# A Power Balance Aware Wireless Charger Deployment Method for Complete Coverage in Wireless Rechargeable Sensor Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Section

#### 2.1. Problem Statement

#### 2.1.1. The WRSN Model

_{1},s

_{2},…,s

_{X}}. The data sink node, which is labeled as Y, exists in Z. The locations of the sensors and data sink node are known. The objective is to deploy M directional chargers. Define M directional chargers as A = {a

_{1},a

_{2},…,a

_{M}}. Each charger can be deployed anywhere in the region, and a given directional charger can be placed at any angle. Assume that the chargers must cover a subset of the sensors S, regardless of the fact that all the chargers have the same charging efficiency. Past studies have failed to consider the existence of the data sink node and this defect can have significant impacts on the wireless charger deployment. In Section 2.1.3, we will explain the difference between charging efficiency and charging demand. According to the study by Garey and Johnson [16], the deployment of the chargers is a nonlinear programming problem, so it belongs to the category of non-deterministic polynomial-time hard (NP hard) problems and thus it is challenging to solve the charger deployment problem.

#### 2.1.2. Coverage Area and Charging Power Model

_{i}with angle θ charges the sensor s

_{j}, and the radius is r. The arrow is directed out from the face of the charger. $\overrightarrow{\mathrm{v}}$ is a unit vector of the charger face, and $\stackrel{\rightharpoonup}{u}$ is the vector formed by the charger and the sensor. According to Ai and Abouzeid [13], the following Equation can be used to determine whether the sensor s

_{j}is in the coverage area.

_{i},s

_{j}) as the Euclidean distance between the location of charger a

_{i}and the location of sensor s

_{j}, and $\alpha =\frac{\xi {W}_{0}{G}_{a}{G}_{s}}{{C}_{W}}{(\frac{\lambda}{4\pi})}^{2}$ where λ is the wavelength, and ζ is the efficiency of the rectifier. G

_{a}and G

_{s}are the antenna gain of the charger and the sensor, respectively. W

_{0}is the charging power, C

_{w}is the antenna polarization loss, and β is the compensation parameter for short range transmission [17]. When the distance is larger than the charging radius r, the charging power is 0. According to Dai et al. [18], although the total power may not be precise, the total power can still be expressed as a function of the following Equation (3).

#### 2.1.3. Charging Efficiency and Charging Demand

_{max}is the charging efficiency limit of the sensor. In fact, for different brands of sensors, the values of σ and W

_{max}are different. Here, however, we assume that all sensors have the same σ and W

_{max}in the sensing region. From Equation (4), we can know that the charger does not have unlimited charging efficiency, so there is no instant charging in this setting. The sensors must have enough charge to maintain their normal operations, and if there is excess charge power, the excess power will be wasted. In a real environment, the charging demand of a sensor is related to the distance between the sensor and the data sink node, so we define charging demand as Equation (5). If the distance to the sink node is f, the charging demand is γ and μ is added/subtracted when the distance is greater/smaller than f.

#### 2.2. Method

#### 2.2.1. Area Discretization of Charging Power and Charging Demand

#### 2.2.2. Searching for Minimal Dominating Set

**Definition 1.**

_{i}and S

_{j}, if S

_{i}⊆ S

_{j}, then S

_{j}dominates S

_{i}.

**Definition 2.**

_{j}in the region, then S

_{j}will be determined to be the optimal dominating set in the region.

Algorithm 1: Calculate the Coverage Dominating Sets of Given Sensors |

Input: Directional charger a_{i}, angle θ and the sensor set S_{i} with the locations of all the sensors |

Output: Coverage Dominating Sets CDSs |

Step 1. Calculate the included angle between the sensor and the charger; |

Step 2. Sort sensors using angles, from small to large. θ_{1} ≤ θ_{2} ≤…≤ θ_{X} with respect to sensors s_{1}, s_{2},…s_{X}; |

Step 3. Set a parameter S_{t} to record the current coverage set. Two parameters, θ_{min} and θ_{max}, record the minimum and maximum of the current coverage set. Initialize θ_{min} = θ_{1}, θ_{max} = θ_{1}; S_{t} = {s_{1}}; |

Step 4. Starting from second sensors, j = 2; |

Step 5. Rotate the chargers until no more sensors can be covered. |

While (θ_{j} − θ_{min} ≤ θ){ |

θ_{max} = θ_{j}; |

Add s_{j} to S_{t}; |

j = (j + 1) mod X; |

} |

Step 6. Add the current S_{t} to the cover dominating sets CDSs; |

Step 7. Let θ_{max} = θ_{j} and add s_{j} to S_{t}; |

Step 8. Remove the sensor from the smallest angle until the current set can be covered again |

While(θ_{max} − θ_{min} > θ){ |

From the S_{t} to remove the minimum angle sensor, and θ_{min} update into the current minimum angle value; |

If(θ_{min}=θ_{1}){ |

End;} |

} |

Go to Step 5. |

_{2}, s

_{3}} and then discovers three more dominating sets, {s

_{3}, s

_{4}}, {s

_{4}, s

_{5}, s

_{6}}, and {s

_{5}, s

_{6}, s

_{7}, s

_{8}}.

Algorithm 2: Search for the Best Dominating Set |

Input: Coverage Dominating Sets CDSs |

Output: The Best Dominating Set BDS |

Step 1: Calculate the total power P and the charging efficiency W of all Coverage Dominating Sets CDSs using Equations (3) and (4) |

Step 2: Calculate the charging demand H of all Coverage Dominating Sets CDSs using Equation (5) |

Step 3: Select the best dominating set BDS with minimum H-W |

## 3. Results

#### 3.1. Description of Parameter Setting and Comparisons

_{1}= 80, β

_{1}= 40, d

_{1}= 16, and θ

_{1}= 90. The upper bound value of the average charge demand f

_{up}= 30 m and f

_{low}= 10 m. W

_{max}= 40 mW, σ = 1, μ = 4 mW, and γ = 28 mW.

_{2}= 70, β

_{2}= 50, d

_{2}= 10, and θ

_{2}= 360.

#### 3.2. Experimental Results

## 4. Discussion

#### 4.1. Effects of the Sensor Quantify on Charger Quantity

#### 4.2. Effects of the Adjustment Coefficient of Charging Demand μ

#### 4.3. Effects of the Charging Angle θ

#### 4.4. Charging Efficiency Performance in the Real Case of Each Sensor

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 8.**(

**a**) The deployment of PBAD (power balance aware deployment) approach; (

**b**) The deployment of the RPRO (random position and random orientation) approach; (

**c**) The deployment of the LMD (location merging design) approach.

Name | Type | Contribution | Power Balance Aware |
---|---|---|---|

Energy provisioning in wireless rechargeable sensor networks. [10] | Omnidirectional charger | Physical characteristics of wireless charging | No |

Mobility-aware charger deployment for wireless rechargeable sensor networks. [11] | Omnidirectional charger | A sensing model for optimization of mobile deployment | No |

Optimized charger deployment for wireless rechargeable sensor networks. [12] | Omnidirectional charger | Charger sleep scheduling | No |

Minimizing charging delay in wireless rechargeable sensor networks. [15] | Omnidirectional charger | Robot stop locations in discrete areas | No |

Approximating optimal visual sensor placement. [13] | Directional charger | Grid linear programming | No |

Deploying directional sensor networks with guaranteed connectivity and coverage. [14] | Directional charger | Considers the fact that the sensors are connected to each other | No |

Symbol | Corresponding Meaning |
---|---|

Z | Selected 2D region |

x | Number of sensors |

s_{j} | The sensor j |

m | The number of chargers |

a_{i} | The charger i |

Y | The data sink node |

θ | Charging angle |

r | Charger radius |

$\overrightarrow{\mathrm{v}}$ | Direction vector of a charger |

$\stackrel{\rightharpoonup}{u}$ | The vector of the charger and sensor |

σ | The factor affecting charging effect |

W_{max} | Upper bound value of charging power |

γ | The average charging demand |

μ | The charging demand adjustment coefficient |

f | The specific distance which influences the charging demand |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Lin, T.-L.; Li, S.-L.; Chang, H.-Y.
A Power Balance Aware Wireless Charger Deployment Method for Complete Coverage in Wireless Rechargeable Sensor Networks. *Energies* **2016**, *9*, 695.
https://doi.org/10.3390/en9090695

**AMA Style**

Lin T-L, Li S-L, Chang H-Y.
A Power Balance Aware Wireless Charger Deployment Method for Complete Coverage in Wireless Rechargeable Sensor Networks. *Energies*. 2016; 9(9):695.
https://doi.org/10.3390/en9090695

**Chicago/Turabian Style**

Lin, Tu-Liang, Sheng-Lin Li, and Hong-Yi Chang.
2016. "A Power Balance Aware Wireless Charger Deployment Method for Complete Coverage in Wireless Rechargeable Sensor Networks" *Energies* 9, no. 9: 695.
https://doi.org/10.3390/en9090695