Innovative hybrid grey wolf-particle swarm optimization for calculating transmission line parameter

Abstract

Transmission line (TL) parameters, particularly capacitance, are important for ensuring the efficient and reliable operation of power systems. As power networks become increasingly complex, accurately determining TL parameters faces certain challenges, particularly capacitance, which involves overcoming computational challenges due to bundling configurations. The interconnected nature of modern TL and the use of advanced technologies further add to the complexity. Effective estimation requires advanced measurement techniques and sophisticated computational tools. Recently, optimization techniques have become widely used for TL parameter calculations. However, traditional methods struggle with challenges like limited exploration and slow convergence. To address these issues, this research introduces a hybrid algorithm called HGWPSO, which combines the Grey Wolf Optimizer (GWO) with Particle Swarm Optimization (PSO). The main objective is to enhance the exploitation ability of GWO with the exploration capability of PSO, maximizing the strengths of both variants. Initially, to verify the efficiency of HGWPSO, CEC_19 benchmark functions are utilized to evaluate its performance. Secondly, the main focus of this study is to calculate TL parameters such as capacitance considering two, three, and four-bundle conductors using the HGWPSO algorithm and comparing its performance with other optimization techniques. According to the obtained result, the average percentage reduction for HGWPSO is 0.15 % in test case 1, 4.85 % in test case 2, and 2.84 % in test case 3, compared to others. It shows that the HGWPSO has better performance than other methods regarding convergence speed, and ability to locate the global optimum. Finally, experimental analysis confirms the superiority of the HGWPSO in accurately estimating TL capacitance for different bundle conductor configurations, obtaining lower average values, and effectively addressing the characteristic complexities in the TL parameter.

List of Acronyms

DE	Differential Evolution
GA	Genetic Algorithm
FOX	Fox Optimizer
PSODE	Particle Swarm Optimization Differential Evolution
PSOGA	Particle Swarm Optimization Genetic Algorithm
PSOACO	Particle Swarm Optimization Ant Colony Optimization
ADRC-SMC	Active Disturbance Rejection Control Sliding Mode Control
iAILC	Indirect Adaptive Iterative Learning Control
WOA	Whale Optimization Algorithm
IMFO	Improved Moth Flame Optimization
MFO	Moth Flame Optimization
SSA	Salp Swarm Algorithm
SCA	Sine Cosine Algorithm
MWOA	Modified Whale Optimization Algorithm
OWOA	Oppositional Based Whale Optimization Algorithm
HSSASCA	Hybrid Salp Swarm Algorithm with Sine Cosine Algorithm
SOS	Symbiotic Organisms Search
SPBO	Student PSYCHOLOGY-BASED OPTIMIZATION
FFO	Firefly Algorithm
HMFPSO	Hybrid Moth Flame Particle Swarm Optimization

1. Introduction

TL capacitance is a very important parameter in the design and operation of electrical power systems. It plays a main role in maintaining voltage regulation and stability by ensuring voltage levels remain within acceptable limits and stabilizing voltage fluctuations. Capacitance helps in reactive power compensation, enhancing power transfer capability and efficiency by reducing transmission losses and improving power factor. It also affects signal propagation speed and integrity, significant for communication and data transmission networks. Additionally, capacitance influences the line charging current, which is significant for system design and operation, particularly for high-voltage and long-distance lines. Proper insulation design and management of dielectric losses, impacted by capacitance, are essential for maintaining transmission efficiency. Optimizing line capacitance can lead to cost savings and environmental benefits by reducing energy losses. Moreover, these TL parameters are important for the effective management and operation of power systems. Traditional TL parameters have relied on direct measurements, simplified analytical models, and regression-based methods to estimate TL parameters such as resistance, inductance, capacitance, and conductance. Techniques like the lumped-parameter model and the distributed-parameter model are widely used, that calculate TL parameters based on physical line data including configuration, dimensions, and materials. However, these traditional methods have a few limitations [1] including inaccuracies in parameter estimation that may affect system reliability and efficiency. To calculate load flow, develop secure systems, locating faults and accurate identification of TL parameters is very important [2,3]. These parameters are often unknown or have changed due to aging, environmental factors, and operating conditions in existing lines, even though they may be available for new installations. Conventional methods estimate TL parameters based on physical characteristics like line size, composition, sag, tower height, and soil characteristics [[4], [5], [6]] but these methods often lead to inaccuracies due to various assumptions. Recently, different estimation methods, such as frequency or time-domain analysis, have been developed to improve accuracy [7,8]. In time-domain methods, TL is modeled using cascaded π networks that include series resistance, inductance, mutual inductance, and shunt capacitance, similar to transformer modeling [9]. The estimation of resistance, inductance, and capacitance at the fundamental frequency, as presented in Ref. [10], supports relay coordination and protection scheme design. However, the complexity of data processing in this method makes it unsuitable for practical field applications. Additionally, another technique utilizes synchronized voltage and current measurements at both terminals of the line during fault conditions, as presented in Ref. [11] however, this technique requires high-speed sensors, and the estimated parameters are impacted by the fault type. Then, open-circuit voltage and short-circuit current measurements another approach is presented in Ref. [12] still this technique is not well-suited for practical applications. With the rise of smart power networks, there is a critical need for reliable, and accurate estimation of TL parameters. Metaheuristic and evolutionary algorithms offer robust solutions for optimization problems, including parameter estimation. These algorithms, not reliant on derivatives, minimize premature convergence issues and can exploit chaotic maps to enhance performance. Nowadays different optimization techniques have been analyzed to estimate TL parameters; however, they have certain challenges like uncertain convergence. Therefore, simpler and more easily implemented optimization methods without the need for gradient definitions are proposed. GWO and PSO are very common methods that are widely used in power systems, the literature also presents other efficient techniques [[13], [14], [15]].

However, there is still a need for algorithms that necessitate minimal numerical solutions and tuning parameters, aiming for an efficient combination [16,17]. There is no population-based variation that has been shown to solve all kinds of optimization problems [18]. Therefore, advancements in computational techniques have led to hybrid optimization methods to improve the accuracy and efficiency of TL parameters. These methods are selected for their simplicity, fast convergence, and efficiency in finding the global optimum. Despite this, GWO and PSO face challenges like difficulty exploring the search space, premature convergence, and lack of adaptability in TL parameter problems.

Recently, hybrid optimization techniques are a powerful technique for solving optimization problems, offering a variety of advantages over traditional techniques. The goal of these hybrid algorithms is to reduce the chance of being stuck in local optima. PSO is one of the most popular variants of hybrid approaches. Similarly, due to its simplicity, speed of convergence, and capacity to find the global optimum. GWO is also widely employed in hybrid approaches inspired by the natural leadership hierarchy and hunting strategy of grey wolves, the GWO is a recently developed metaheuristic. The technique has demonstrated efficiency in addressing many challenges such as optimal power flow difficulties [19], economic dispatch problems [20], flow shop scheduling problems [21], time forecasting [22], feature selection [23], and optimal design of double-layer grids [24]. The convergence performance of the GWO has been improved by several algorithms, such as improved binary GWO [25], parallelized GWO [26], integration of GWODE [27], hybrid GWOGA [28], and GWO with FOX [29]. Different research works have been conducted on combining PSODE [30] with other metaheuristics, such as PSOGA [31], and PSOACO [32]. A hybrid controller named ADRC-SMC is introduced to enhance control-loop performance and stability. Its parameters are optimally tuned using a metaheuristic slime mould algorithm, which reduces heuristic reliance and enables a fair comparison with the traditional ADRC algorithm [33]. Additionally, an iAILC scheme is proposed to improve the P-type controller for linear and nonlinear systems. This scheme incorporates an adaptive mechanism that adjusts the learning gain in real-time based on input-output measurements, enhancing the system's efficiency by learning from set points [34].

2. Literature review

This section has provided a detailed overview of the literature on TL parameter estimation, highlighting various problem-solving approaches, key studies, and challenges. It has also outlined the motivation for current research, identified gaps in existing knowledge, and emphasized the main contributions of the proposed work. Through these efforts, the field of TL parameter estimation continues to evolve, driven by the need for accurate, efficient, and adaptable solutions in the face of increasingly complex transmission systems.

2.1. Related work

In [35] the authors propose a method for estimating TL parameters in the presence of non-Gaussian measurement noise to identify, classify, and localize fault events in transmission power systems. In Ref. [36] an MWOA is proposed to identify TL parameters. The authors in Ref. [37] suggest a method for parameter identification using machine learning and deep learning techniques, specifically support vector regression. Furthermore, the author in Ref. [38] presents a method for real-time parameter estimation of 500 kV power TL using a PMU. In Ref. [39], a research study proposes a method for TL parameter estimation utilizing voltage-current data and the WOA. The authors in Refs. [40,41] employ synchronized measurements and Kalman filtering with PSO, respectively, to estimate frequency-dependent characteristics and electrical parameters of TLs Additionally, the author in Ref. [42] calculates TL parameters using the whale optimization algorithm. However, there different optimization methods such as GWO [43], and IMFO [44] have shown higher efficiency in TL parameter problems. In Ref. [45], the author presented parameter estimation of a long TL formulated it as an optimization problem, and solved using an improved PSO algorithm. The obtained results are compared with various benchmark functions and well-known optimization techniques Furthermore, a range of benchmark methods [[46], [47], [48], [49], [50]] including SSA and SCA, in addition to established optimization techniques have been utilized in various power system applications.

Solving the TL parameter estimation problem, such as capacitance, involves multiple challenges such as the geometric complexity of transmission lines, which requires precise calculation of TL parameters between different bundle conductors. Additionally, environmental factors like temperature, humidity, and pollution can influence capacitance and are difficult to model accurately. Traditional methods, such as analytical and empirical approaches, use mathematical models based on the physical properties and geometrical configuration of the conductors to estimate TL parameters. However, these traditional methods have few challenges leading to less accurate and reliable TL parameter estimation.

In response to these challenges, the HGWPSO algorithm offers a comprehensive and effective solution for the accurate estimation of TL parameters, such as capacitance, particularly in complex scenarios involving different bundle conductors. By combining the exploration strengths of GWO with the exploitation efficiency of PSO, the HGWPSO provides a balanced and adaptive approach that enhances both the accuracy of parameter estimation. This method represents a significant advancement in the modeling and management of modern power systems, ensuring reliable and efficient operation even under challenging conditions. The exploration of HGWPSO for optimizing TL parameters is motivated by its significant advantages.

• •Simplicity, ease of understanding
• •Few control parameters simplify the tuning process
• •Better convergence characteristics enable it to find optimal solutions
• •Designed to perform a global search, making it effective in finding global optima and avoiding local minima.

The algorithm integrates strategies for both exploration and exploitation, maintaining a balance by exploring new regions of the solution space while also exploiting known solutions. However, there are certain limitations, such as.

• •Increased computational complexity
• •Parameter sensitivity
• •Risk of premature convergence

2.2. Motivation and incitement

The proposed research works to address the issue of TL parameter estimation in power transmission systems, a critical factor in maintaining the balance between electricity generation and distribution. The main challenge lies in achieving both accuracy and reliability in assessing power system performance. Current methods for estimating TL parameters in power systems face a few challenges, including system complexity, non-linearity, measurement inaccuracies, environmental factors, and the need for significant computational resources and time. To overcome these challenges and improve the efficiency of power systems, there is a need for advanced control and optimization techniques. Additionally, there is a lack of validation and assessment of optimization methods across different power systems. While various optimization techniques have demonstrated promising results in specific scenarios, their performance in large power system applications such as renewable energy integration, economic dispatch, and unstable systems remains underexplored. It is important to evaluate the effectiveness of these methods under different conditions to ensure their applicability in real-world power systems.

Moreover, developing efficient and time-effective optimization approaches for TL parameters is crucial for ensuring the reliability of power grids. This research aims to explore optimization techniques that address existing gaps and improve power system performance. The study introduces the HGWPSO method as a solution to these research gaps and proposes it as an enhanced optimization technique for TL parameter estimation in power systems. The objective is to assess the efficiency of HGWPSO through TL parameters, ensuring that parameter changes remain within acceptable limits. This research aims to make a significant contribution to the field of power systems by developing an optimized technique for TL parameters.

2.3. Research gap

From the reviewed literature, the optimal value of capacitance for different bundle conductors (such as two, three, and four bundles) is primarily determined using a single objective function for the transmission system. However, the effects of different bundle configurations, along with detailed statistical analysis, have not been comprehensively considered particularly capacitance. Additionally, some studies did not employ hybrid optimization methods, which are often the most effective for parameter estimation in TL systems. Furthermore, most of the reviewed literature lacked a detailed analysis of the effects of different algorithms on the TL parameter problem.

2.4. Main contribution of the proposed work

There have been various studies on TL parameter estimation using optimization techniques to enhance accuracy and improve system quality. The main challenge is how to optimally integrate these techniques. The main contribution of this work is the optimization of TL capacitance across various test cases using the HGWPSO algorithm, detailed as follows.

• •Validation and effectiveness of HGWPSO are verified on CEC_19 benchmark functions.
• •Application of HGWPSO to address TL parameter estimation problems considering two-, three-, and four-bundle conductors, specifically to capacitance.
• •Comparative analysis of HGWPSO is compared with GWO [51], PSO [52], MFO [53], WOA [54], and SCA [55] in terms of convergence curve and performance analysis.

3. Capacitance parameter estimation modeling

This section presents the modeling of capacitance parameter estimation, including considerations for different bundle conductors, along with the objective function and constraints.

3.1. Problem description

The capacitance per unit length for a TL can be expressed in terms of the geometry of the conductors and their arrangement. For bundled conductors, the capacitance is influenced by the number of conductors in the bundle and their physical distance the capacitance per unit length capacitance can be approximated by Ref. [56].

Assume that.

C = capacitance per unit length

ε = permittivity of the medium.

D = distance between the center of the bundles.

${\mathrm{r}}_{\text{eq}}$ = geometric mean radius of the conductors for capacitance

$$C=\frac{2\pi ϵ}{\mathit{ln}\frac{D}{{\mathrm{r}}_{\text{eq}}}}$$ (1)

3.2. Consideration of different bundle configuration

Assume that.

$d_s$ is the bundle spacing

R is the sub-radius of the conductors.

• •Two bundle conductors

$${\mathrm{r}}_{\text{eq}}=\sqrt{\left(d_s\right)\left(R\right)}$$ (2)

• •Three bundle conductors

$${\mathrm{r}}_{\text{eq}}=\sqrt[3]{{\left(d_s\right)}^2\left(R\right)}$$ (3)

• •Four bundle conductors

$${\mathrm{r}}_{\text{eq}}=1.90\left(\sqrt[4]{{\left(d_s\right)}^2\left(R\right)}\right)$$ (4)

3.3. Optimization problem modeling

The objective is to minimize the optimal value of capacitance based on the TL parameters using HGWPSO and the objective function for capacitance estimation with bundle conductors can be defined as.

$$\text{Minimize}f\left(C\right)$$ (5)

Constraints ensure that the optimization parameters remain within physically feasible and operational limits as follows.

• •Physical limitations on D (minimum spacing between the conductors)
• •Material properties (maximum allowable conductor radius)
• •The ɛ must be within a specified range based on environmental conditions.

This can be formulated mathematically as

$$\begin{array}{c}\text{Subject}\text{to:}\\ D_{\mathit{min}}\le \mathrm{D}\le D_{\mathit{max}}\\ \begin{array}{c}{r}_{\mathit{min}}\le \mathrm{r}\le r_{\mathit{max}}\\ {ϵ}_{\mathit{min}}\le ϵ\le {ϵ}_{\mathit{max}}\end{array}\end{array}$$ (6)

Where.

• •f(C) is the objective function to be minimized
• •$d_{\mathit{min}},d_{\mathit{max}},r_{\mathit{min}},r_{\mathit{max}},{ϵ}_{\mathit{min}}\text{and}{ϵ}_{\mathit{max}}\text{are}\text{the}\text{lower}\text{and}\text{upper}\text{bounds}\text{for}\text{the}\text{design}\text{parameters}$.

The optimization problem can be formulated as follows.

$$Minimizef\left(C\right)=\frac{2\pi ϵ}{\mathit{ln}\frac{D}{{\mathrm{r}}_{\text{eq}}}}$$

$$\text{Subject}\text{to}:\begin{array}{c}{D}_{\mathit{min}}\le \mathrm{D}\le D_{\mathit{max}}\\ \begin{array}{c}{r}_{\mathit{min}}\le \mathrm{r}\le r_{\mathit{max}}\\ {ϵ}_{\mathit{min}}\le ϵ\le {ϵ}_{\mathit{max}}\end{array}\end{array}$$

$${\mathrm{r}}_{\text{eq}}=\{\begin{array}{c}\sqrt{\left(d_s\right)\left(R\right)},\text{For}\text{two}\text{bundle}\text{conductor}\\ \sqrt[3]{{\left(d_s\right)}^2\left(R\right)},\text{For}\text{three}\text{bundle}\text{conductor}\\ 1.90\left(\sqrt[4]{{\left(d_s\right)}^2\left(R\right)}\right),\text{For}\text{four}\text{bundle}\text{conductor}\end{array}$$ (7)

4. Methodology

This section provides a detailed description of GWO and PSO, including their mathematical modeling. While both techniques are highly efficient, they have certain limitations. To address these, a hybrid approach, HGWPSO, is proposed. This method combines the strengths of GWO and PSO to overcome their limitations. Additionally, a flowchart and pseudocode of HGWPSO are provided to clarify the algorithm steps.

GWO is an optimization algorithm inspired by the hierarchical structure and hunting strategies observed in grey wolves which consist of alpha(α), beta(β), and delta(δ) [51]. In GWO, a group of candidate solutions, similar to the positions of wolves within a pack, undergoes iterative refinement to optimize a TL parameter. The algorithm mimics the social dynamics of grey wolves, encompassing features like leadership and cooperation, to dynamically explore and exploit the search space. Through iterative adjustments based on individual experiences and interactions among wolves, GWO strives to converge toward optimal or near-optimal solutions across diverse optimization problems. The following mathematical equations have been developed to facilitate this work.

$$\overrightarrow{U}\left(\mathrm{t}+1\right)=\overrightarrow{{\mathrm{U}}_{\mathrm{p}}}\left(\mathrm{t}\right)-\overrightarrow{\mathrm{V}}.\overrightarrow{\mathrm{W}}$$ (8)

$$\overrightarrow{W}=\left|\overrightarrow{Z}.\overrightarrow{U_p}\left(t\right)-\overrightarrow{Z}\left(t\right)\right|$$ (9)

$$\overrightarrow{\mathrm{V}}=2\overrightarrow{\mathrm{a}}.\overrightarrow{{\mathrm{r}}_1}-\overrightarrow{\mathrm{a}}$$ (10)

$$\overrightarrow{\mathrm{Z}}=2\overrightarrow{{\mathrm{r}}_2}$$ (11)

$$\{\begin{array}{c}\overrightarrow{W_{\alpha }}=\left|\overrightarrow{Z_1}-\overrightarrow{U_{\alpha }}\left(t\right).\overrightarrow{U_t}\right|\\ \overrightarrow{W_{\beta }}=\left|\overrightarrow{Z_2}-\overrightarrow{U_{\beta }}\left(t\right).\overrightarrow{U_t}\right|\\ \overrightarrow{W_{\delta }}=\left|\overrightarrow{Z_3}-\overrightarrow{U_{\delta }}\left(t\right).\overrightarrow{U_t}\right|\end{array}$$ (12)

$$\{\begin{array}{c}\overrightarrow{U_1}=\left|\overrightarrow{U_{\alpha }}\left(t\right)-\overrightarrow{V_1}.\overrightarrow{W_{\alpha }}\right|\\ \overrightarrow{U_2}=\left|\overrightarrow{U_{\beta }}\left(t\right)-\overrightarrow{V_2}.\overrightarrow{W_B}\right|\\ \overrightarrow{U_3}=\left|\overrightarrow{U_{\delta }}\left(t\right)-\overrightarrow{V_2}.\overrightarrow{W_{\delta }}\right|\end{array}$$ (13)

Where.

• •$\overrightarrow{U}$ is the position of the grey wolf and t is the iteration
• •$\overrightarrow{U_{\mathrm{p}}}$ is the position of the prey and $\overrightarrow{W}$ is the distance between the wolf and the prey
• •$a$ is the control parameter and $\overrightarrow{V}$ and $\overrightarrow{Z}$ are the coefficients

PSO is a computational optimization method inspired by the coordinated movements observed in social creatures, such as birds flocking or fish schooling [52]. In PSO, a group of potential solutions, referred to as particles, explores the search space of a TL parameter problem. Each particle dynamically updates its position and velocity, drawing from its encounters and those of nearby particles. By iteratively updating their positions according to mathematical equations that balance exploration and exploitation, the swarm collectively converges toward optimal or near-optimal solutions. PSO is widely used in various fields for solving optimization problems where traditional methods are computationally complex. In the PSO, each member of the swarm adjusts its position within the global search space using two mathematical equations. In PSO each particle has its position according to the following equations.

$$g_i^{t+1}={\mathrm{\omega }g}_i^t+k_1{m}_1+\left({pbest}_i^t-x_i^t\right)+k_2{m}_2+\left({gbest}_i^t-x_i^t\right)$$ (14)

$$x_i^{t+1}=x_i^t+q_i^{t+1}$$ (15)

Where.

• •${\mathit{g}}_{\mathit{i}}^{\mathit{t}}$ is the velocity of ith particles in the population
• •${\mathit{x}}_{\mathit{i}}^{\mathit{t}}$ is the position of the ith particle in the population
• •${\mathit{p}\mathit{b}\mathit{e}\mathit{s}\mathit{t}}_{\mathit{i}}^{\mathit{t}}$ is the self-best solution and ${\mathit{g}\mathit{b}\mathit{e}\mathit{s}\mathit{t}}_{\mathit{i}}^{\mathit{t}}$ is the best solution
• •${\mathit{m}}_{\mathbf{1}}\mathit{a}\mathit{n}\mathit{d}{\mathit{m}}_{\mathbf{2}}$ are the vectors consisting of dimensional search space uniformly distributed on [0, 1]
• •${\mathit{k}}_{\mathbf{1}}\mathit{a}\mathit{n}\mathit{d}{\mathit{k}}_{\mathbf{2}}$ are the constant value that controls the influence of the social and cognitive components
• •ω is the inertia weight

However, both GWO and PSO suffer from premature convergence and difficulties in balancing exploration and exploitation. The main difference is that GWO has stronger exploration capabilities but may not exploit solutions effectively, while PSO has stronger exploitation capabilities but risks diversity loss and premature convergence. These limitations motivate the development of hybrid algorithms like HGWPSO to improve performance.

To address these limitations, HGWPSO is proposed [57] by the combination of GWO and PSO. By combining the exploration capabilities of GWO with the exploitation strengths of PSO, HGWPSO offers different advantages, including enhanced exploration-exploitation balance, improved convergence rate, and better solution quality. In HGWPSO the positions of the initial three agents in the search space are adjusted according to Eq. (16). The exploration and exploitation of the GWO within the search space are regulated by the inertia constant, instead of using complex mathematical formulations. The modified equations are defined as follows

$$\{\begin{array}{c}\overrightarrow{W_{\alpha }}=\left|\overrightarrow{Z_1}-\overrightarrow{U_{\alpha }}\left(t\right)-w.\overrightarrow{U_t}\right|\\ \overrightarrow{W_{\beta }}=\left|\overrightarrow{Z_2}-\overrightarrow{U_{\beta }}\left(t\right)-w.\overrightarrow{U_t}\right|\\ \overrightarrow{D_{\delta }}=\left|\overrightarrow{Z_3}-\overrightarrow{U_{\delta }}\left(t\right)-w.\overrightarrow{U_t}\right|\end{array}$$ (16)

To integrate PSO and GWO variants, the velocity and update equations are proposed as follows:

$$g_i^{t+1}=\mathrm{\omega }.\left({g_i^t+k}_1{m}_1\left(U_1-x_i^t\right){+k}_2{m}_2\left(U_2-x_i^t\right){+k}_3{m}_3\left(U_3-x_i^t\right)\right)$$ (17)

$$x_i^{t+1}=x_i^t+q_i^{t+1}$$ (18)

Combining GWO with PSO can effectively balance exploration and exploitation, leading to improved optimization performance for estimating TL parameters like capacitance. Following are the detailed steps for the HGWPSO algorithm.

Step 1

Initialization

Initialize the population of candidate solutions for both GWO and PSO. For GWO, this involves setting up the grey wolves with random positions in the search space. For PSO, initialize particles with random positions and velocities. Define the problem's parameters, such as the number of bundle conductors and the range of possible capacitance values.

Step 2

Fitness function evaluation

Evaluate the fitness of each candidate solution. The fitness function calculates TL capacitance. This involves calculating the capacitance for each bundle conductor configuration.

Step 3

GWO operators

Apply GWO operators to the current population. This involves simulating the hunting behavior of grey wolves, where each wolf (candidate solution) is updated based on the positions of α, β, and δ wolves (best solutions). This step helps in exploring and exploiting the search space by guiding solutions towards the optimal region.

Step 4

Call PSO particles

Update the positions and velocities of PSO particles based on their own best-known positions and the global best-known position. The PSO algorithm will use these updates to refine the candidate solutions, helping to explore the solution space more effectively.

Step 5

Update new and wolf position

Evaluate the fitness of the updated positions for both wolves and particles and update the wolves if better solutions are found. Additionally, update the personal best and global best positions in PSO.

Step 6

Convergence check

Check if the algorithm has converged or if the stopping criteria have been met. This typically involves evaluating whether the improvement in fitness scores is below a certain threshold or if the maximum number of iterations has been reached. If convergence has not been achieved, return to the update steps; otherwise, proceed to the final step.

Step 7

Best optimal value obtained

Output the best solution found by the hybrid algorithm, which provides the estimated capacitance parameters for the TL with different bundle conductors. This solution should ideally be close to the actual capacitance values.

The flowchart and pseudo-code of the proposed algorithm are presented in Algorithm 1 and Fig. 1 presented, respectively.

Algorithm 1

pseudo code of the HGWPSO

1	Initialization
2	Set the initial values for parameters a, V, ω, and Z
3	Calculate the fitness function for each agent using Eq. (16).
4	While the while (t < maximum number of iteration)
5	for each search agent
6	Adjust the velocity and position according to Eq. (17).
7	end for
8	Adjust parameters a, V, ω, and Z
9	Assess the fitness values for all search agents
10	Update the position of each search agent
11	t = t + 1
12	end While
13	Return the best optimal solution

Fig. 1.

HGWPSO flowchart.

5. Experimental setup and evaluation

This section shows the simulation and experimentation of TL parameter settings for the HGWPSO. The performance of HGWPSO is verified using the CEC_19 benchmark functions, specifically from CEC_19_F1 to CEC_19_F10. The verification process includes a comprehensive statistical analysis that evaluates computational time, best, average, and worst values, as well as rank and standard deviation. Following the verification, the application of HGWPSO to TL capacitance estimation is explored. This includes testing with different bundle configurations. Case 1 (two bundles), case 2 (three bundles), and case 3 (four bundles). The results from these test cases are compared with those obtained from highly cited algorithms to assess the performance of HGWPSO. Additionally, a box plot representation of the algorithm's performance is provided. This visualization helps in understanding the distribution and variability of the results. Finally, the section discusses the practical implications of HGWPSO in real-world applications, highlighting its potential impact and benefits for power system analysis and management.

HGWPSO algorithm's performance is sensitive to its parameter settings, which significantly affect its optimization effectiveness. Key parameters include population size, crossover rate, mutation rate, and number of iterations.

• •Population size: Larger populations generally enhance the algorithm's exploration capabilities but may lead to increased computational costs and slower convergence. However, smaller populations can speed up convergence but might result in premature convergence.
• •Crossover rate: This parameter determines the frequency of crossover operations in the hybridization process. A higher crossover rate improves exploration by generating diverse solutions, whereas a lower rate focuses on exploiting known solutions, potentially improving local optimization but risking reduced global search capability.
• •Mutation rate: The mutation rate affects the algorithm's ability to escape local optima and maintain diversity. High mutation rates can enhance exploration but may destabilize the search process if not balanced appropriately.
• •Number of iterations: Increasing the number of iterations generally leads to better convergence and solution quality, but this comes at the cost of higher computational time. Optimal iteration settings balance the trade-off between solution quality and computational efficiency.

The proposed algorithm is executed in MATLAB R2019a on a computer featuring an Intel(R) Core (TM) i5-7200U CPU @ 2.50 GHz, 16 GB of RAM, and a 64-bit Windows 10 operating system. The performance of the HGWPSO algorithm is compared with other established algorithms like GWO, PSO, MFO, WOA, and SCA. The optimal solutions for these TL parameters are identified based on the average and standard deviation. Table 1, Table 2 show the initial setup of the different optimization algorithms, along with their respective ranges and tuning parameters, respectively.

Table 1.

Initial setup of the different optimization algorithms.

GWO	Number of grey wolves	30
	Maximum iteration	100
	Stopping criteria	Maximum iteration
	Boundary limit	[-10,10]
PSO	Number of particles	30
	Maximum iteration	100
	Stopping criteria	Maximum iteration
	Boundary limit	[-10,10]
MFO	Population size	30
	Maximum iteration	100
	Stopping criteria	Maximum iteration
WOA	Number of search agent	30
	Maximum iteration	Maximum iteration
	Stopping criteria	Stopping criteria
SCA	Population size	30
	Maximum number of iterations	100
	Stopping criteria	Maximum number of iterations

Table 2.

Tuning parameters for the algorithm.

Algorithms	Parameter	Value
GWO	Control parameter (a)	Linear reduction from 2 to 0
PSO	Cognitive constant	1.8
	Social constant	2
	Minimum inertia weight wmin	0.1
	Maximum inertia weight wmax	0.9
MFO	Logarithmic spiral	0.75
WOA	a1 in location updating	[0, 2]
	a2 in location updating	[-2, −1]
	Constan b	1
SCA	Constant a	2
	Random number r1	(0–2)
	Random number r2	Random(0,2π)
	Random number r3	(0–2)

The computational efficiency of the HGWPSO demonstrates significant improvement compared to different established algorithms. Particularly, HGWPSO achieved a runtime of 0.8986 s, outperforming standard algorithms such as GWO (1.5331 s), PSO (1.9001 s), MFO (1.4731 s), WOA (1.5835 s), and SCA (2.0761 s). These results highlight the potential of HGWPSO in scenarios where processing time is critical, particularly for high-dimensional problems. This reduced runtime suggests that the hybridization of GWO and PSO in the proposed research work is not computationally complex and leads to an effective balance between exploration and exploitation, making it more suitable for the TL parameter problem. This hybrid approach achieves faster convergence, better solutions, and greater robustness across different optimization problems, making the HGWPSO algorithm a better choice for such problems.

5.1. Verification of HGWPSO with CEC_19 benchmark functions

In this subsection, the capability of HGWPSO to solve optimization problems is evaluated using the CEC_19 test suite, which comprises 10 benchmark functions. A detailed description of these benchmark test functions is provided in Table 3.

Table 3.

Description of CEC-19 benchmark function [58].

Function	Name of function	Range	Dimension
CEC_19_F1	Storn's Chebyshev Polynomial Fitting Problem	[–8192, 8192]	9
CEC_19_F2	Inverse Hilbert Matrix Problem	[−16,384, 16,384]	16
CEC_19_F3	Lennard–Jones Minimum Energy Cluster	[−4, 4]	18
CEC_19_F4	Rastrigin's Function	[−100, 100]	10
CEC_19_F5	Griewangk's Function	[−100, 100]	10
CEC_19_F6	Weierstrass Function	[−100, 100]	10
CEC_19_F7	Modified Schwefel's Function	[−100, 100]	10
CEC_19_F8	Expanded Schaffer's F6 Function	[−100, 100]	10
CEC_19_F9	Happy Cat Function	[−100, 100]	10
CEC_19_F10	Ackley Function	[−100, 100]	10

According to Table 4, in the CEC_19 (F1-F10) evaluation, the HGWPSO shows better performance compared to other algorithms in the F1, F2, F3, F4, F6, F7, and F8 test functions. However, HGWPSO demonstrated average performance in F5, F9, and F10. These findings emphasize the substantial improvement of HGWPSO over the traditional GWO and PSO, particularly in terms of enhancing their ability to avoid local optima and converge towards the global optimum during optimization. Additionally, Table 4 indicates that HGWPSO obtained the top position among all optimization algorithms. GWO and PSO obtained the second and third positions across most CEC_19 functions, with WOA, MFO, and SCA. Furthermore, the standard deviation values of HGWPSO for most CEC_19 functions remain within the boundary limits, indicating better stability and a more focused search for optimal solutions compared to the original GWO and PSO. This improvement is attributed to HGWPSO's balanced approach to exploration and exploitation, utilizing GWO's diverse search capabilities, and PSO's refinement techniques, ultimately enhancing convergence speed, solution quality, and robustness against local optima. Moreover, HGWPSO is the variant of both GWO and PSO, making it adaptable to various optimization problems while benefiting from GWO's stability and PSO's robustness. The combination of GWO social hierarchy and PSO social learning mechanisms results in a powerful hybrid algorithm that effectively combines the strengths of GWO and PSO, thereby making HGWPSO a more efficient and effective optimization technique offering improved performance across the CEC_19 functions.

Table 4.

Statistical analysis of contemporary CEC_19 benchmark function.

Function name	Algorithm	Best Value	Worst	Mean	Standard Deviation	Rank	Computational Time (s)
CEC_19_F1	HGWPSO	1.3251E+06	3.1380E+07	1.6353E+07	2.1252E+07	1	5.501072
	GWO	2.5749E+08	4.3021E+08	3.4385E+08	1.2213E+08	6	9.369108
	PSO	1.0277E+08	2.3047E+08	1.6662E+08	9.0296E+07	5	8.248431
	MFO	4.6550E+07	4.9965E+07	4.8257E+07	2.4146E+06	4	6.275473
	WOA	3.6142E+07	4.5866E+07	4.1004E+07	6.8758E+06	3	6.308210
	SCA	5.8715E+06	7.9876E+06	6.9296E+06	1.4963E+06	2	6.124272
CEC_19_F2	HGWPSO	17.3591	17.5187	17.4389	0.1128	1	3.178741
	GWO	33.8702	130.2731	82.0717	68.1672	5	5.278978
	PSO	65.7111	66.4721	66.0916	0.5381	6	3.258050
	MFO	18.3057	75.0761	46.6909	40.1427	4	3.264839
	WOA	17.5810	18.5295	18.0552	0.6707	2	4.187779
	SCA	17.6815	17.9685	17.8250	0.2029	3	3.319909
CEC_19_F3	HGWPSO	12.7062	12.7082	12.7072	0.0014	1	3.046684
	GWO	12.7112	12.7112	12.7112	0	4	3.270995
	PSO	12.7082	12.7121	12.7101	0.0027	2	3.325209
	MFO	12.7082	12.7112	12.7097	0.0022	2	3.352050
	WOA	12.7082	12.7112	12.7097	0.0022	2	3.117072
	SCA	12.7086	12.7109	12.7098	0.0016	3	3.223765
CEC_19_F4	HGWPSO	7.4038E+04	7.4040E+04	7.4039E+04	1.2447	1	3.036598
	GWO	7.4039E+04	7.4039E+04	7.4039E+04	0.0928	2	3.202304
	PSO	7.4040E+04	7.4041E+04	7.4040E+04	0.5592	3	3.372872
	MFO	7.4041E+04	7.4041E+04	7.4041E+04	0.4308	4	3.326384
	WOA	7.4040E+04	7.4041E+04	7.4040E+04	0.7933	3	3.348539
	SCA	7.4039E+04	7.4040E+04	7.4039E+04	0.4564	2	3.239401
CEC_19_F5	HGWPSO	8.7911	8.7912	8.7911	2.8526E-05	1	3.161033
	GWO	8.7913	8.7913	8.7913	6.0692E-05	2	3.214263
	PSO	8.7913	8.7914	8.7913	5.7784E-05	2	3.562514
	MFO	8.7913	8.7913	8.7913	2.1660E-05	2	3.996284
	WOA	8.7913	8.7913	8.7913	1.2055E-05	2	3.298638
	SCA	8.7911	8.7912	8.7912	6.0494E-05	1	4.165703
CEC_19_F6	HGWPSO	11.2461	13.1407	12.1934	13.1407	1	3.071204
	GWO	14.0105	15.7173	14.8639	15.7173	5	3.548133
	PSO	14.6755	15.1435	14.9095	15.1435	6	3.487778
	MFO	13.2260	13.5718	13.3989	13.5718	2	3.302131
	WOA	13.2439	13.4569	13.3504	13.4569	3	3.472476
	SCA	13.6037	14.1872	13.8955	14.1872	4	3.442269
CEC_19_F7	HGWPSO	5.6133E+03	5.6138E+03	5.6135E+03	0.3161	1	3.135879
	GWO	5.6135E+03	5.6135E+03	5.6135E+03	0.0264	2	3.265127
	PSO	5.6137E+03	5.6141E+03	5.6139E+03	0.2832	4	3.656632
	MFO	5.6141E+03	5.6142E+03	5.6141E+03	0.0970	6	3.325706
	WOA	5.6138E+03	5.6138E+03	5.6138E+03	0.0220	5	3.373470
	SCA	5.6136E+03	5.6137E+03	5.6137E+03	0.0541	3	3.286131
CEC_19_F8	HGWPSO	6.8293	7.8575	7.3434	0.7270	1	3.225505
	GWO	7.6633	8.2889	7.9761	0.4424	6	4.118287
	PSO	7.5313	8.1204	7.8259	0.4166	4	4.871498
	MFO	7.6090	7.6764	7.6427	0.0477	5	3.309874
	WOA	7.4969	8.0982	7.7976	0.4252	3	3.946609
	SCA	7.1695	7.8946	7.5320	0.5127	2	3.388648
CEC_19_F9	HGWPSO	6.4553E+03	6.4554E+03	6.4553E+03	0.0268	1	3.150594
	GWO	6.4553E+03	6.4554E+03	6.4553E+03	0.0127	1	3.270221
	PSO	6.4554E+03	6.4555E+03	6.4555E+03	0.0335	2	3.385383
	MFO	6.4554E+03	6.4555E+03	6.4554E+03	0.0703	2	3.259365
	WOA	6.4554E+03	6.4554E+03	6.4554E+03	0.0034	2	3.259847
	SCA	6.4553E+03	6.4553E+03	6.4553E+03	0.0119	1	6.328135
CEC_19_F10	HGWPSO	20.6131	21.0177	20.8154	0.2861	1	3.170710
	GWO	20.7892	21.1719	20.9805	0.2706	2	5.888577
	PSO	21.0343	21.1643	21.0993	0.0920	5	3.101598
	MFO	20.7499	20.9260	20.8380	0.1245	3	3.195944
	WOA	20.7914	20.9931	20.8923	0.1426	4	3.304461
	SCA	20.6131	21.1317	20.9562	0.2483	1	3.274696

In Table 4, the computational time for GWO, PSO, WOA, MFO, and SCA across 30 runs is presented for all CEC_19 functions. Particularly, GWO obtained a higher computational time compared to PSO, WOA, MFO, and SCA, the proposed HGWPSO demonstrates lower computational time than the GWO and PSO. These results underscore HGWPSO successful reduction in the computational time of GWO and PSO, significantly enhancing their efficiency.

The obtained result shows that HGWPSO is the best optimizer for CEC_19_F1 through CEC_19_F4, and CEC_19_F6 through CEC_19_F9, outperforming the competitor algorithms. The analysis indicates that the HGWPSO outperforms by effectively balancing exploration and exploitation, earning it the top rank in optimizing the CEC 2019 test suite. The performance results of the HPSOGWO approach, alongside competitor algorithms, are presented in Table 4.

The convergence of HGWPSO across the CEC_19 functions is presented in Fig. 2. Across the CEC_19_F1- CEC_19_F10 test functions, HGWPSO showed better convergence compared to the other algorithms, thereby reducing the time needed for exploration and exploitation to locate the optimal global solution. This result highlights the significant improvement of HGWPSO exploration and exploitation capabilities through the integration of GWO and PSO. However, HGWPSO demonstrates slower convergence during the initial iterations for F1, F5, and F8, as presented in Fig. 2. However, as the iterations approached 100, HGWOPSO improved its convergence rate and accuracy, eventually reaching the optimal global solution. Additionally, the HGWPSO outperforms other optimization algorithms, securing the highest position among all compared methods in both optimal value and computational time for the CEC_19 functions. Additionally, boxplots demonstrated multiple runs to showcase the distribution of attained optimal values, offering insights into the consistency of the hybrid GWOPSO. The convergence curve and box plots are valuable tools for evaluating the performance of the proposed approach on the CEC_19 benchmark functions. In these box plots, the highest and lowest values are depicted by the top and bottom plots, respectively, while the rectangular box denotes the interquartile range where half of the data (50 %) is concentrated. These visual representations explain the variation of optimal values across different runs, providing strategies to minimize the risk of local optima. Particularly, the median for the HGWOPSO has a tendency towards the lower quartile, suggesting a relatively high possibility of achieving the minimum fitness function value. Fig. 2 shows the convergence characteristics and box plots of the benchmark functions using different algorithms. Therefore, it is concluded that HGWPSO demonstrated better performance in terms of both convergence and speed compared to established algorithms such as GWO, PSO, WOA, MFO, and SCA.

Fig. 2.

Convergence characteristics of the algorithms and box plot for the algorithms.

From the obtained results, it is confirmed that the HGWPSO algorithm provides better solutions for high-dimensional problems by combining GWO's exploration with PSO's exploitation strengths. However, as dimensionality increases, the search space becomes more complex, which can lead to slower convergence and a higher risk of getting trapped in local optima. While HGWPSO can improve solution quality through its hybrid approach, its performance in high-dimensional scenarios consistently depends on careful parameter tuning and may require modifications to handle the increased computational demands and maintain efficiency.

5.2. Application of HGWPSO in TL capacitance

In this section, the application of HGWPSO in calculating TL parameters, specifically capacitance, for configurations involving two, three, and four bundle conductors is presented. TL parameters, including capacitance, are important for understanding the behavior and performance of power transmission systems, with capacitance playing a significant role in overall system stability. Accurately calculating capacitance for configurations with multiple conductors can be complex and computationally intensive. The HGWPSO approach provides a better solution to this optimization problem by combining the strengths of GWO and PSO. This allows the algorithm to effectively optimize the calculation process, leading to more accurate and efficient results, which is particularly beneficial for complex optimization scenarios. Fig. 3 shows the application process of the proposed HGWPSO for TL capacitance.

Case 1

Two bundle capacitances

To evaluate the efficiency of HGWPSO compared to GWO, SCA, PSO, WOA, MFO, and SCA algorithms, a test system involving a two-bundle conductor with capacitance is used. The optimal value obtained for this case is 0.2209. The results are compared with those from the other algorithms to assess the advantages of the HGWPSO method. Comparative results for the two-bundle conductor using HGWPSO and other techniques are presented in Table 5, showing improved performance in identifying the optimal solution. To further evaluate HGWPSO, the results are obtained from more than 30 trials. The convergence characteristics of the capacitance algorithms for the two bundles over iteration cycles are shown in Fig. 4a.

Case 2

Three bundle capacitances

To assess the efficiency of HGWPSO, a capacitance problem involving three bundles of conductors is considered. The optimal value in this test system is 0.0192. The capacitance values for the three bundles, obtained using various techniques such as GWO, SCA, PSO, WOA, MFO, and SCA, are presented in Table 5. A comparative analysis of Table 5 demonstrates that the HGWOPSO obtained better solutions compared to others. Fig. 3b shows the convergence curve results with the number of generations during the optimization process achieved by the proposed hybrid GWOPSO approaches. Fig. 4b shows that the proposed HGWPSO method obtained a better convergence speed when compared to GWO and PSO.

Case 3

Four bundle capacitances

In this case, a capacitance problem involving a system with four bundles is examined to assess the efficiency of the HGWPSO method. To verify the efficiency of the proposed method, the results are compared with those obtained from GWO, SCA, PSO, WOA, and MFO methods. The optimal values obtained from these methods are presented in Table 5, showing that HGWPSO consistently produces lower values compared to GWO and the other methods. This indicates a significant performance improvement of HGWPSO. Fig. 4c illustrates the convergence scenario for the four-bundle capacitance using the proposed HGWPSO method. The convergence analysis suggests that HGWPSO converges earlier than PSO, highlighting its superior exploration and exploitation capabilities. These results contribute to a better understanding of the efficiency and convergence characteristics of HGWPSO.

Fig. 3.

The application procedure of HGWPSO for TL parameter estimation.

Table 5.

Convergence analysis of capacitance.

Algorithm	HGWPSO	GWO	PSO	MFO	WOA	SCA
Case 1
Best value	0.2209	0.2271	0.2286	0.2264	0.2300	0.2308
% Decrease		0.0225	0.0292	0.0193	0.0355	0.0391
Worst value	0.2678	0.2632	0.2885	0.2827	0.2815	0.2762
Average value	0.2459	0.2472	0.2452	0.2486	0.2391	0.2379
SD	0.0218	0.0247	0.0270	0.0287	0.0173	0.0215
Rank	1	3	4	2	5	6
Computational time (s)	0.862887	0.959313	0.902922	0.921724	0.878248	0.903898
Case 2
Best value	0.0192	0.0314	0.0595	0.0245	0.0363	0.0373
% Decrease		0.6354	2.098	0.2760	0.8906	0.9427
Worst value	0.0342	0.0479	0.0784	0.0380	0.0411	0.0491
Average value	0.0242	0.0369	0.0658	0.0290	0.0379	0.0412
SD	0.0072	0.0079	0.0091	0.0065	0.0023	0.0057
Rank	1	3	6	2	4	5
Computational time (s)	0.862074	0.894462	0.915541	0.872266	0.898639	0.897972
Case 3
Best value	0.0010	0.0074	0.0648	0.0545	0.0372	0.0667
% Decrease		0.6400	0.6380	0.5350	0.3620	0.6570
Worst value	0.0148	0.0164	0.0793	0.0698	0.0591	0.0795
Average value	0.0710	0.0445	0.0596	0.0696	0.0104	0.0056
SD	0.0061	0.0105	0.0073	0.0070	0.0043	0.0066
Rank	1	2	5	4	3	6
Computational time (s)	0.850817	1.186497	0.872551	0.875557	0.882953	0.871636

Fig. 4.

Convergence characteristics of the algorithms.

According to Tables 5 and in test case 1, the average percentage reduction rates are 0.0225 % for GWO, 0.0292 % for PSO, 0.0193 % for MFO, 0.0355 % for WOA, and 0.0391 % for SCA. In test case 2, the reduction rates are 0.6354 % for GWO, 2.098 % for PSO, 0.2760 % for MFO, 0.8906 % for WOA, and 0.9427 % for SCA. In test case 3, the reduction rates are 0.6400 % for GWO, 0.6380 % for PSO, 0.5350 % for MFO, 0.3620 % for WOA, and 0.6570 % for SCA. The proposed method performs better performance due to the combination of GWO's strong local search abilities with PSO's robust global search capabilities.

According to the obtained result presented in Table 6, HGWPSO provides a better solution for estimating the capacitance of TL with different bundled conductors. Bundled conductors introduce complexities such as mutual capacitance interactions, and environmental influences, which make accurate parameter estimation challenging. HGWPSO addresses these challenges by combining the exploration strength of GWO with the fast convergence of PSO. GWO explores a broad solution space, while PSO fine-tunes the results to achieve optimal accuracy. This hybrid approach is particularly well-suited for TL parameter estimation, ensuring precise capacitance values that directly influence power system stability, efficiency, and performance. The accurate estimation of these values is important for the optimal functioning of power transmission systems, especially with bundled conductor configurations. The results of HGWPSO in three test cases emphasize its effectiveness as compared to others. In test case 1, the best optimal value is 0.2209, in test case 2 the best optimal value is 0.0192, and in test case 3 the best optimal value is 0.0010. These optimal values are important as they demonstrate HGWPSO's ability to provide precise capacitance estimates across different configurations. This accuracy is important for maintaining the reliable operation of power systems, ensuring that the TL capacitance is optimized for various operating conditions. The results also show the algorithm's flexibility and robustness in addressing different challenges posed by bundled conductors.

Table 6.

Comparative analysis of different algorithms.

Algorithms	Best value	Best value	Best value
	Case 1	Case 2	Case 3
HGWPSO	0.2209	0.0192	0.0010
GWO	0.2271	0.0314	0.0074
PSO	0.2286	0.0595	0.0648
MFO	0.2264	0.0245	0.0545
WOA	0.2300	0.0363	0.0372
SCA	0.2308	0.0363	0.0667
HSSASCA [1]	0.2234	0.0220	0.0050
SSA [1]	0.2241	0.0225	0.0059
SOS [1]	0.2240	0.0227	0.0057
SPBO [1]	0.2242	0.0228	0.0056
FFO [1]	0.2245	0.0226	0.0052
HMFPSO [17]	0.2210	0.0216	0.0047
MWOA [36]	0.2248	0.0237	0.0051
OWOA [36]	0.2298	0.0250	0.0065
IMFO [44]	0.2240	0.0229	0.0049

In addition to these boxplots play an important role in evaluating the performance of optimization algorithms, particularly when considering complex problems such as the calculation of TL parameters. In this scenario, algorithms such as GWO, SCA, PSO, WOA, MFO, and HGWPSO are assessed, with a focus on understanding how the HGWPSO approach performs compared to other algorithms. The statistical result obtained using HGWPSO shows better performance compared to GWO, SCA, PSO, WOA, and MFO, as presented in Fig. 5. Another advantage of HGWPSO over other methods is the significant reduction in average CPU execution time. These combined results highlight the importance of the HGWPSO technique.

Fig. 5.

Box plot representation of algorithms.

5.3. Practical implication in real-world applications

This subsection explores the practical implications of TL parameter estimation and highlights how it influences transmission system performance, cost efficiency, and grid stability.

Accurate parameter estimation, particularly of TL capacitance, is important for precise system modeling and optimal power system performance. It directly impacts voltage profiles, reactive power flows, and overall system behavior. By considering various bundle configurations in capacitance estimation, engineers can achieve better accuracy, leading to improved simulations that reflect real-world conditions. This enhances predictions of system performance under different operating scenarios. Accurate capacitance estimation helps in the effective design of compensation strategies, minimizing reactive power flow, reducing energy losses, and improving voltage regulation. Precise capacitance values enable optimal placement of capacitors and reactors, enhancing transmission network efficiency.

Moreover, accurate estimations contribute to cost efficiency by preventing oversizing equipment, optimizing component sizing, and facilitating better maintenance assessments, ultimately reducing long-term operational costs. They also play a significant role in managing reactive power and maintaining voltage stability, which is vital for preventing disruptions during peak loads or disturbances. As power grids evolve, understanding capacitance characteristics with different bundle configurations becomes increasingly important for planning and integrating new or modified transmission lines. Accurate estimations help adapt systems to varying environmental conditions, ensuring reliability and effectiveness. In the field of smart grids, where real-time monitoring and control are integral, precise TL capacitance values enable the dynamic optimization of power flows. Accurate parameter estimation is important for enhancing system modeling, improving cost efficiency, ensuring grid stability, and supporting the successful operation of advanced power systems. By applying these implications, grid operators can make informed decisions for a more efficient and reliable power grid.

6. Conclusion and future research

In this research work, TL parameters are analyzed using the HGWPSO method. This approach effectively addresses the complexities associated with TL parameters. Initially, the efficiency of HGWPSO is validated using CEC_19 benchmark functions. Then, TL parameters such as capacitance are evaluated for configurations involving two, three, and four-bundle conductors. The importance of TL parameters cannot be ignored. Understanding TL parameters plays a crucial role in the efficient and reliable operation of power systems. The experimental analysis demonstrates that the HGWPSO achieves the lowest average values, with 0.15 % for test case 1 and 4.85 % for test case 2. For test case 3, the HGWPSO obtained 2.84 %, compared to other methods. The comparison between GWO, PSO, MFO, WOA, and SCA indicates the significance of selecting an appropriate optimization technique for the accurate calculation of TL parameters. However, using HGWPSO for accurately estimating TL parameters presents certain issues with capacitance, considering different bundle conductors. Initially, the hybrid approach combining GWO and PSO needs more computational resources and time, particularly for complex bundle conductor configurations. Then, HGWPSO involves multiple parameters for both GWO and PSO, requiring careful tuning to achieve optimal performance, and increasing the overall complexity. Finally, accurately modeling the capacitance for two, three, and four-bundle conductors is complex due to complex mutual coupling and geometrical arrangements. Ease of implementation, and flexibility of the algorithm while addressing these limitations are important areas for future research and development, especially for large applications in TL parameters. The effectiveness of the HGWPSO method indicates future work for further investigation.

• •Future work in accurately calculating TL parameters using an HGWPSO approach should focus on enhancing algorithm efficiency, improving modeling techniques, and ensuring scalability and flexibility.
• •Incorporating advanced computational methods, robust initialization techniques, and validation against real-world data will help overcome current limitations and improve the practical applicability of this hybrid optimization technique in complex power systems.

Funding detail

This work is supported by the innovation teams of ordinary Universities in Guangdong Province (2021KCXTD038, 2023KCXTD022), Key Laboratory of Ordinary Universities in Guangdong Province (2022KSYS003), China University Industry, University, and Research Innovation Fund Project (2022XF058), Key Discipline Research Ability Improvement Project of Guangdong Province (2021ZDJS043, 2022ZDJS068), Special Projects in Key Fields of Ordinary Universities in Guangdong Province（2022ZDZX3011，2023ZDZX2038）， Chaozhou Engineering Technology Research Center，Chaozhou Science and Technology Plan Project（202102GY17，202201GY01）, and the Quality Engineering Project of Hanshan Normal University (HSJYS-KC22719).

CRediT authorship contribution statement

Muhammad Suhail Shaikh: Writing – review & editing, Writing – original draft, Validation. Haoyue Lin: Methodology, Investigation. Gengzhong Zheng: Software, Resources, Project administration. Chunwu Wang: Project administration, Methodology, Investigation. Yifan lin: Software, Resources, Project administration. Xiaoqing Dong: Writing – review & editing, Supervision, Methodology.

Declaration of competing interest

None.