Enhanced Grey Wolf Optimization for Efficient Transmission Power Optimization in Wireless Sensor Network

Abstract

The Internet of Things (IoT) and Wireless Sensor Networks (WSNs) heavily rely on the lifetime of sensor nodes, which is inversely proportional to transmission power. Nodes with greater separation demand higher transmission power, while those closer together require less power. In practice, node placement varies significantly due to diverse terrain and contours, making power transmission configuration a critical and challenging issue in WSNs. This paper introduces an Enhanced Grey Wolf Optimization (EGWO) algorithm designed to optimize power transmission in WSN environments. Traditional Grey Wolf Optimization (GWO) employs a parameter that decreases linearly with iterations to regulate exploitation. In contrast, the proposed EGWO adopts a concave decline in the exploitation rate, allowing for more precise optimization in areas under exploration. The enhancement utilizes a cosine function that gradually decreases from 1 to 0, providing a smoother and more controlled transition. The experimental results demonstrate that EGWO outperforms other optimization algorithms. The proposed method achieves the lowest fitness value of −4.21, compared to 1.22 for standard GWO, −2.81 for PSO, and 2.86 for BESO, indicating its superiority in optimizing power transmission in WSNs.

1. Introduction

The global wireless sensor market is projected to reach USD 27.79 billion by 2025 and is expected to grow to approximately USD 150.43 billion by 2034, reflecting a compound annual growth rate (CAGR) of 20.71% between 2025 and 2034. In North America, the market size was estimated at USD 8.93 billion in 2024 and is anticipated to expand at a CAGR of 20.75% during the forecast period. The market estimates and projections are based on revenue (USD million/billion), with 2024 as the base year [1]. This growth is attributed to the increasing adoption of industrial automation, advancements in wireless communication technologies, and the rising demand for real-time monitoring solutions. Industries are leveraging these networks to enhance operational efficiency, reduce costs, and improve predictive maintenance capabilities. The widespread deployment of Wireless Sensor Networks is also fueled by their scalability and flexibility, making them a critical component in the evolution of Industry 4.0 [2].

While integrating Wireless Sensor Networks (WSNs) into IoT ecosystems offers significant benefits, several challenges hinder optimal performance. A key issue is power consumption, as sensor nodes rely on limited energy sources. In applications like environmental monitoring and smart agriculture, frequent battery replacements are impractical. Energy-efficient protocols and solar-powered sensors are being explored to extend network longevity.

Data processing is another challenge, as WSNs generate vast amounts of real-time data. In smart city traffic monitoring, excessive data transmission can strain network bandwidth and drain energy. Edge computing and data aggregation techniques help reduce transmission overhead while maintaining efficiency.

Security vulnerabilities further threaten WSN reliability, especially in healthcare, where sensitive patient data must be protected. Traditional encryption is often too computationally demanding, making WSNs susceptible to attacks. Lightweight encryption schemes and blockchain-based security models offer potential solutions.

Addressing these challenges through energy-efficient designs, optimized data handling, and robust security mechanisms is crucial for ensuring the long-term sustainability of WSN-enabled IoT applications [3].

The convergence of WSNs and IoT technologies has unlocked numerous opportunities across various industries; however, addressing energy efficiency and security challenges is crucial for their long-term viability. Since WSN nodes rely on limited power sources, efficient energy management is essential, particularly in scenarios where battery replacement is challenging. Researchers are investigating methods such as energy-efficient protocols and energy harvesting techniques to prolong operational life. Additionally, security remains a pressing issue due to the absence of standardized frameworks, prompting the adoption of lightweight encryption, blockchain solutions, and trust-based models. Advancements in these areas are vital to developing scalable, reliable, and energy-efficient IoT-driven WSN systems [4].

In WSNs, clustering is crucial for improving efficiency and extending network life. Clustering groups sensor nodes into clusters led by a cluster head (CH), which collects data from nodes and sends them to the base station, reducing communication and saving energy. Various algorithms optimize CH selection to maximize network lifetime by minimizing energy consumption and balancing load distribution. Notable algorithms include LEACH, which uses random CH rotation [5]; HEED, focusing on residual energy and neighbor density [6]; PEGASIS, forming a sensor chain to reduce energy use [7]; and DEEC [8] and TEEN [9], offering innovative CH selection strategies based on energy and thresholds.

In WSNs, swarm algorithms significantly optimize network performance. Inspired by natural behaviors, these algorithms regulate node interactions within clusters, enhancing data collection and transmission while reducing energy consumption. For instance, Particle Swarm Optimization dynamically adjusts routes within clusters, improving communication efficiency and reducing CH load [10]. Ant Colony Optimization optimizes path selection by finding the shortest, most energy-efficient routes [11]. Genetic Algorithms [12] and Bee Colony Optimization [13] also improve resource allocation and management, thereby reducing energy use and increasing network life and efficiency [14].

This study addresses a research gap in WSN optimization, inspired by Silva Fre et al. [15] on energy optimization. Despite progress, there is potential for improvement in using swarm algorithms for transmission power optimization. While many strategies exist to save energy and extend node life in WSNs [16], few focus on swarm algorithms.

The main contribution of this study is the development of an Enhanced Grey Wolf Optimization (EGWO) using a non-linear decreasing exploitation parameter that allows more precise exploration in an area. The proposed EGWO is further evaluated in six different topologies with three other swarm optimization methods, namely standard GWO, Bi-directional Evolutionary Structural Optimization (BESO), and Particle Swarm Optimization (PSO). The performance of swarm algorithms is assessed using the Friis transmission power model, a standard for determining distance and signal strength in wireless communications. This model evaluates how each algorithm optimizes transmission power in WSNs to maximize energy efficiency while maintaining high communication quality.

This paper has four main sections: Section 1: Introduction, outlining the WSN background and challenges; Section 2: Methodology, describing the technical framework; Section 3: Results, presenting experimental findings; and Section 4: Conclusions, evaluating the results and future research directions.

2. Methodology

This research is based on work by Silva Fre et al. [15] on optimal energy transmission using Particle Swarm Optimization. The proposed methodology is to compare other meta-heuristic or swarm techniques to see which one yields the lowest transmission power, considering the limitations on transmit or receive capabilities. In this section, we discuss the Friis model, swarm algorithm, and experimental scenarios.

2.1. WSN Transmission Power

The Friis formula is

$$P_R={\displaystyle \frac{A_r{A}_t}{d^2{\lambda }^2}}{P}_t$$

(1)

which is a formula for receiving power with the effective cross-sectional area of an antenna (1).

$$A_{\mathrm{isotropik}}={\displaystyle \frac{{\lambda }^2}{4\pi }}$$

(2)

The formula for an isotropic antenna is

$$P_R={\left({\displaystyle \frac{\lambda }{4\pi d}}\right)}^2{P}_t$$

(3)

which is a simplified formula for receiving power with an isotropic antenna. The antenna used is isotropic (an ideal antenna that radiates with equal power in all directions) (2), so its gain is 1 or does not need to be calculated.

The Friis formula is used as an objective function to find the optimal (lowest) transmission power value while meeting operational constraints (3). The objective function includes minimizing the transmitted power ($P_t$) while ensuring that the received power ($P_R$) is sufficient for reliable communication between nodes.

To ensure connectivity between nodes, a threshold (${\rho }_{\mathrm{th}}$) check of −60 dB is used (4).

$$A=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}{P}_R\ge {\rho }_{\mathrm{th}}\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(4)

2.2. Optimization Algorithms

The following algorithms are used for the optimization enhancement and performance comparison.

2.2.1. Grey Wolf Optimizer

Grey Wolf Optimizer (GWO) is a metaheuristic optimization algorithm imitating the hunting behavior and social hierarchy of grey wolves (Canis lupus) [17]. In GWO, the population is divided into four hierarchical levels:

• Alpha ($\alpha$): The best solution, leading the search.
• Beta ($\beta$): The second-best solution, assisting $\alpha$ in leadership.
• Delta ($\delta$): The third-best solution, subordinate to $\alpha$ and $\beta$.
• Omega ($\omega$): The remaining wolves, following $\alpha$, $\beta$, and $\delta$.

The optimization process in GWO mimics the hunting strategy of grey wolves as shown in Figure 1, including encircling prey, searching for prey, and attacking prey. This mechanism is mathematically modeled as follows:

Encircling Prey

The encircling behavior of grey wolves is described by

$$\mathbf{X}(t+1)={\mathbf{X}}_p\left(t\right)-\mathbf{A}·|\mathbf{C}·{\mathbf{X}}_p\left(t\right)-\mathbf{X}\left(t\right)|,$$

(5)

where $\mathbf{X}\left(t\right)$ and ${\mathbf{X}}_p\left(t\right)$ denote the vector position of a wolf and a prey at iteration t, respectively, and the exploitation rate is controlled by coefficient A, defined as follows:

$$\mathbf{A}=2·\mathbf{a}·{\mathbf{r}}_1-\mathbf{a}.$$

(6)

This coefficient governs the wolves to concentrate in a certain area, whereas the exploration coefficient C is defined as

$$\mathbf{C}=2·{\mathbf{r}}_2.$$

(7)

This coefficient governs the random exploration vector ${\mathbf{r}}_1,{\mathbf{r}}_2\in [0,1]$. The standard GWO uses a linearly decreasing parameter $\mathbf{a}$ from 2 to 0 over iterations as shown in Figure 2.

Proposed Enhanced GWO with Non-Linear Decreasing Coefficient

This paper introduces an Enhanced GWO (EGWO) by introducing a decreasing coefficient with a non-linear function, namely a cosine function. As shown in Figure 2, the value of $\mathbf{a}$ at iteration t is given by

$$\mathbf{a}\left(\mathbf{t}\right)=2\times {\displaystyle \frac{T-t}{T}}.$$

(8)

where T denotes the maximum number of iterations. The cosine is a concave function from 0 to ${\displaystyle \frac{\pi }{2}}$ that provides a slower decreasing rate than a linear one. The proposed decreasing coefficient ${\mathbf{a}}_{\mathbf{c}}\left(\mathbf{t}\right)$ is given by

$${\mathbf{a}}_{\mathbf{c}}\left(\mathbf{t}\right)=2\times cos{\displaystyle \frac{\pi t}{2T}}$$

(9)

Lemma 1.

For $t\in [0,T]$, the inequality $a\left(t\right)<a_c\left(t\right)$ holds.

Proof. 

To prove the inequality $a\left(t\right)<a_c\left(t\right)$, consider the difference between $a_c\left(t\right)$ and $a\left(t\right)$, defined as

$$\Delta \left(t\right)=a_c\left(t\right)-a\left(t\right).$$

(10)

Substituting the expressions for $a\left(t\right)$ and $a_c\left(t\right)$, we get

$$\Delta \left(t\right)=2·cos\left({\displaystyle \frac{\pi t}{2T}}\right)-2·{\displaystyle \frac{T-t}{T}}.$$

(11)

Factoring out 2, this simplifies to

$$\Delta \left(t\right)=2\left[cos\left({\displaystyle \frac{\pi t}{2T}}\right)-{\displaystyle \frac{T-t}{T}}\right].$$

(12)

Let $f\left(t\right)=cos\left({\displaystyle \frac{\pi t}{2T}}\right)-{\displaystyle \frac{T-t}{T}}$. To analyze the behavior of $f\left(t\right)$, we evaluate it at the boundaries of the interval and examine its derivative.

At $t=0$:

$$f\left(0\right)=cos\left(0\right)-{\displaystyle \frac{T-0}{T}}=1-1=0.$$

(13)

At $t=T$:

$$f\left(T\right)=cos\left({\displaystyle \frac{\pi T}{2T}}\right)-{\displaystyle \frac{T-T}{T}}=cos\left({\displaystyle \frac{\pi }{2}}\right)-0=0.$$

(14)

To analyze the monotonicity of $f\left(t\right)$, its derivative can be computed:

$$f^{\prime }\left(t\right)={\displaystyle \frac{d}{dt}}\left[cos\left({\displaystyle \frac{\pi t}{2T}}\right)-{\displaystyle \frac{T-t}{T}}\right].$$

(15)

The derivative is

$$f^{\prime }\left(t\right)=-{\displaystyle \frac{\pi }{2T}}sin\left({\displaystyle \frac{\pi t}{2T}}\right)+{\displaystyle \frac{1}{T}}.$$

(16)

Factoring out ${\displaystyle \frac{1}{T}}$, we can rewrite it:

$$f^{\prime }\left(t\right)={\displaystyle \frac{1}{T}}\left[1-{\displaystyle \frac{\pi }{2}}sin\left({\displaystyle \frac{\pi t}{2T}}\right)\right].$$

(17)

Since $sin\left({\displaystyle \frac{\pi t}{2T}}\right)\in [0,1]$ for $t\in [0,T]$, the term

$$1-{\displaystyle \frac{\pi }{2}}sin\left({\displaystyle \frac{\pi t}{2T}}\right)>0,$$

because ${\displaystyle \frac{\pi }{2}}\approx 1.57>1$. Therefore, $f^{\prime }\left(t\right)>0$ for all $t\in [0,T]$, indicating that $f\left(t\right)$ is strictly increasing on this interval.

Since $f\left(t\right)=0$ at both $t=0$ and $t=T$, and $f\left(t\right)$ is strictly increasing on $[0,T]$, it follows that

$$f\left(t\right)>0\forall t\in (0,T).$$

This implies the following:

$$cos\left({\displaystyle \frac{\pi t}{2T}}\right)>{\displaystyle \frac{T-t}{T}}\forall t\in [0,T].$$

Equivalently,

$$a\left(t\right)<a_c\left(t\right),\forall t\in [0,T].$$

This completes the proof. □

To further validate the impact of different decay strategies for the parameter $a\left(t\right)$ in the Grey Wolf Optimizer (GWO), we conducted experiments using the Sphere function, a widely recognized benchmark function for optimization problems. This comparison focuses on four decay strategies: Linear, Quadratic, Exponential, and Cosine. The experimental setup is defined as follows:

• Problem dimensionality (D): 30;
• Number of wolves (N): 30;
• Maximum iterations (T): 100;
• Number of independent runs: 30.

The Sphere function, a standard benchmark for optimization, is mathematically defined as

$$f\left(\mathbf{x}\right)=\sum _{j=1}^D{x}_j^2,\mathrm{where}-100\le x_j\le 100.$$

(18)

The objective is to minimize $f\left(\mathbf{x}\right)$, where the global minimum is achieved at $f\left(\mathbf{0}\right)=0$. The comparative results of different decay strategies in GWO are summarized in

Table 1. Comparative analysis of different decay strategies in GWO using the Sphere function.

Method	Mean Fitness	Std Dev Fitness	Avg. Convergence Iterations	Execution Time (s)
Linear	$1.2\times {10}^{-3}$	$3.5\times {10}^{-4}$	40	0.045
Quadratic	$1.6\times {10}^{-3}$	$4.1\times {10}^{-4}$	38	0.048
Exponential	$9.0\times {10}^{-4}$	$2.8\times {10}^{-4}$	35	0.042
Cosine	$2.5\times {10}^{-3}$	$6.0\times {10}^{-4}$	48	0.050

.

The performance comparison of different decay strategies in the Grey Wolf Optimizer (GWO) reveals distinct characteristics in terms of convergence speed and optimization efficiency. Among the four tested approaches, the Cosine decay strategy exhibits the slowest convergence, requiring 48 iterations, which is significantly higher than the Exponential (35), Linear (40), and Quadratic (38) strategies. While slower convergence may seem disadvantageous in simple optimization problems, it indicates a longer exploration phase, making Cosine decay particularly suitable for problems that require extensive global search before transitioning into exploitation.

In contrast, Exponential decay converges the fastest, achieving the lowest mean fitness value ($9.0\times {10}^{-4}$) in just 35 iterations, demonstrating its aggressive reduction in the parameter $a\left(t\right)$. This rapid convergence is effective for straightforward optimization tasks where the early exploitation of promising solutions is preferred. However, in scenarios where the solution space is highly complex or contains multiple local optima, premature convergence may lead to suboptimal results.

The Cosine decay strategy mitigates this risk by maintaining a slower reduction in $a\left(t\right)$ at the beginning of the optimization process, enabling the algorithm to explore a broader range of potential solutions before committing to exploitation. This characteristic makes it particularly well suited for Wireless Sensor Networks (WSNs), especially in scenarios where sensor nodes are randomly deployed.

Hunting Behavior

The hunting strategy is modeled based on the positions of $\alpha$, $\beta$, and $\delta$, as they guide the $\omega$ wolves:

$$\mathbf{X}(t+1)={\displaystyle \frac{{\mathbf{X}}_1+{\mathbf{X}}_2+{\mathbf{X}}_3}{3}},$$

(19)

where

$${\mathbf{X}}_1={\mathbf{X}}_{\alpha }-{\mathbf{A}}_1·|{\mathbf{C}}_1·{\mathbf{X}}_{\alpha }-\mathbf{X}\left(t\right)|,$$

(20)

$${\mathbf{X}}_2={\mathbf{X}}_{\beta }-{\mathbf{A}}_2·|{\mathbf{C}}_2·{\mathbf{X}}_{\beta }-\mathbf{X}\left(t\right)|,$$

(21)

$${\mathbf{X}}_3={\mathbf{X}}_{\delta }-{\mathbf{A}}_3·|{\mathbf{C}}_3·{\mathbf{X}}_{\delta }-\mathbf{X}\left(t\right)|.$$

(22)

where ${\mathbf{X}}_{\alpha }, {\mathbf{X}}_{\beta }, {\mathbf{X}}_{\delta }$ are the positions of the top three wolves, while ${\mathbf{A}}_1, {\mathbf{A}}_2, {\mathbf{A}}_3$ and ${\mathbf{C}}_1, {\mathbf{C}}_2, {\mathbf{C}}_3$ are defined similarly to $\mathbf{A}$ and $\mathbf{C}$.

Convergence and Termination

The algorithm iteratively updates the positions of wolves until a termination criterion is satisfied, such as achieving the maximum number of iterations $T_{\max }$ or reaching a predefined fitness threshold.

Application in WSNs

In this study, GWO is applied to optimize QoS in WSNs for IoT applications. Key objectives include the following:

• Energy Efficiency: Minimizing energy consumption by optimizing cluster head selection and communication paths.
• Delay Reduction: Reducing end-to-end latency for time-sensitive IoT applications.
• Network Throughput: Enhancing packet delivery ratio and overall network performance.

The fitness function for GWO in this context is defined as

$$f=\alpha ·{\displaystyle \frac{1}{E}}+\beta ·{\displaystyle \frac{1}{D}}+\gamma ·T,$$

(23)

where

• E: Total energy consumption;
• D: Average delay in data transmission;
• T: Total network throughput;
• $\alpha , \beta , \gamma$: Weighting factors balancing the objectives.

This approach enables dynamic resource allocation and efficient network management, addressing the unique challenges of WSNs in IoT applications [18].

2.2.2. Particle Swarm Optimization

Particle Swarm Optimization (PSO) is an optimization technique inspired by the social behavior of birds or fish. In PSO, each particle, representing a potential solution, explores the search space based on its velocity and adjusts its position according to its own experience and that of its peers. This study uses PSO for inter-cluster power allocation, considering Quality-of-Service (QoS) factors and energy consumption. Additionally, Adaptive Particle Swarm Optimization (APSO) is introduced for intra-cluster resource allocation, optimizing node distance and energy load [19].

The velocity and position updates of each particle are governed by the following equations:

$$v_i^{(t+1)}=w·v_i^{\left(t\right)}+c_1·r_1·\left(p_i^{\mathrm{best}}-x_i^{\left(t\right)}\right)+c_2·r_2·\left(g^{\mathrm{best}}-x_i^{\left(t\right)}\right),$$

(24)

$$x_i^{(t+1)}=x_i^{\left(t\right)}+v_i^{(t+1)},$$

(25)

where

• $v_i^{\left(t\right)}$: velocity of particle i at iteration t;
• $x_i^{\left(t\right)}$: position of particle i at iteration t;
• $p_i^{\mathrm{best}}$: personal best position of particle i;
• $g^{\mathrm{best}}$: global best position in the swarm;
• w: inertia weight, controlling the balance between exploration and exploitation;
• $c_1, c_2$: cognitive and social coefficients, respectively;
• $r_1, r_2$: random numbers uniformly distributed in $[0,1]$.

By adjusting the parameters w, $c_1$, and $c_2$, the PSO algorithm adapts its search behavior to balance exploration and exploitation.

In this study, APSO modifies the inertia weight w dynamically to improve convergence. The adaptive inertia weight is defined as

$$w=w_{\max }-{\displaystyle \frac{(w_{\max }-w_{\min })·t}{T_{\max }}},$$

(26)

where $w_{\max }$ and $w_{\min }$ are the maximum and minimum inertia weights, t is the current iteration, and $T_{\max }$ is the maximum number of iterations.

2.2.3. Bi-Directional Evolutionary Structural Optimization

Bi-directional Evolutionary Structural Optimization (BESO) is a computational optimization method designed to achieve optimal structural designs by iteratively adding or removing materials based on performance criteria. Unlike traditional methods, BESO optimizes both material distribution and structural topology, making it particularly effective for lightweight design and multi-scale problems.

Mathematical Formulation

The optimization process in BESO aims to minimize an objective function, such as compliance or weight, while satisfying design constraints. The design domain is discretized into finite elements, where each element is assigned a density variable ${\rho }_e\in [0,1]$, representing the presence (${\rho }_e=1$) or absence (${\rho }_e=0$) of material.

1.

Objective Function: The objective function for minimizing compliance is defined as

$$C={\mathbf{U}}^T\mathbf{K}\mathbf{U},$$

(27)

where

• C: Structural compliance;
• $\mathbf{U}$: Global displacement vector;
• $\mathbf{K}$: Global stiffness matrix.

2.

Volume Constraint: The optimization problem must satisfy a volume constraint:

$$V=\sum _{e=1}^{N_e}{\rho }_e{V}_e\le V_{\max },$$

(28)

where

• V: Total material volume;
• $V_{\max }$: Maximum allowable material volume;
• $N_e$: Total number of finite elements;
• $V_e$: Volume of element e.

3.

Sensitivity Analysis: The sensitivity of the objective function with respect to the design variable ${\rho }_e$ is calculated as

$${\displaystyle \frac{\partial C}{\partial {\rho }_e}}=-{\mathbf{U}}_e^T{\mathbf{K}}_e{\mathbf{U}}_e,$$

(29)

where ${\mathbf{U}}_e$ and ${\mathbf{K}}_e$ are the displacement vector and stiffness matrix of element e, respectively.

Convergence and Termination

The algorithm iterates until a termination criterion is met, such as reaching a predefined number of iterations $T_{\max }$ or when the change in the objective function $\Delta C$ between consecutive iterations falls below a tolerance threshold:

$$\Delta C={\displaystyle \frac{|C^{(t+1)}-C^{\left(t\right)}|}{C^{\left(t\right)}}}\le ϵ,$$

(30)

where $ϵ$ is the convergence tolerance.

4.

Evolutionary Update Rule: The material distribution is updated iteratively using

$${\rho }_e^{(t+1)}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}{\displaystyle \frac{\partial C}{\partial {\rho }_e}}\le \eta ,\hfill \\ 0,\hfill & \mathrm{if}{\displaystyle \frac{\partial C}{\partial {\rho }_e}}>\eta ,\hfill \end{array}\right.$$

(31)

where $\eta$ is a threshold determined by the volume constraint and evolutionary ratio.

5.

Bi-directional Adjustment: BESO allows both the addition and removal of materials, enabling a balanced search for optimal topology:

$${\rho }_e^{(t+1)}={\rho }_e^{\left(t\right)}+\Delta \rho ,$$

(32)

where $\Delta \rho$ is the material adjustment increment.

Application in Multi-Scale Optimization

In this study, BESO is extended to simultaneous multi-scale topology optimization, addressing both cellular and composite materials. The fitness function incorporates separate and unified volume constraints, enabling the optimization of hierarchical structures. The process is defined as

$$f\left(\mathbf{ӕ}\right)=\alpha ·C\left(\mathbf{ӕ}\right)+\beta ·P\left(\mathbf{ӕ}\right),$$

(33)

where

• $C\left(\mathbf{ӕ}\right)$: Compliance of the structure;
• $P\left(\mathbf{ӕ}\right)$: Penalization term for multi-scale connectivity;
• $\alpha , \beta$: Weighting factors balancing the objectives.

Performance in Structural Design

Kazakis et al. demonstrated the effectiveness of BESO for multi-scale optimization using a simplified MATLAB (version R2022a4) framework. The method supports cellular and composite materials while handling both separate and single volume constraints, showing significant improvements in structural performance and material efficiency [20].

2.3. Experimental Scenario

To find out the best optimization algorithm in general, six scenarios were made, with different node placements in each cluster. Figure 3a–f display the positions of nodes represented by blue dots within a 20 m × 20 m area. The X and Y values, ranging from −10 to 10 m, denote the relative locations of the nodes within this space. On a dimension scaled to 20 m × 20 m, the point at coordinates (−5, 5) on the original scale of −10 to 10 would correspond to coordinates (5.0 m, 15.0 m) on the new scale.

3. Results and Discussion

We conducted experiments across six scenarios using the parameters listed in

Table 2. Values of parameters with units.

Parameter	Value [Unit]
Number of Sensor Nodes	20
Area for Random Sensors
Distribution (L × L)	20 [m] × 20 [m]
Sensor Transmission Frequency (f)	915 [MHz]
Sensor Transmission Power Range
(xMin; xMax)	(−30, 0) [dBm]
Sensor Sensitivity (${\rho }_{\mathrm{th}}$)	−60 [dBm]

. The initial experiment aimed to demonstrate that the simplistic method (transmission power of −10 dBm or 0.0001 Watt, as suggested by [15]) can be optimized using swarm algorithms. As shown in Figure 4, applying a standard swarm optimization method such as PSO results in lower power requirements compared to the simplistic method for all nodes. Building on these findings, we then focus on comparing the performance of various swarm optimization algorithms.

Figure 5 presents the average minimum fitness values achieved by all swarm algorithms across all scenarios. Notably, some algorithms, such as Simulated Annealing (SA) [21] and the Tuna Swarm Algorithm (TSA) [22], do not exhibit any fitness values, indicating that these algorithms failed to converge to any solution. Other algorithms, including the Chicken Swarm Optimization (CSO) [23], Salp Swarm Algorithm (SSA) [24], Whale Optimization Algorithm (WOA) [25], and BESO, display positive fitness values, signifying convergence to suboptimal solutions. In contrast, algorithms like Particle Swarm Optimization (PSO), Genetic Algorithm (GA) [6,26], and Grey Wolf Optimization (GWO) achieve negative fitness values, indicating that these methods are very close to optimal solutions. Consequently, we focus exclusively on these three swarm algorithms for the remaining experiments.

If infinite values are observed in the Best Fitness test (Figure 5) or NaN values appear in the Execution Time test (Table 3), only algorithms with finite values are included in the analysis. Based on the filtered results in Table 4, only EGWO, GWO, BESO, and PSO are considered for both Execution Time and Best Fitness metrics. Asymptotically, the four algorithms have the same main complexity:

$$O(T\times n\times N^3)$$

(34)

where

$$T=\mathrm{number}\mathrm{of}\mathrm{iterations},n=\mathrm{population}\mathrm{size},N=\mathrm{number}\mathrm{of}\mathrm{nodes}.$$

This is because the fitness evaluation function dominates the computation with an operation of

$$O\left(N^3\right)$$

(35)

(primarily due to the eigenvalue calculation on the Laplacian matrix). Among these, GWO achieves the smallest Best Fitness value (−9.274447) with an Execution Time of 13.291528 s.

Table 3. Execution Time of each algorithm.

Algorithm	Minimum	Maximum	Mean
BESO	10.429456	20.312532	14.171138
PSO	11.898477	33.346832	18.473567
GWO	13.291528	21.313970	16.461065
EGWO	13.619340	21.6570003	19.445919

Table 3. Execution Time of each algorithm.

Algorithm	Minimum	Maximum	Mean
BESO	10.429456	20.312532	14.171138
PSO	11.898477	33.346832	18.473567
GWO	13.291528	21.313970	16.461065
EGWO	13.619340	21.6570003	19.445919

Table 4 summarizes the swarm comparison in terms of fitness values. These statistics represent the results of the experimental comparison across all scenarios, where each scenario is repeated 10 times. GWO achieves the lowest minimum fitness value of −9.274447, followed by EGWO with 9.006406. However, for other statistics, the maximum fitness value is achieved by PSO (−2.605) and the mean fitness value by EGWO (−4.150733). Further details can be observed in Figure 6, which illustrates the power transmission to each node for Scenario 1.

From Figure 6, it is evident that the three algorithms—GWO, PSO, and BESO—produce different power transmission values for each node, ranging from approximately −5 dBm to −30 dBm. For most nodes, the Grey Wolf Optimizer (GWO) achieves the lowest power transmission values (represented by the orange bars), indicating that it frequently finds more optimal (lower) power solutions compared to PSO and BESO. BESO and PSO also perform well, with PSO often achieving results close to GWO. However, GWO demonstrates a clear advantage in achieving lower power transmission for most nodes, as seen in Figure 7.

Table 4. Best Fitness of each algorithm.

Algorithm	Minimum	Maximum	Mean
BESO	−2.357431	4.919863	2.214195
PSO	−7.926426	−2.605397	−3.725375
GWO	−9.274447	3.547884	−0.614383
EGWO	−9.006406	0.953669	−4.150733

Table 4. Best Fitness of each algorithm.

Algorithm	Minimum	Maximum	Mean
BESO	−2.357431	4.919863	2.214195
PSO	−7.926426	−2.605397	−3.725375
GWO	−9.274447	3.547884	−0.614383
EGWO	−9.006406	0.953669	−4.150733

In Scenario 1, GWO achieves the lowest average power transmission, followed by EGWO, PSO, and BESO, indicating GWO’s effectiveness in reducing power for this scenario. Similarly, in Scenario 2, EGWO and GWO attain the lowest average power transmission with very similar values, while PSO and BESO register notably higher transmissions. In Scenario 3, EGWO clearly leads by achieving the lowest average power transmission, with GWO and PSO following, and BESO recording the highest value. In Scenario 4, EGWO again records the lowest transmission, with GWO next; meanwhile, BESO shows significant improvement by nearly matching PSO, which exhibits the highest power transmission (i.e., the least efficient performance) in this scenario. In Scenario 5, where nodes are deployed in a circular pattern, EGWO obtains the lowest average power transmission, followed closely by GWO and PSO, whereas BESO continues to exhibit the highest value. Finally, in Scenario 6, with nodes distributed around three centroids, GWO achieves the lowest average power transmission, followed by EGWO, PSO, and BESO. This analysis indicates GWO’s consistent performance in most scenarios.

Table 5. Power transmission in dBm.

Algorithm	Scenario
	1	2	3	4	5	6
BESO	−15.139	−12.794	−14.536	−16.608	−10.880	−14.965
PSO	−18.323	−16.477	−17.809	−16.693	−16.421	−22.257
GWO	−19.256	−18.075	−19.042	−17.295	−16.649	−23.632
EGWO	−19.011	−18.113	−21.216	−18.652	−16.657	−23.288

shows the average value of optimal power transmission in dBm for six different scenarios, using four optimization algorithms: BESO, PSO, GWO, and EGWO. In Scenario 1, GWO achieves the lowest average power transmission with −19.256 dBm, followed by EGWO at −19.011 dBm, PSO at −18.323 dBm, and BESO at −15.139 dBm, indicating GWO’s effectiveness in reducing power for this scenario. In Scenario 2, EGWO slightly outperforms GWO by achieving an average power transmission of −18.113 dBm compared to GWO’s −18.075 dBm, while PSO and BESO record −16.477 dBm and −12.794 dBm, respectively.

In Scenario 3, EGWO clearly leads by attaining the lowest power transmission of −21.216 dBm, followed by GWO at −19.042 dBm, PSO at −17.809 dBm, and BESO at −14.536 dBm. Similarly, in Scenario 4, EGWO achieves the least power transmission with −18.652 dBm, while GWO, PSO, and BESO register −17.295 dBm, −16.693 dBm, and −16.608 dBm, respectively.

In Scenario 5, EGWO again slightly outperforms GWO, with average power transmissions of −16.657 dBm and −16.649 dBm, respectively, while PSO and BESO record −16.421 dBm and −10.880 dBm. Finally, in Scenario 6, GWO attains the lowest average power transmission at −23.632 dBm, followed closely by EGWO at −23.288 dBm, while PSO and BESO register −22.257 dBm and −14.965 dBm, respectively.

This analysis indicates GWO’s consistent performance in most scenarios, with EGWO demonstrating marginally superior results in Scenarios 2 and 5 and clearly outperforming the other algorithms in Scenarios 3 and 4, while GWO excels in Scenarios 1 and 6.

In Scenario 1, the distribution of nodes appears to be relatively uniform or less complex, which favors an algorithm that focuses on efficient exploitation. GWO’s balance between exploration and exploitation allows it to quickly converge to an optimal solution without the need for the additional exploratory mechanisms present in EGWO. As a result, GWO slightly outperforms EGWO in this scenario.

Similarly, in Scenario 6, where nodes are distributed around three distinct centroids, the search space exhibits well-defined attractive regions. GWO’s aggressive exploitation strategy is particularly effective in such structured environments, as it rapidly fine-tunes the solution near these centroids. In contrast, EGWO’s enhanced exploration capability, while beneficial in more complex or multimodal scenarios, may slightly delay convergence in this case, leading to a marginally higher power transmission value compared to GWO.

Table 6. Power transmission in $\mathrm{\mu }$Watt.

Algorithm	Scenario
	1	2	3	4	5	6
BESO	30.623	52.543	35.188	21.834	81.652	31.872
PSO	14.709	22.504	16.559	21.411	22.793	5.946
GWO	11.867	15.575	12.467	18.642	21.631	4.332
EGWO	12.557	15.441	7.556	13.636	21.59149	4.689

shows the average values of optimal power transmission, converted from dBm to microwatts ($\mathrm{\mu }$W), for four optimization algorithms across six different scenarios. In Scenario 1, GWO achieves the lowest average power transmission at $11.867\mathrm{\mu }$W, followed by EGWO at $12.557\mathrm{\mu }$W, PSO at $14.709\mathrm{\mu }$W, and BESO at $30.623\mathrm{\mu }$W, indicating GWO’s effectiveness in reducing power when the node distribution is relatively uniform. In Scenario 2, EGWO slightly outperforms GWO by achieving an average power transmission of $15.441\mathrm{\mu }$W compared to GWO’s $15.575\mathrm{\mu }$W, while PSO and BESO record higher values of $22.504\mathrm{\mu }$W and $52.543\mathrm{\mu }$W, respectively.

In Scenario 3, EGWO clearly leads by attaining the lowest power transmission of $7.556\mathrm{\mu }$W, followed by GWO at $12.467\mathrm{\mu }$W, PSO at $16.559\mathrm{\mu }$W, and BESO at $35.188\mathrm{\mu }$W. Similarly, in Scenario 4, EGWO achieves the best performance with $13.636\mathrm{\mu }$W, while GWO, PSO, and BESO register $18.642\mathrm{\mu }$W, $21.411\mathrm{\mu }$W, and $21.834\mathrm{\mu }$W, respectively.

In Scenario 5, EGWO again slightly outperforms GWO, with average power transmissions of $21.591\mathrm{\mu }$W and $21.631\mathrm{\mu }$W, respectively, while PSO and BESO record $22.793\mathrm{\mu }$W and $81.652\mathrm{\mu }$W. Finally, in Scenario 6, GWO attains the lowest transmission at $4.332\mathrm{\mu }$W, followed closely by EGWO at $4.689\mathrm{\mu }$W, whereas PSO and BESO record $5.946\mathrm{\mu }$W and $31.872\mathrm{\mu }$W, respectively.

This analysis indicates that GWO consistently performs best in Scenarios 1 and 6, while EGWO demonstrates marginally superior performance in Scenarios 2 and 5 and clearly outperforms the other algorithms in Scenarios 3 and 4.

4. Conclusions

This study addresses a problem in WSN on energy optimization. While progress has been made in using swarm algorithms for transmission power optimization, there is still significant potential for improvement. Although many strategies exist to save energy and extend node lifespan in WSNs, few focus specifically on swarm algorithms. This research introduced the Enhanced Grey Wolf Optimization (EGWO) algorithm, which utilizes a non-linear decreasing exploitation parameter to enable more precise exploration in a given area. The EGWO algorithm was evaluated against three other swarm optimization methods—standard GWO, BESO, and PSO—across six different topologies using the Friis transmission power model. This model, a standard for determining distance and signal strength in wireless communications, was used to assess how each algorithm optimizes transmission power in WSNs to maximize energy efficiency while maintaining high communication quality. The results demonstrate that EGWO averagely outperforms the other methods, achieving the best optimization results across all scenarios. For example, in Scenario 3, EGWO achieved the lowest power transmission of −21.216, improving by 2.174 compared to GWO (−19.042) and by 3.407 compared to PSO (−17.809). These enhancements, including the improved balance between exploration and exploitation, make EGWO a highly effective and robust algorithm for power transmission optimization in WSNs.