the antenna via the ground into the ground system. The second process is the H-field loss, which is caused by the current induced in the ground by the tangential magnetic field of the vertical down-lead current flowing toward the antenna bottom [24, 26].In this study, theH-field loss and E-field loss of the antenna are calculated, respectively. Firstly, according to the current distribution of the antenna, the tangential component of the earth surface magnetic field Ht near the antenna can be obtained. Then, the surface current density caused on the earth surface is
(3)Jt=ez×Ht,where ez is the normal vector of the interface. Ignoring the displacement current, all the current flows back to the antenna through the earth, and the tangential electric field on the surface is
(4)Et=Zeez×Ht,where Ze is the surface impedance of the earth. Then, the loss power of the earth is
(5)PH=Re∫AJt⋅EtdA=∫ARe⋅Ht2dA,where Re is the surface resistance of the earth and A is the area of the near field of the antenna.The earth with ground screen can be equivalent to the earth without ground screen with equivalent surface impedanceZg. When the distance between the conductors of the ground screen is less than the skin depth, the equivalent surface impedance Zg is equal to the parallel connection of the individual earth surface impedance Ze and the impedance per unit area of the ideal conductor ground screen in free space Zs. And the real part RH′ of Zg is the equivalent surface resistance of the ground screen area. If there is a dense ground screen on the ground with low conductivity, the equivalent surface resistance is [24]
(6)RH′≈2.1×10−9⋅s2f3/2σ1/2lgs/πd2,where s is the distance between ground screen conductors, f is the operating frequency, σ is the earth conductivity, and d is the diameter of ground screen conductors. Then, the H-field loss resistance is
(7)RH=1Io2∫ARH′⋅Ht2dA,where I0 is the input current of the antenna system.Then, theE-field loss is calculated. Ez is the vertical electric field component at the ground surface, and the displacement current perpendicular to the ground is Jd=ωε0Ez; the E-field loss power is
(8)PE=∫AJd2⋅RE′dA=∫ARE′⋅ωε02Ez2dASo theE-field loss resistance is
(9)PE=1Io2∫ARE′⋅ωε02Ez2dAwhere RE′ is the effective series resistance per unit area of the ground network area, ω is the operating angular frequency, ε0 is the vacuum permittivity, and Io is the input current of the antenna. And the total ground loss resistance is
(10)Rg=RH+REThen, the radiation efficiency of the antenna is(11)η=RrRr+Rg+Rcwhere Rr is the radiation resistance and Rc is the loss resistance of conductors.


## 2.3. The Influence of Top Load Catenaries’ Sag on the Antenna Electrical Performance

By modifying the sag of the NWC antenna top load catenaries, the models of the thirteen-tower umbrella antenna with sag from 4% to 8% are constructed. In order to reduce the ground loss resistance and improve the electrical performance of the antenna, a radiant ground screen with a radius of 1600 meters is laid, the number of conductors is 360, and the total length of the ground screen is5.76∗105 meters. The parameters of the antenna system are shown in Table 2. In addition, the operating frequency changes between 16 and 28 kHz.Table 2
The parameters of the antenna system model.
ParametersValueCenter tower height (m)387.4Inner ring tower height (m)358.1Inner ring radius (m)647.4Outer ring tower height (m)303.6Outer ring radius (m)1257.6Sag of top load wires4%-8%Conductor conductivity (S∗m−1)0.001Earth conductivity (S∗m−1)5.8×107Input power (kW)2000Figures3 and 4 describe the radiation resistance, radiation reactance, and radiation efficiency of the NWC antenna model with different top load cable sags, respectively. The simulation results show that the radiation resistance of the antenna falls dramatically when the top load catenaries’ sag increases from 4% to 8%, with an average reduction of 17% in the frequency range of 16-28 kHz. The antenna’s radiation efficiency is also reduced by around 2% as a result of this. The variation of the antenna radiation reactance with the antenna top load catenaries’ sag is minimal, which is compatible with the variation of the antenna static capacitance under various top load sags shown in Table 3.Figure 3
The radiation resistance and reactance of the antenna with different sags. (a) Radiation resistance. (b) Radiation reactance.
(a)(b)Figure 4
The radiation efficiency of the antenna with different sags.Table 3
The simulated results of the antenna’s static capacitance with different sags.
Sag0.040.050.060.070.08Static capacitance (nF)150.9152.3153.6155.2156.8


## 3. Optimal Design of the Top Load of a Thirteen-Tower Umbrella Antenna

The influence of the top load cables’ sag on the antenna radiation efficiency has been studied above. The result shows that, if the cable stress condition allows, the cables’ sag should be reduced as much as possible to improve the radiation efficiency of the antenna. Next, the location distribution of the top load cables will be analyzed. Unlike the cable sag, which is a single linear variable, the location distribution of the cables is nonlinear multidimensional variables. It is difficult to obtain the optimal value of the location distribution using traditional analytical calculation. Therefore, particle swarm optimization (PSO) is introduced in this study to optimize the top load.


### 3.1. Inertia Weight of PSO

The inertia weightw is one of the most important parameters of PSO. The algorithm’s global search capability increases when the value of w is increased, whereas the algorithm’s local search capability improves as the value of w is decreased. This study employed an inertia weight that changes dynamically with the number of iterations to prevent becoming stuck in a local optimum while retaining a strong global search capability [27].
(12)wt=wstart−wstart−wendtT2,where wstart is the initial values of inertia weight, wstart=0.9, and wend is the final value of inertia weight, wend=0.4. Figure 5 depicts the variation of w with the iterations, and w varies slowly and takes on bigger values in the early iterations, allowing the algorithm to maintain a strong global search capability, whereas w changes quicker and takes on smaller values in the later iterations, allowing the algorithm to increase its local search capability.Figure 5
The variation of inertia weightw with the iterations.


### 3.2. Antenna Optimization Model

The position distribution of the eight top load cables of a single diamond-shaped top load can have a significant impact on the antenna’s electrical performance when the cables’ sag is constant. As shown in Figure6, the optimization model of the top load cables’ position distribution is established. In order to eliminate the influence of cables’ sag on the results, in this part of the optimization calculation, the cables are straight; that is, the sag is equal to 0.Figure 6
The optimization model of the top load cables.In order to maintain the stability of the antenna, the cables on both sides of the dotted line of the diamond-shaped top load always remain symmetrical during the optimization, so it is only necessary to optimize the position distribution of the four top load cables on one side of the dotted line. Take the point O in Figure6 as the coordinate origin and the distances of the four top load cables to the origin as X1, X2, X3, and X4, respectively, which are the four dimensions of particle position.SinceX1, X2, X3, and X4 are the distances from the four top load cables to the coordinate origin O, respectively, and the distance between two adjacent wires cannot be too small, the minimum distance between two wires is set to 5 meters in this study. The following boundary condition restrictions are determined for the four dimensions of the particle’s position:
(13)5≤Xi≤Xmax,i=1,2,3,4,Xi−Xi+1≥5,i=1,2,3,where Xmax is the maximum of the position of the top load cables, and Xmax is 332 m.


### 3.3. Results Obtained by Single Objective PSO

The position distribution of the top load cables of the NWC antenna is shown in Table4, when the cables’ sag is set to zero. The radiation resistance operating at 19.8 kHz is 0.2400 Ω, and the static capacitance is 151.25 nF. The two values are used as the comparison benchmark for subsequent optimization.Table 4
The position distribution of the NWC antenna’s top load cables.
X1 (m)X2 (m)X3 (m)X4 (m)Value of the NWC antenna322.5300.4199.667.6This study mainly focuses on the radiation resistance and static capacitance of the thirteen-tower umbrella antenna. Therefore, the radiation resistance and static capacitance, which are solved by the FEKO software, are used as fitness functions of PSO to solve their maximum values. The population size of PSO is 20, and the max iterations are set to 100.The workflow of PSO with a single objective is shown in Figure7 [20], and the optimization results of the position distribution of the top load cables are shown in Figures 8–11.Figure 7
The workflow of the single objective PSO.Figure 8
The fitness degree of radiation resistance.Figure 9
The individual optimal values of radiation resistance and the corresponding position distribution of the top load cables.Figure 10
The fitness degree of static capacitance.Figure 11
The individual optimal values of static capacitance and the corresponding position distribution of the top load cables.According to the optimization results, the optimal solution of the antenna radiation resistance is 0.2478, which is 3.3% higher than the benchmark value of the NWC antenna model. And the optimal solution of antenna static capacitance is 153.6 nF, which is 1.7% higher than the benchmark value of the NWC antenna model. Figures9 and 11 show that the individual optimal solutions of radiation resistance for 20 particles are basically 0.2477, and the individual optimal solutions of static capacitance are basically 153.1 nF. There is a significant difference in the position distribution of the antenna top load cables when the antenna achieves the optimal solution of radiation resistance and static capacitance, respectively. As shown in Table 5, when the antenna’s radiation resistance achieves the optimal solution, the top load cables’ position dispersion is significantly narrower than that when the static capacitance achieves the optimal solution.Table 5
The average position distribution of the antenna top load cables corresponding to various antenna optimal solutions.
X1 (m)X2 (m)X3 (m)X4 (m)Position distribution of the radiation resistance’s optimal solution240.9214.2173.795.4Position distribution of the static capacitance’s optimal solution331.3290.7199.576.5When the antenna radiation resistance obtains the optimal solution of 0.2478Ω, the antenna’s static capacitance is 136.55 nF, which is 9.7% lower than the benchmark value of the NWC antenna model, and when the antenna’s static capacitance obtains the optimal solution of 153.6 nF, the antenna’s radiation resistance is 0.2383 Ω, which is 0.7% lower than the benchmark value of the NWC antenna model. This indicates that the radiation resistance and static capacitance have some contradictions in the optimization search, and it is difficult for the single objective PSO to complete the simultaneous optimization of the two parameters.


### 3.4. Results Obtained by MOPSO

Since it is difficult for single objective PSO to optimize both parameters of radiation resistance and static capacitance at the same time, it is necessary to optimize the antenna top load by MOPSO. There are two general ways to realize MOPSO: one is to transform multiple fitness functions into a single fitness function by weighted summation and the other is to look for the Pareto optimal front so as to obtain a set of feasible solutions [28, 29]. Since the two fitness functions in this study contradict each other, after experimenting with a series of different weighting combinations, the ideal optimization result is still not found. In this study, MOPSO based on the Pareto optimality is used to optimize the antenna top load.The workflow of a MOPSO algorithm is shown in Figure12, and the obtained Pareto optimal front in the target space is shown in Figure 13. In the noninferior solution set, the radiation resistance and static capacitance of the antenna show a negative correlation, which also reflects that the optimization of these two parameters is contradictory to each other. In this study, a total of 9 optimization solutions with both radiation resistance and static capacitance better than the NWC antenna model are found, and the top load cables’ position distribution is shown in Table 6.Figure 12
The workflow of MOPSO.Figure 13
The Pareto optimal front for the optimization of the NWC antenna model.Table 6
The optimization solutions with both radiation resistance and static capacitance better than the NWC antenna model.
No.Radiation resistance (Ω)Static capacitance (nF)X1 (m)X2 (m)X3 (m)X4 (m)10.2400152.12325.7273.4194.055.820.2402152.00324.1284.8194.771.230.2402151.83324.2285.4212.789.140.2405151.76323.6271.2198.971.450.2406151.68322.5281.4189.876.660.2406151.68322.6278.4193.669.970.2407151.60322.2272.2181.958.980.2408151.51321.8270.0186.865.190.2409151.31320.1277.5194.670.3From the 9 optimization solutions in Table6, it can be seen that the position X1 of the 1st top load cable is basically the same as the NWC antenna, the position X2 of the 2nd top load cable changes by about 7%, and the positions X3 and X4 of the 3rd and 4th top load cables change by about 3%, which result in a slight improvement of the radiation resistance and static capacitance. Of course, we can also focus more on the radiation resistance or the static capacitance according to the requirements of actual antenna design and filter the desired optimization solution from the noninferior solution set in Figure 13.The optimization of the NWC antenna model using the above MOPSO has achieved better results. To further verify the optimization performance of the algorithm, the algorithm is now considered to optimize the thirteen-tower umbrella antenna with each diamond-shaped top load containing 10 cables without a priori knowledge. The optimization model for the individual top load of the antenna is shown in Figure14.Figure 14
Optimization model of the top load containing 10 cables.The new antenna model is optimized by the MOPSO algorithm in this paper. The dimension of the particle position is 6, where 5 dimensions represent the position and the other dimension represents the sag of the top load cables. The population size is set to 30, the max iterations are set to 100, and the inertia weight remains as equation (12). The obtained Pareto optimal front is shown in Figure 15. An optimized solution is selected from the noninferior solution set, and the location distribution of the antenna’s top load cables is shown in Table 7. The radiation resistance of the antenna is 0.2545 Ω, and the static capacitance is 160.8 nF, both of which are improved by more than 6% over the baseline value of the NWC antenna model.Figure 15
The Pareto optimal front for the optimization of the top load containing 10 cables.Table 7
One optimization solution for the antenna with 10 wires in each diamond-shaped top load.
Radiation resistance (Ω)Static capacitance (nF)X1 (m)X2 (m)X3 (m)X4 (m)X5 (m)Sag0.2545160.8322.7278.9219.0136.551.00.04


## 3.1. Inertia Weight of PSO

The inertia weightw is one of the most important parameters of PSO. The algorithm’s global search capability increases when the value of w is increased, whereas the algorithm’s local search capability improves as the value of w is decreased. This study employed an inertia weight that changes dynamically with the number of iterations to prevent becoming stuck in a local optimum while retaining a strong global search capability [27].
(12)wt=wstart−wstart−wendtT2,where wstart is the initial values of inertia weight, wstart=0.9, and wend is the final value of inertia weight, wend=0.4. Figure 5 depicts the variation of w with the iterations, and w varies slowly and takes on bigger values in the early iterations, allowing the algorithm to maintain a strong global search capability, whereas w changes quicker and takes on smaller values in the later iterations, allowing the algorithm to increase its local search capability.Figure 5
The variation of inertia weightw with the iterations.


## 3.2. Antenna Optimization Model

The position distribution of the eight top load cables of a single diamond-shaped top load can have a significant impact on the antenna’s electrical performance when the cables’ sag is constant. As shown in Figure6, the optimization model of the top load cables’ position distribution is established. In order to eliminate the influence of cables’ sag on the results, in this part of the optimization calculation, the cables are straight; that is, the sag is equal to 0.Figure 6
The optimization model of the top load cables.In order to maintain the stability of the antenna, the cables on both sides of the dotted line of the diamond-shaped top load always remain symmetrical during the optimization, so it is only necessary to optimize the position distribution of the four top load cables on one side of the dotted line. Take the point O in Figure6 as the coordinate origin and the distances of the four top load cables to the origin as X1, X2, X3, and X4, respectively, which are the four dimensions of particle position.SinceX1, X2, X3, and X4 are the distances from the four top load cables to the coordinate origin O, respectively, and the distance between two adjacent wires cannot be too small, the minimum distance between two wires is set to 5 meters in this study. The following boundary condition restrictions are determined for the four dimensions of the particle’s position:
(13)5≤Xi≤Xmax,i=1,2,3,4,Xi−Xi+1≥5,i=1,2,3,where Xmax is the maximum of the position of the top load cables, and Xmax is 332 m.


## 3.3. Results Obtained by Single Objective PSO

The position distribution of the top load cables of the NWC antenna is shown in Table4, when the cables’ sag is set to zero. The radiation resistance operating at 19.8 kHz is 0.2400 Ω, and the static capacitance is 151.25 nF. The two values are used as the comparison benchmark for subsequent optimization.Table 4
The position distribution of the NWC antenna’s top load cables.
X1 (m)X2 (m)X3 (m)X4 (m)Value of the NWC antenna322.5300.4199.667.6This study mainly focuses on the radiation resistance and static capacitance of the thirteen-tower umbrella antenna. Therefore, the radiation resistance and static capacitance, which are solved by the FEKO software, are used as fitness functions of PSO to solve their maximum values. The population size of PSO is 20, and the max iterations are set to 100.The workflow of PSO with a single objective is shown in Figure7 [20], and the optimization results of the position distribution of the top load cables are shown in Figures 8–11.Figure 7
The workflow of the single objective PSO.Figure 8
The fitness degree of radiation resistance.Figure 9
The individual optimal values of radiation resistance and the corresponding position distribution of the top load cables.Figure 10
The fitness degree of static capacitance.Figure 11
The individual optimal values of static capacitance and the corresponding position distribution of the top load cables.According to the optimization results, the optimal solution of the antenna radiation resistance is 0.2478, which is 3.3% higher than the benchmark value of the NWC antenna model. And the optimal solution of antenna static capacitance is 153.6 nF, which is 1.7% higher than the benchmark value of the NWC antenna model. Figures9 and 11 show that the individual optimal solutions of radiation resistance for 20 particles are basically 0.2477, and the individual optimal solutions of static capacitance are basically 153.1 nF. There is a significant difference in the position distribution of the antenna top load cables when the antenna achieves the optimal solution of radiation resistance and static capacitance, respectively. As shown in Table 5, when the antenna’s radiation resistance achieves the optimal solution, the top load cables’ position dispersion is significantly narrower than that when the static capacitance achieves the optimal solution.Table 5
The average position distribution of the antenna top load cables corresponding to various antenna optimal solutions.
X1 (m)X2 (m)X3 (m)X4 (m)Position distribution of the radiation resistance’s optimal solution240.9214.2173.795.4Position distribution of the static capacitance’s optimal solution331.3290.7199.576.5When the antenna radiation resistance obtains the optimal solution of 0.2478Ω, the antenna’s static capacitance is 136.55 nF, which is 9.7% lower than the benchmark value of the NWC antenna model, and when the antenna’s static capacitance obtains the optimal solution of 153.6 nF, the antenna’s radiation resistance is 0.2383 Ω, which is 0.7% lower than the benchmark value of the NWC antenna model. This indicates that the radiation resistance and static capacitance have some contradictions in the optimization search, and it is difficult for the single objective PSO to complete the simultaneous optimization of the two parameters.


## 3.4. Results Obtained by MOPSO

Since it is difficult for single objective PSO to optimize both parameters of radiation resistance and static capacitance at the same time, it is necessary to optimize the antenna top load by MOPSO. There are two general ways to realize MOPSO: one is to transform multiple fitness functions into a single fitness function by weighted summation and the other is to look for the Pareto optimal front so as to obtain a set of feasible solutions [28, 29]. Since the two fitness functions in this study contradict each other, after experimenting with a series of different weighting combinations, the ideal optimization result is still not found. In this study, MOPSO based on the Pareto optimality is used to optimize the antenna top load.The workflow of a MOPSO algorithm is shown in Figure12, and the obtained Pareto optimal front in the target space is shown in Figure 13. In the noninferior solution set, the radiation resistance and static capacitance of the antenna show a negative correlation, which also reflects that the optimization of these two parameters is contradictory to each other. In this study, a total of 9 optimization solutions with both radiation resistance and static capacitance better than the NWC antenna model are found, and the top load cables’ position distribution is shown in Table 6.Figure 12
The workflow of MOPSO.Figure 13
The Pareto optimal front for the optimization of the NWC antenna model.Table 6
The optimization solutions with both radiation resistance and static capacitance better than the NWC antenna model.
No.Radiation resistance (Ω)Static capacitance (nF)X1 (m)X2 (m)X3 (m)X4 (m)10.2400152.12325.7273.4194.055.820.2402152.00324.1284.8194.771.230.2402151.83324.2285.4212.789.140.2405151.76323.6271.2198.971.450.2406151.68322.5281.4189.876.660.2406151.68322.6278.4193.669.970.2407151.60322.2272.2181.958.980.2408151.51321.8270.0186.865.190.2409151.31320.1277.5194.670.3From the 9 optimization solutions in Table6, it can be seen that the position X1 of the 1st top load cable is basically the same as the NWC antenna, the position X2 of the 2nd top load cable changes by about 7%, and the positions X3 and X4 of the 3rd and 4th top load cables change by about 3%, which result in a slight improvement of the radiation resistance and static capacitance. Of course, we can also focus more on the radiation resistance or the static capacitance according to the requirements of actual antenna design and filter the desired optimization solution from the noninferior solution set in Figure 13.The optimization of the NWC antenna model using the above MOPSO has achieved better results. To further verify the optimization performance of the algorithm, the algorithm is now considered to optimize the thirteen-tower umbrella antenna with each diamond-shaped top load containing 10 cables without a priori knowledge. The optimization model for the individual top load of the antenna is shown in Figure14.Figure 14
Optimization model of the top load containing 10 cables.The new antenna model is optimized by the MOPSO algorithm in this paper. The dimension of the particle position is 6, where 5 dimensions represent the position and the other dimension represents the sag of the top load cables. The population size is set to 30, the max iterations are set to 100, and the inertia weight remains as equation (12). The obtained Pareto optimal front is shown in Figure 15. An optimized solution is selected from the noninferior solution set, and the location distribution of the antenna’s top load cables is shown in Table 7. The radiation resistance of the antenna is 0.2545 Ω, and the static capacitance is 160.8 nF, both of which are improved by more than 6% over the baseline value of the NWC antenna model.Figure 15
The Pareto optimal front for the optimization of the top load containing 10 cables.Table 7
One optimization solution for the antenna with 10 wires in each diamond-shaped top load.
Radiation resistance (Ω)Static capacitance (nF)X1 (m)X2 (m)X3 (m)X4 (m)X5 (m)Sag0.2545160.8322.7278.9219.0136.551.00.04


## 4. Conclusions

This study analyzes and optimizes the top load of a VLF thirteen-tower umbrella antenna. The radiation efficiency of the antenna decreases with the increase of the top load catenaries’ sag, so the sag should be reduced as much as possible in the case of cable tension allows. The influence of the top load cables’ position distribution on the antenna electrical performance is more complex, and the traditional analytical calculation method is difficult to find the optimal design of the top load. A MOPSO algorithm combined with the Pareto optimality is proposed to optimize the design of the top load of the thirteen-tower umbrella antenna. While keeping all the other structures of the antenna unchanged, the optimization algorithm can increase the radiation resistance and static capacitance of the antenna by more than 6% at the same time by only increasing the cables of each diamond-shaped top load from 8 to 10.



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*Source: 1792918-2023-03-11.xml*