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# Bio-inspired optimization algorithms applied to rectenna design

- Menglong He
^{1}Email author, - Zhao Wang
^{1}, - Mark Leach
^{1}, - Zhenzhen Jiang
^{1}and - Eng Gee Lim
^{1}

**Received:**7 September 2017**Accepted:**30 November 2017**Published:**3 January 2018

## Abstract

A comparative study of the use of bio-inspired optimization technologies including the Cuckoo Search (CS) algorithm, the Differential Evolution (DE) algorithm, and Quantum-behaved Particle Swarm Optimization (QPSO) in the design of microstrip patch antennas for use in RF energy harvesting systems is presented. Radio frequency (RF) energy harvesting is considered as an eco-friendly energy source and has become a focus of intense research especially for use in distributed sensor networks. In a RF energy harvesting system, the antenna is responsible for capturing RF signals over a certain frequency band, and it is a vital element in determining the performance of the RF energy harvester. In this paper, a new mathematical weighted evaluation model involving antenna efficiency, center frequency, and bandwidth is proposed to evaluate the performance of a rectangular microstrip patch antenna (RMPA) for a RF harvesting system based on both the transmission-line model and the cavity model. With the evaluation model as the objective function, bio-inspired optimization approaches are utilized to determine the geometrical parameters of the optimal antenna based on given constraints. Moreover, the optimised designs of an antenna for harvesting energy from the Global System for Mobile Communications (GSM) frequency band are proposed via the mathematical model and bio-inspired optimization approaches using simulations. Furthermore, a comparative study of the DE, CS, and QPSO techniques is conducted via the evaluation of the properties of the antenna designs.

## Keywords

- RF harvesting system
- Microstrip antenna
- Cuckoo Search (CS) algorithm
- Differential Evolution (DE) algorithm
- Quantum-behaved particle swarm optimization (QPSO)

## Background

With the advance of technologies including the Internet of Things (IoT) and wearable electronics, the demand for mobile electrical devices has surged. Battery depletion has become a fundamental bottleneck which limits the performance of these devices [1]. Considering the conventional fact that batteries have to be replaced or replenished manually after depletion, deeper implications exist for devices such as implantable heart pumps for which the replacement or recharging of the battery by cable is inconvenient and high-cost [2]. RF energy harvesting technology provides an alternative to this and has recently received significant attention in the research community demonstrating its potential as a sustainable energy source for low power electronics [3–7].

Average RF power density in London [8]

Band name | Frequency band | Average power density |
---|---|---|

GSM1800 (BTx) | 1805 MHz ∼ 1880 MHz | 84 nW/ |

GSM1800 (MTx) | 1710 MHz ∼ 1785 MHz | 0.5nW/ |

GSM900(BTx) | 925 MHz ∼ 960MHz | 36nW/ |

GSM900(MTx) | 880 MHz ∼ 915MHz | 0.45 nW/ |

3G (BTx) | 2110 MHz ∼ 2170 MHz | 12 nW/ |

3G (MTx) | 1920 MHz ∼ 1980 MHz | 0.46 nW/ |

WiFi | 2400 MHz ∼ 2500 MHz | 6 nW/ |

Summary of performance of variety of up-to-date antenna design

Literature | Antenna type | Designed band | Antenna gain | Return loss |
---|---|---|---|---|

M. Arrawatia, et al. [39] | Differential Microstrip | 0.87 GHz ∼ 1.05 GHz | 8.5 dBi | 17.2984 dB |

S. Ghosh, et al. [40] | Microstrip | 0.935 GHz ∼ 0.96 GHz | 4.0947 dB | 12.82 dB |

H. Saghlatoon, et al. [41] | Planar Monopole | 0.6 GHz ∼ 1.5 GHz | 3.432 dBi | 27.5 dB |

A. Dadgarpour, et al. [42] | Bow-tie | 2.5 GHz ∼ 3.9 GHz | ≥ 10.9 dBi | 27.68 dB |

M. W. Zeng, et al. [10] | Fractal Loop | 1.73 GHz ∼ 1.84 GHz | 3.2 dBi | 31.25 dB |

M. Arrawatia, et al.[11] | Triangular Monopole | 0.85 GHz ∼ 1.94 GHz | ≥ 2 dBi | 19.085 |

J. Wen, et al.[13] | Dipole | 1.7 GHz ∼ 3.6 GHz | 9.05 dBi | 33.75 dB |

D. Yang, et al.[14] | Planar Quasi-Yagi | 3.15 GHz ∼ 10.65 GHz | 7.6 dBi | 24.714 dB |

R. Maher, et al. [15] | Planar | 2.1 GHz ∼ 7GHz | 11.4585 dBi | 44.1499 |

J. Y. Li, et al. [16] | Dipole | 2.48 GHz ∼ 9.51GHz | ≥ 6dBi | 19.08 |

K. P. Esselle, et al. [17] | Microstrip | 4 GHz ∼ 9.5GHz | ≥ 7.4 dBi | 35 |

W. Han, et al. [43] | Circular Microstrip | 6.12 GHz ∼ 6.84 GHz | 8.7 dBi | 14 dB |

## Method

In a RF energy harvesting system, the antenna, which is responsible for receiving RF signals over a certain frequency band, is an important element in the design of the RF energy harvester. The design of the antenna involves several parameters, some of which induce contradictory modifications of antenna performances. In this paper, a new mathematical weighted evaluation model involving antenna efficiency, center frequency, and bandwidth is proposed to evaluate the performance of a rectangular microstrip patch antenna (RMPA) based on both the transmission-line model and the cavity model. With the mathematic modeling of a RMPA, three bio-inspired optimization technologies including the Cuckoo Search (CS) algorithm, the Differential Evolution (DE) algorithm, and Quantum-behaved Particle Swarm Optimization (QPSO)are used to optimize the design of RMPA with certain constraints. The simulation processes and designed antennas’ performances are also presented and compared. With the evaluation model as the objective function, bio-inspired optimization approaches are utilized to determine the geometrical parameters of the optimal antenna based on given constraints.

The paper is organized as follows:“Mathematic modeling of RMPA”section introduces the basic architecture and properties of a RMPA, and a mathematical model is proposed based on the cavity and transmission line models. Additionally, the objective function and constraints for optimization are derived. The“Bio-inspired algorithms for antenna design” section describes the current implementation of the optimal antenna design algorithms including QPSO, CS, and DE. Finally, based on the performance of designed antennas, the properties of three bio-inspired algorithms are discussed and compared.

## Mathematic modeling of RMPA

*P*

_{ out }) over input power (

*P*

_{ in }). Conventionally, the harvested RF power from the rectenna is expressed as follows:

where the *η*_{
a
},*η*_{
m
} and *η*_{
r
} respectively represent the efficiency of receiving antenna, impedance matching network, and rectifying circuit. The magnitude of *P*_{
T
} is correlated with the design frequency band of antenna. Accordingly, the optimum antenna design for a RF harvesting system requires two features: high antenna efficiency and appropriate frequency band. In more detail, the performance of the antenna used for a RF energy harvester mainly depends on the antenna gain, bandwidth, return loss, and center frequency. However, there is a trade-off between antenna size and performance.

### Architecture of a RMPA

*L*

_{ d }) as shown in Fig. 2, and it consists of three layers including patch, substrate, and ground plane. Normally the patch and microstrip feed line are fabricated on the upper surface of the dielectric substrate, and a metal ground plane is placed on the bottom. The four most popular feeding configurations are: the microstrip line, coaxial probe, aperture coupling, and proximity coupling, among which the inset microstrip feed line has the simplest configuration to implement and control [9] By far, the rectangular patch is the most widely used microstrip antenna, and can be modeled and analyzed by both transmission line model and cavity theory. The mathematical formulations of the RMPA have been demonstrated and derived in the following section, where the notations used in the model are as shown in Tables 3 and 4.

Nomenclature of important parameters

F | Center frequency of antenna |
A
| Antenna efficiency of antenna |

B | Band width of antenna |
| Return loss of antenna |

| Input power of rectenna |
| Output power of rectenna |

| Efficiency of antenna |
| Efficiency of matching circuit |

| Efficiency of rectifier circuit |
| Transmitted power from source |

| Width of RMPA |
| Length of RMPA |

| Feed width of RMPA |
| Inset feed length |

| Notch width |
| Vacuum dielectric constant |

| Minimum constraint of antenna size |
| Maximum constraint of antenna size |

| Vacuum permeability |
| Antenna input impedance |

| Voltage reflection coefficient | VSWR | Voltage standing wave ratio |

| Quality factor due to radiation losses |
| Quality factor due to conduction losses |

| Quality factor due to dielectric losses |
| Quality factor due to surface waves |

| Total quality factor | tan | Loss tangent of the substrate material |

| Antenna gain |
| The speed of EM wave |

| Spatial angular frequency of wave |
| Absolute gain |

| Radiation power of antenna |
| Maximum radiation intensity |

| Directivity of antenna with single slot |
| Directivity of array factor AF |

| Efficient bandwidth ratio |
| Aimed lower conner frequency |

| Aimed upper conner frequency |
| Designed lower conner frequency |

| Designed upper conner frequency |
| Conductivity of patch |

| Directivity of antenna |
| ntenna Radiation efficiency |

Nomenclature of important parameters

\(\hbar \) | Planck’s constant |
---|---|

| Potential energy distribution |

Mbest | Best position of particle in the direction d |

| Contraction-expansion coefficient |

| Local attractor of particle j in direction d |

| Gaussian distributed number within [0,1] |

| Random value within [0,1] |

### Problem formulation

*w*

_{1}

*w*

_{2}

*w*

_{3}] is a weight matrix. Moreover,

*δ*

*F*,

*δ*

*B*,

*δ*

*A*

_{ e }, and

*δ*

*R*

_{ L }shows the deviation between the performance of designed antenna and the performance of the optimal antenna. The inequality geometry constraints are five groups of inequations which can be obtained from the architecture of RMPA

### Input impedance of a RMPA

*Z*

_{ in }can be obtained from the equivalent transmission line circuit as shown in Fig. 3. The admittance of the antenna with two slots (

*Y*

_{1}and

*Y*

_{2}) is given by [24–26]:

where *G*_{1} = *G*_{2}, *B*_{1} = *B*_{2} and owing to that the two slots of the antenna are identical.

*P*

_{ rad }is defined as the radiated power which can be calculated by [25, 26]:

*G*

_{12}is [9, 24, 26].

*J*

_{0}is the Bessel function of the first kind of order zero, and accordingly the input resistance for the inset feed case can be approximately expressed as [26]:

### Resonant frequency of A RMPA

*ε*

_{ eff }and effective patch width

*L*

_{ eff }are introduced to take account of the fringing effect. Accordingly, the resonant frequency of the dominant mode of the RMPA is derived as [9, 24, 25].

*ε*

_{ eff }and effective patch length

*L*

_{ eff }can be calculated by:

*Δ*

*L*is the normalized extension of the length and given as:

### Antenna gain of a RMPA

Hence, antenna gain depends on radiation efficiency as well as directivity. The two parameters are calculated separately as follows.

*G*

_{ t }/

*l*and

*K*can be calculated by [9].

*Q*

_{ c }and the loss by surface wave

*Q*

_{ sw }can be ignored. Thus, the total loss can be approximately expressed as:

*e*

_{ cd }, the radiation efficiency of an antenna, can be derived through the quality factor as:

*D*

_{0}is defined as [24]:

*D*

_{ AF }is calculated as follows [9]:

### Antenna efficiency of a RMPA

*I*

*R*

^{2}losses is used to consider losses at the input terminals and within the antenna. Thus, the antenna efficiency can be written as:

*Γ*which can be calculated using:

*Ω*, the return loss can be expressed by Eq. (34), The input independence of the RMPA has also been calculated using Eq. (12).

### Bandwidth of a RMPA

*e*

_{ BW }is proposed mathematically as:

## Bio-inspired algorithms for antenna design

### CS algorithm for antenna design

*P*

_{ a }. The general system-equation of the CS is based on lévy flight as [29]:

*x*

^{ t }is the current solution, and

*x*

^{t+1}is the newly generated solution. Additionally, the \(l\acute {e}vy(\lambda)\) follows the \(l\acute {e}vy\) distribution [29].

*α*follows the Mantegna Algorithm in which the step size can be obtained by [29, 30]:

*β*is set to 1, the above two formula reduce to Eq. (46).

Therefore, the overall procedure of optimizing the antenna design for a RF harvesting system is shown as Algorithm 1, which has been implemented by MATLAB and Python.

### Differential evolution for RMPA design

*j*th parameter in the

*i*th individual at generation zero is given by [33].

*V*

_{i,G}are the mutant solutions generated by the current solutions including \(X_{r_{1}^{i},G}\), \(X_{r_{2},G}\), and \(X_{r_{3},G}\). Here F is a positive control factor. After the mutation, the crossover operation is applied to generate more potential new solutions based on

*V*

_{i,G}and

*X*

_{i,G}[32]:

Finally, the locally optimal solutions will be selected and the global optimum value can be found. Based on the four steps mentioned, the overall pseudo code also implemented in MATLAB and Python is shown as Algorithm 2.

### QPSO algorithm for RMPA design

The evolutionary particle swarm optimization (PSO) is a global search technique with incomparable advantages in searching speed and precision. It was originally introduced by Kennedy and Eberhar [34–36]. The basic idea of PSO is inspired by the social behavior of interactions between members including birds and fish. There are three main attractive features of PSO: robust search ability, fast computation and easy implementation [34, 35, 37]. In addition to its advantages, it has a slow solution fine-tuning ability of the solution, which sucks the solution towards the locally optimum value.

Accordingly, QPSO, as the modified PSO technology, was proposed to enhance the global search ability. The essential difference between the QPSO and PSO is that the movement of particles follows the principles of quantum mechanics instead of Newtonian mechanics.

During the whole procedure, the local optimal value would be updated. The overall process is described in Algorithm 3

## Results

**W**was set to [0.3,0.4,0.3]. Some assumed antenna properties have been shown in Table 5.

Some assumed properties of designed antenna

Parameter | Value |
---|---|

Height of substrate | 1.588mm |

Dielectric constant | 4.4 |

Loss tangent | 0.008 |

### Optimal antenna for GSM1800

^{ T M }i5 duo PC with 3.10 GHz CPU and 8 GB RAM.

Parameters setting of algorithms QPSO, CS, and DE

Algorithms | Parameter | Value |
---|---|---|

QPSO |
| 0.75 |

Max iterations | 500 | |

Population size | 20 | |

CS |
| 0.3 |

Number of nest | 20 | |

Max generation | 500 | |

| 1 | |

DE | Mutation strategy | DE/rand-to-best/1 |

Crossover strategy | binomial crossover | |

F | 0.5 | |

CR | 0.3 | |

Numbers of population | 20 | |

Max generation | 500 |

### Test case 1: antenna size should be less than 100 mm

Summary of the properties of designed antenna based on QPSO, CS, and DE for antenna size less than 100 mm

Property | DE | CS | QPSO |
---|---|---|---|

Antenna gain (dBi) | 3.01 | 2.812 | 2.932 |

Antenna effciency | 0.445 | 0.425 | 0.435 |

Efficient bandwidth ratio | 0.3463 | 0.3263 | 0.3383 |

Patch width (mm) | 99.711 | 90.529 | 96.483 |

Patch length (mm) | 37.448 | 37.664 | 37.571 |

Inset feed length (mm) | 9.032 | 27.279 | 27.354 |

Notch width (mm) | 0.256 | 0.2574 | 0.2568 |

Run time (s) | 66.2068 | 100.621 | 59.156 |

### Test case 2: antenna size should be less than 150mm

Comparison between the properties of designed antenna based on QPSO, CS, and DE for antenna size less than 150 mm

Property | DE | CS | QPSO |
---|---|---|---|

Antenna gain (dBi) | 3.82 | 3.78 | 3.736 |

Antenna effciency | 0.4902 | 0.45 | 0.483 |

Efficient bandwidth ratio | 0.425 | 0.4115 | 0.411 |

Patch width (mm) | 150 | 147.1 | 149.04 |

Patch length (mm) | 37.294 | 37.285 | 37.75 |

Inset feed length (mm) | 2.2649 | 29.21 | 3.23 · 10 |

Notch width (mm) | 0.2549 | 0.2549 | 0.0374 |

Run time (s) | 92.181 | 99.606 | 66.2068 |

## Discussion

## Conclusion

The general anatomy of an RF energy harvester has been explained and examined. Some state of the art designs for receiving antenna including both narrow-band and broad band antennas have been introduced. In the second part, a mathematical weighted evaluation model involving antenna efficiency, center frequency, bandwidth, and gain was proposed to evaluate the performance of a RMPA for a RF harvesting system.The heuristic optimization approaches CS, DE, and QPSO were introduced and then utilized to give optimal designs for a GSM1800 receiving antenna. The proposed optimization algorithms successfully achieved optimal solutions under two different antenna size constraints. The overall comparison between QPSO, DE, and CS showed that the DE based optimization design approach provides the best solution and hence has the best globally optimum value search ability. This can significantly enhance antenna performance. Moreover, among the three optimization approaches, the CS algorithm has the best robustness. Furthermore, the simulations using QPSO based algorithm indicate its superiority displaying a faster convergence speed than DE and CS in the application of solving a complex electromagnetic problem.

## Declarations

### Acknowledgements

Not applicable.

### Funding

Not applicable.

### Availability of data and materials

All data generated or analyzed during this study are included in this paper and reference list.

### Ethics approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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