# VKECE-3D: Energy-Efficient Coverage Enhancement in Three-Dimensional Heterogeneous Wireless Sensor Networks Based on 3D-Voronoi and K-Means Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

- To improve the network homogeneity, a secondary deployment of nodes using a highly destructive polynomial mutation strategy is proposed based on the idea of mutation characteristics;
- After the elbow method determines the quantity of clusters k, the nodes after secondary deployment are divided into several clusters using K-means, and the optimal sensing radius of the computational unit is calculated using 3D-Voronoi partition to lower the quantity of active nodes and improve the network’s QoS;
- Network data communication is divided into single-hop communication and multi-hop communication. The polling working mechanism is established according to the distance of the centroid from near to far. All nodes outside the unit centroid are dormant to lower the nodes’ energy consumption and lengthen the network life cycle.

## 2. Related Works

## 3. System Model and Term Definition

#### 3.1. Network Model

**Hypothesis**

**1.**

**Hypothesis**

**2.**

**Hypothesis**

**3.**

**Hypothesis**

**4.**

**Hypothesis**

**5.**

#### 3.2. Perceptual Model

#### 3.3. Energy Consumption Model

#### 3.4. Description of Connectivity

#### 3.5. Related Concepts

**Definition**

**1 (Joint sensing probability).**

**Definition**

**2 (Coverage).**

**Theorem**

**1.**

## 4. Energy-Efficient Coverage Enhancement Algorithm VKECE-3D

#### 4.1. Highly Destructive Polynomial Mutation Strategy

#### 4.2. K-Means Clustering

- Select $k$ sample data points at random from the sample data set as the primary cluster center $a={a}_{1},{a}_{2},{a}_{3},\dots ,{a}_{k}$;
- The Euclidean distance is used to compute the distance between the sample data ${x}_{i}$ and the center of each cluster, and the clusters with the smallest distance between them are partitioned;
- After one round of division, recalculate the cluster center of each cluster ${a}_{j}$ using Formula (14);$${a}_{j}=\frac{1}{\left|{C}_{i}\right|}{\displaystyle \sum _{x\in {C}_{i}}x},$$
- Repeat the above two operations until the partition result is unchanged.

#### 4.3. 3D-Voronoi Partition

- The distance from each center of mass to its 3D-Voronoi vertex is calculated, and the cell’s optimal perceptual radius is confirmed by the greatest distance between them.
- Quantity of 3D-Voronoi cells is equal to the minimum quantity of active nodes needed by the network to achieve the QoS standards. The best deployment position of the active node is defined as the cell’s centroid in 3D-Voronoi space.

#### 4.4. Multi-Hop Communication and Polling Working Mechanism

#### 4.5. VKECE-3D Algorithm

Algorithm 1: VKECE-3D |

Input: Location of N sensor nodes; Maximum number of iterations ${T}_{max}$Output: Deployment Solutions1: In the target area R, a random deployment method is taken to initialize N sensor nodes. 2: Secondary deployment using highly destructive polynomial mutation strategy nodes 3: for (t = 1, 2…,${T}_{max}$)4: Determine the clustering algorithm hyperparameter $k$ using the elbow method 5: Cluster the nodes using K-means and find the clustering centers 6: Perform 3D-Voronoi partitioning based on clustering centers 7: The optimal perceptual radius of a cell is defined as the farthest distance from the centroid to the vertex of the 3D-Voronoi cell 8: Within each 3D-Voronoi cell, the node closest to the center of mass moves to the center of mass and starts working, while the rest of the nodes are dormant 9: Verify network connection, Kruskal’s algorithm constructs a minimal spanning tree 10: If the coverage of the network ${R}_{cov}$ is higher than 90% and the connection of the network satisfies the criteria, stop the loop; otherwise, continue 11: Return best deployment solutions 12: The multi-hop communication and polling working mechanisms start to take effect. |

#### 4.6. Complexity Analysis

## 5. Simulation Result Analysis

#### 5.1. Experimental Environment

#### 5.2. Method of Comparison

- GHND [20]: GHND (Grid-based Hybrid Network Deployment) is a representative algorithm to ensure energy-efficient deployment schemes based on grid hybrid deployment to upgrade the energy efficiency and load balancing of WSNs. In the network, a technique of splitting and merging is proposed to enhance the uniformity of sensors. The network can enhance the best load balance state by merging the low-density adjacent areas and splitting the high-density adjacent areas. Benefits in load balancing, network life cycle, and efficiency are all high when using this strategy.
- CSPM [39]: CSPM (Cuckoo Search with highly disruptive Polynomial Mutation) is initially designed to improve the problem of scheduling algorithms in cloud computing environments tending to fall into local optima. Its application to WSN node deployments improves node uniformity and thus the coverage of the network.
- CS [43]: CS (Cuckoo Search Algorithm) is a representative of a traditional swarm-intelligence algorithm to deal with 3D problems in ascending dimensions and is a bionic optimization algorithm that simulates the reproductive characteristics of the aggressive nest-seeking and egg-hatching of cuckoos. It has the advantages of setting fewer parameters, a simple process, and fast convergence and has good robustness for many optimization problems. It has been widely used in industrial design and in other practical problems.
- MACHPS [13]: An example of a swarm intelligence algorithm combined with a sleep-wake mechanism is MACHPS (Mobile Assisted Coverage Hole Patching Scheme). It has the potential to vastly improve network coverage and extend the useful life of a network. Before anything else, a scattershot network of sensor nodes is placed across the region of interest and left there to remain motionless or in a sleep state. Second, we partition the whole network into squares called “grids,” and we calculate the coverage percentage for each grid. As a candidate grid, we choose the one with the lowest coverage. Particle swarm optimization (PSO) is then utilized to estimate the mobile locations of sensor nodes, waking up dormant mobile sensors to fill in the coverage hole.
- IPSO-IRCD [44]: IPSO-IRCD is a two-stage coverage optimization strategy that addresses an enhanced particle swarm algorithm with a node coverage schedule. In order to discover potential node placements, IPSO computes the particle fitness value and compares it to the prior ideal value, so avoiding the issue of losing the optimal solution of the original method. To find the best fit between nodes and candidate sites without compromising network coverage, IRCD, an optimization method, takes into account both the coverage increments and the move distance to make its selections. It is a great solution for wireless sensor networks’ coverage redundancy and hole problems.

#### 5.3. Operation Process Simulation

#### 5.3.1. Case 1: Highly Destructive Polynomial Mutation Strategy for the Secondary Deployment of Nodes

#### 5.3.2. Case 2: K-Means Clustering to Determine the Minimum Quantity of Nodes to Be Deployed in the Network

#### 5.3.3. Case 3: Moving the Sensor Nodes to the Optimal Position and Verifying Network Connectivity

#### 5.3.4. Case 4: Multi-Hop Communication and Polling Working Mechanism to Lengthen the Network’s Life Cycle

#### 5.4. Performance Testing

#### 5.4.1. Coverage Performance Testing

#### 5.4.2. Proportion of Dead Nodes in Different Rounds

#### 5.4.3. Activity Node Ratio

## 6. Conclusions and Future Directions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Schematic of Voronoi’s optimal sensing radius coverage: (

**a**) Two-dimensional case; (

**b**) Three-dimensional case.

**Figure 3.**Network uniformity effect diagram: (

**a**) Chaotic mapping; (

**b**) Highly destructive polynomial mutation strategy.

**Figure 4.**3D-Voronoi partition process diagram: (

**a**) Original 3D-Voronoi partition; (

**b**) Display centroid 3D-Voronoi partition.

**Figure 7.**Original and secondary deployment diagrams: (

**a**) Random deployment of nodes effect diagram of the small network; (

**b**) Highly destructive polynomial mutation strategy effect diagram of the small network; (

**c**) Random deployment of nodes effect diagram of the medium-sized network; (

**d**) Highly destructive polynomial mutation strategy effect diagram of the medium-sized network; (

**e**) Random deployment of nodes effect diagram of the large network; (

**f**) Highly destructive polynomial mutation strategy effect diagram of the large network.

**Figure 8.**The Elbow method determines the clustering hyperparameter $k$: (

**a**) the small network; (

**b**) the medium-sized network; (

**c**) the large networks.

**Figure 9.**K-means clustering process: (

**a**) The small networks: clustering raw data and the center of mass; (

**b**) The small networks: K-means clustering results; (

**c**) The medium-sized networks: clustering raw data and the center of mass; (

**d**) The medium-sized networks: K-means clustering results; (

**e**) The large networks: clustering raw data and the center of mass; (

**f**) The large networks: K-means clustering results.

**Figure 10.**3D-Voronoi partition results and minimal spanning tree: (

**a**) The small networks: 3D-Voronoi partition results and center of mass; (

**b**) The small networks: minimal spanning tree; (

**c**) The medium-sized networks: 3D-Voronoi partition results and center of mass; (

**d**) The medium-sized networks: minimal spanning tree; (

**e**) The large networks: 3D-Voronoi partition results and center of mass; (

**f**) The large networks: minimal spanning tree.

**Figure 11.**Different rounds of coverage in small networks: (

**a**) Network runs 50 rounds; (

**b**) Network runs 5000 rounds; (

**c**) Network runs 10,000 rounds; (

**d**) Network runs 15,000 rounds.

**Figure 12.**Different rounds of coverage in medium-sized networks: (

**a**) Network runs 50 rounds; (

**b**) Network runs 5000 rounds; (

**c**) Network runs 10,000 rounds; (

**d**) Network runs 15,000 rounds.

**Figure 13.**Different rounds of coverage in large networks: (

**a**) Network runs 50 rounds; (

**b**) Network runs 5000 rounds; (

**c**) Network runs 10,000 rounds; (

**d**) Network runs 15,000 rounds.

**Figure 14.**Coverage of our proposed algorithm vs. comparison algorithms: (

**a**) the small network, N = 100; (

**b**) the medium-sized network, N = 250; (

**c**) the large networks, N = 500.

**Figure 15.**Node mortality of our proposed algorithm vs. comparison algorithms: (

**a**) the small network, N = 100; (

**b**) the medium-sized network, N = 250; (

**c**) the large networks, N = 500.

**Figure 16.**Determining the optimal ratio of activity nodes: (

**a**) Coverage of different activity node ratios; (

**b**) Total network energy consumption of different activity node ratios.

Notations | Descriptions |
---|---|

R | target area |

${R}_{s}$ | sensing range |

${R}_{c}$ | communication range |

${R}_{cov}$ | coverage |

$k$ | quantity of K-means algorithm cluster |

SSE | sum of the squared errors |

Q | 3D-Voronoi cell |

BS | distributed base station |

Parameter List | Value of Parameter |
---|---|

Size of the target area | 50 × 50 × 50 ${\mathrm{m}}^{3}$, 200 × 200 × 200 ${\mathrm{m}}^{3}$, 600 × 600 × 600 ${\mathrm{m}}^{3}$ |

Iteration number | 250 |

Number of nodes deployed randomly | 100, 250, 500 |

Initial energy of every sensor | 5 J |

Threshold battery power | 0.02 J |

Circuit unit energy consumption $\left({E}_{elec}\right)$ | 50 nJ/bit |

Free space channel parameter (${\epsilon}_{fs}$) | 10 pJ/bit/m^{3} |

Multi-path channel parameter (${\epsilon}_{amp}$) | 0.0013 pJ/(bit/m^{4}) |

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## Share and Cite

**MDPI and ACS Style**

Gou, P.; Guo, B.; Guo, M.; Mao, S.
VKECE-3D: Energy-Efficient Coverage Enhancement in Three-Dimensional Heterogeneous Wireless Sensor Networks Based on 3D-Voronoi and K-Means Algorithm. *Sensors* **2023**, *23*, 573.
https://doi.org/10.3390/s23020573

**AMA Style**

Gou P, Guo B, Guo M, Mao S.
VKECE-3D: Energy-Efficient Coverage Enhancement in Three-Dimensional Heterogeneous Wireless Sensor Networks Based on 3D-Voronoi and K-Means Algorithm. *Sensors*. 2023; 23(2):573.
https://doi.org/10.3390/s23020573

**Chicago/Turabian Style**

Gou, Pingzhang, Baoyong Guo, Miao Guo, and Shun Mao.
2023. "VKECE-3D: Energy-Efficient Coverage Enhancement in Three-Dimensional Heterogeneous Wireless Sensor Networks Based on 3D-Voronoi and K-Means Algorithm" *Sensors* 23, no. 2: 573.
https://doi.org/10.3390/s23020573