I. Introduction
Environmental issues resulting from the power frequency electromagnetic fields of ultra-high voltage transmission lines have gained increasing prominence in China due to the widespread development of ultra-high voltage transmission line projects in the country. Simultaneously, there has been a steady rise in public concern for environmental preservation. Occasionally, issues related to electromagnetic radiation in transmission lines have given rise to complaints and disputes that have severely impeded the development of local economies and power grid construction. Four key factors contribute to the environmental problems arising from power transmission lines: radio interference, audible noise, power frequency magnetic field, and power frequency electric field. Among these, the public has expressed the most concern about the effects of power frequency electric fields and power frequency magnetic fields [1–3]. Measures such as raising the height of the conductor during installation, logically spacing the conductors, and optimizing the phase sequence and conductor arrangement can be implemented to lower the electromagnetic field value below overhead transmission line. However, these activities are bound to raise project costs. Research has shown that the electromagnetic field intensity beneath shielding wires can be successfully lowered, which can, in turn, lower project construction investment costs. Considering this context, it is crucial to develop a strategy that accounts for several erection parameters and shielding efficacy to improve the electric field environment in the target area.
A number of complex considerations must be paid attention to for optimizing the electric field strength in multi-target areas under transmission lines. These include the target area’s field strength value, the height of the shielded wire erection, the distance between wires, and the cost of the erection [4, 5]. The shielding effect in response to a single component, such as the location and quantity of shielding wires, is the primary focus of the current study. Only a few works of literature have examined this issue, considering a broad perspective that accounts for multiple relevant elements. The literature [1] predicted and investigated the relationship between changes in the quantity and location of shielding lines and the shielding effect using backpropagation neural networks. The problem of shielding lines was solved by implementing the weighted TOPSIS (approximation to ideal solution sorting) approach and shielding line position modeling analysis [2]. With regard to optimizing erections, [3] addressed the shielding line erection problem using the analytic hierarchy technique to examine the relationship between the shielding line position and the shielding effect. Furthermore, in [6], a molecular differential evolution method was employed to address the multi-objective optimization issue of shielding line erection, resulting in the establishment of a shielding effect prediction model based on a novel weighted and regularized extreme learning machine algorithm. Meanwhile, a numerical simulation model of the ion flow field of DC lines was constructed in [7], using the Kaptzov hypothesis to achieve an optimal installation plan for shielding lines while accounting for the radius and height variables. By creating models of lines and metal clothes drying racks, Li et al. [8] investigated the mitigation and enhancement impacts of different shielded wire erection techniques on induced electricity on clothes drying racks. Furthermore, Liang et al. [9] employed transmission lines, offline greenhouses, and human body models to compute the body’s induced potential and discharge current, compare several shielded wire erection scenarios, and derive the ideal shielded wire and shielding network erection plan. In [10], a shielded wire erection model was established to investigate the relationship between the shielding effect and the shielded wire position by putting the shielded wire erection plan into practice using ANSYS Maxwell software. Moreover, Qiao et al. [11] achieved better results for the shielding wire using the flux line method to forecast the ground electric field and ion current, employing the regional decomposition method to quantitatively examine the shielding effect of shielding wires. With regard to the method of arrangement, Peng et al. [12] took recourse to the finite element calculation method to examine the relationship between the shielding scheme and the power frequency distortion electric field to derive a realistic shielding wire erection plan. Furthermore, the authors of [13] explored the relationship between the number of shielding wires and shielding effectiveness by implementing the charge simulation method and the image method before and after shielding wire installation.
This article suggests using the analytic hierarchy process-improved particle swarm optimization (AHP-IPSO) method to optimize shielding line erection analysis and resolve the conflict between the shielding effect and shielding line erection costs. First, the link between the shielding efficacy of shielding wires and their quantity, erection height, and horizontal spacing is determined using the finite element approach based on ANSYS Maxwell simulation software. Second, the analytic hierarchy procedure is employed to determine the weight value of each area, accounting for the complexity of multi-target areas beneath the transmission line. This allowed for the reduction of the multi-target area into a single target area, thus solving the problem. Lastly, an ideal shielding line installation plan is constructed by applying the modified particle swarm algorithm to optimize the field strength value of the target areas.
II. Shielding Model for Power Frequency Electric Field
1. Influence of the Number of Shielding Wires on the Power Frequency Electric Field
The most important variables that influence the field strength value below the transmission line include the quantity, height of erection, and horizontal spacing of shielding wires. For this study, 500 kV ultra-high voltage transmission lines were employed as the research object to examine the shielding effect of shielding wires on field strength, as illustrated in Fig. 1. Notably, insulated wires (LGJ-240/30 type) were used for the experiment. In Fig. 1, h denotes the wire’s height, and d refers to the wire’s horizontal distance from the line’s center.
The distribution curve of the electric field intensity at 1.5 m above the ground is depicted in Fig. 2. Notably, we identified the distribution of electric field intensity under various numbers of shielding wires by examining the model parameters. Fig. 2 shows that the electric field intensity drops dramatically in the case of a single-circuit transmission line following the installation of a shielded wire. The maximum field intensity appeared 13 m from the line’s center, attaining a symmetrical distribution curve around the line’s center. Table 1 indicates that the maximum electric field intensity of a single horizontal line in the absence of a shielded wire was 4.43 kV/m, which decreased to 3.84 kV/m—13.3% less than the electric field intensity in an unprotected area—on erecting a shielded cable. Once two insulated wires were erected, the highest electric field intensity achieved was 3.49 kV/m, while the maximum electric field intensity value after three shielding wires were erected was 3.17 kV/m. Therefore, in the presence of two and three shielding wires, the maximum electric field intensity value declined by 0.35 kV/m and 0.67 kV/m, respectively, in comparison to using just one protected wire. These results emphasize that insulated wires can be installed to lessen the intensity of the electric field beneath transmission lines in locations with stringent environmental regulations.
2. Effect of Shielding Line Position on the Power Frequency Electric Fields at Target Point and Target Area
The preceding section examined the effects of various numbers of shielding wires on the electric field, determining that three shielding wires produce the most significant shielding impact. Therefore, to investigate the effect of shielding wires on field strength at various heights and spacings, three shielding wires were considered as the study object. The target area was 1.5 m higher than the ground and 20 m from the line’s center, with a 15-m area span. Assuming that the target location beneath the transmission line is located 1.5 m above the ground and 20 m from the line’s center, the influence of the electric field shielding effect on the target points and areas was investigated by varying the height and spacing of the shielding lines. The outcomes of the simulation are displayed in Figs. 3 and 4.
Figs. 3 and 4 imply a roughly negative correlation between changes in the height and horizontal spacing of shielding lines and the average electric field intensity at the target point and target region. The field strength of the target point and target area decreased with an increase in the horizontal distance, with the shielding line’s height remaining constant. The same results were obtained when the shielding line’s horizontal distance was kept constant and the erection height was increased. Therefore, it is established that the field strength value in the target point and target region can be decreased by adjusting the shielding line’s height and horizontal spacing. Effectively, both shielding effect and financial investment can be maximized by the strategic placement of shielding lines.
III. Optimization Strategy for Multi-Target Area Shielding Lines
Ultra-high voltage transmission lines are usually characterized by long transmission distances through extremely complicated corridor environments. They inevitably travel past residential neighborhoods, factories, and schools in certain areas. Consequently, transmission lines generate electromagnetic radiation in these regions. Moreover, these places often have large electric field requirements. Therefore, suppressing the electric field intensity value in these regions would require concentrated action. Notably, the field intensity value in a target area can be successfully suppressed by installing shielding wires underneath transmission lines, as demonstrated by the analysis in the preceding section.
1. Optimizing the Regional Model
When optimizing multi-target areas, the weight of each target area must be carefully taken into account. Furthermore, the electric field environment of the multi-target area must be optimized in accordance with the weight of each target area. Additionally, it is crucial to search for affordable and efficient shielding wire erection options, given the expenses involved. Assuming that homes, factories, and schools located on one side of a transmission line are the three target regions, Fig. 5 displays the top view of the target area and the location of the transmission line. Table 2 lists the regional span distribution and the distance between the target area and the line’s center. Notably, a uniform point evaluation technique was implemented to determine the average value of the electric field intensity in the target area.
2. Shield Optimization Model Construction
We designated Em as the electric field strength value of the multi-objective optimization area. Em is considered the total of the electric field intensity value at each target area and its associated weight. This allowed the field strength optimization problem in a single target region to be derived from the field strength problem in many target areas.
2.1. Shield optimization model
To examine the shielding scheme, a multi-target region located next to a 500 kV ultra-high voltage transmission line was considered, as shown in Figs. 1 and 5, with the number of shielding lines denoted as n, the height of the shielding wire erection being h, and the line spacing being d. The shielded line’s parameters were observed to be connected to the field strength Em in the target area. Therefore, a relationship table between the field strength Em and n, h, and d was constructed using the finite element method. However, it was challenging to find a mathematical function that could accurately characterize the shielding line parameters and field strength Em, since both are highly nonlinear. Therefore, a mapping relationship between the field strength and the shielding line parameters was established to optimize the field strength in the multi-target area. The literature [4] states that it is possible to derive the mapping relationship between shielding line parameters and the field strength Em, with the model function being f, as follows:
In this context, it is imperative to account for the economic cost of erecting shielding wires in addition to adhering to the safety limits for electric field strength. Consequently, a minimal cost model for shielding wires was developed, which can be expressed as follows:
In this formula, C represents the cost of erecting shielding wires, a denotes the cost of erecting per unit height of shielding wires, h represents the erection height, b is the cost per unit length of shielding wires, and s refers to the length of a single shielded wire.
IV. Improved PSO Algorithm based on AHP
1. AHP
American operations researcher Thomas L. Saaty introduced the multi-attribute analytic hierarchy process in 1977 [14–16], which involves using both qualitative and quantitative analytical functions to hierarchize and address a variety of decision-making-related difficulties. The merits of this strategy include its methodical approach, adaptability, and concision. In summary, “decompose first and then synthesize” is the central tenet of AHP. The computation process for AHP involves the following steps: Choose indicator parameters to create a model with a hierarchical structure, build judgment matrices at every level using decision vectors as the basis, determine the judgment matrix’s highest eigenvector and eigenvector value, conduct a one-time test of the judgment matrix, determine whether the weight calculation is complete, determine the relative weight of each indicator, and obtain the required value from the weight computation.
Moreover, the AHP implements four major methods to calculate weights: the least squares method, the eigenvector method, the arithmetic mean method (summation method), and the geometric mean method (square root method). The weight vectors derived from these four computation techniques often closely resemble each other. In this study, the weight value of each target area was determined using the geometric mean approach, which was chosen for the optimization of the multi-target area shielding effect.
The geometric mean method (square root method) can be defined as follows:
1.1. Building a hierarchical model
After integrating the multi-objective area shielding optimization issue into the hierarchical model, the decision-making problem was analyzed using AHP. Fig. 6 displays the hierarchical model. The target area’s population B1 per unit area and personnel exposure time on average per day comprise the middle layer, while the most significant area A is located at the top. Notably, average daily exposure of personnel B2, personnel age has a structural distribution of B3. The target area, consisting mostly of dwellings, schools, and factories, was stationed at the lowest level.
1.3. Overall hierarchical ordering and consistency testing
The consistency index (CI) was calculated using the following equation:
In this formula, λmax represents the maximum value of the judgment matrix.
Furthermore, the consistency ratio (CR) was estimated from the formula below:
In this formula, RI represents the average random consistency index.
1.4. Combined weight of each layer on the target layer
If the previous level contains n factors B1, B2, ... B3, and the corresponding hierarchical weight values are W1, W2, ... W3, respectively, and if the consistency index of certain factors of the C level for the single ordering of Bi is CIi and the corresponding average consequent consistency index is RIi, then the total ordering stochastic consistency ratio of the C level [15]:
2. PSO
2.1. Traditional PSO
Kennedy and Eberhart introduced particle swarm optimization, a novel intelligent optimization technique, in 1995. It is a heuristic algorithm that mimics flock foraging. The algorithm exploits cooperation and competition between particles to deliver intelligent guidance for optimization [17–21]. It uses the location and velocity vectors of each particle to represent it in the particle swarm optimization technique. Notably, a particle’s position vector indicates a potential course of action for solving an issue, while its velocity vector signifies the magnitude and direction of the position change [22]. Assuming that the particle swarm in an N-dimensional target search space consists of m particles, each of which possesses an N-dimensional vector, the location and flight speed state of the i-th particle in generation t can be described as follows:
(10)
In the above formula, Xi(t) and Vi(t) represent the position and velocity vectors of the particle, respectively, while i refers to the sequence number of the particle and t is the current iteration number of the i-th particle. Each particle is expected to modify their location and speed based on both the ideal group position and their optimal individual position. Therefore, the ideal positions for the group and the individual particles may be stated as follows [23, 24]:
In this formula, represents the individual optimal position of the i-th particle and signifies the global optimal position of the particle.
The state of particle i during the t-th iteration can be formulated as Eq. (10). Furthermore, the position and speed of particle i in subsequent iterations can be obtained using Eq. (12):
In this formula, c1 and c2 are expressed as learning factors, and both are positive integers. Meanwhile, r1 and r2 represent random numbers that obey the uniform distribution at the interval [0,1], and w denotes the inertia weight factor. Notably, the velocity vi,j of particle i in each dimension should satisfy −vjmax ≤ vi,j ≤ vjmax, vjmax, which indicates the maximum flight speed of particle i in the j-th dimension space. Usually, is 10%–20% of the j-th dimension’s variable search space [18].
2.2. Improved particle swarm algorithm
The particle swarm settles into a local optimum during the optimization process in the conventional particle swarm technique, since its learning factor and inertia weight remain fixed. This indicates that the limitations of the particle swarm falling into a local optimum can be overcome by varying the inertia weight and the learning factor. However, the effects of increasing the learning factor are negligible. Therefore, in this study, only the inertia weight was enhanced [25, 26]. The formula below expresses the weight factor w:
Here, wmax represents the initial value of the inertia weight factor, which is generally 0.9; wmin denotes the terminal value of the inertia weight factor, which is generally 0.4; and T and t refer to the maximum and current number of iterations, respectively. An increase in the inertia weight factor makes the optimization less likely to enter a local minimum, which is advantageous for global search, while a decrease in the inertia weight factor benefits local search and improves the algorithm’s convergence.
3. AHP–IPSO
Optimizing the field intensity shielding effect in a multi-objective zone is a challenging task. If particle swarm optimization is considered the only method that can find the best solution for this, several objective functions and restrictions must be defined, thus lengthening the solution method and increasing the likelihood of finding only a local optimal solution. By analyzing the two algorithms, the AHP algorithm is used to simplify the objective function and thus optimize the PSO algorithm. The AHP-IPSO algorithm is capable of quickly finding an optimal solution set for multi-objective areas by combining the optimization performance of PSO with the weight-solving performance of AHP. Subsequently, the two algorithms were combined with the objective function. Fig. 7 illustrates a flowchart explaining the enhanced PSO technique using AHP. The proposed technique involves the following precise actions:
1) Establishing the maximum number of iterations, learning factor, initial value of the inertia weight, and penalty factor. Initializing the population’s speed and location.
2) Establishing particle constraints and the value of the objective function.
3) Initializing the global extreme value of the population, the historical Pareto optimal solution set, the global Pareto optimal solution set, and the individual extreme value of each particle.
4) Updating the particle’s position and speed, and computing the inertia weight value (w) for this iteration.
5) Determining the fitness value of each particle and updating the population’s individual extreme values.
6) Updating the previous Pareto optimum solution set and determining the global Pareto optimal solution set for this iteration.
7) Eliminating solutions that are distant from the ideal Pareto optimum solution set from the historical Pareto optimal solution set using the slope approach.
8) Verifying whether there are more past Pareto optimal solution sets than N. If yes, Step 9 should be performed; if not, N solutions should be chosen in accordance with the crowding distance.
9) Verifying whether the algorithm completes the predetermined maximum number of iterations. If this is the case, the iterations end here. The Pareto optimum preface output is obtained, from which the Pareto optimal compromise solution can be chosen. If not, the global Pareto optimal solution set is cleared out and return to Step 4.
V. Analysis of Calculation Example Results
1. Target Area Weight Calculation
1.1. Solution to the judgment matrix weight coefficient
A hierarchical model for addressing multi-objective area shielding optimization issues was developed in Section III-3. Notably, the population density per unit area, the personnel’s average daily exposure time, and the age distribution of the personnel were the three evaluation parameters selected for evaluating electric field intensity in a target area. A pairwise comparison matrix A composed of the three standards versus the aim was established by referring to expert opinions. The population per unit area, the average daily exposure time of staff, and the personnel age are represented using the structural distribution judgment matrices B1, B2, and B3, respectively [3]:
This paper adopted the geometric mean method (square root method) to solve each judgment matrix and obtain the total weight value, as indicated by Eq. (6). By computing the aforementioned judgment matrix in MATLAB, the maximum characteristic root and the maximum eigenvalue of matrix A were determined. The characteristic root was found to be λ = 3.000, and the maximum eigenvalues of the judgment matrices B1, B2, and B3 were λB1 = 3.005, λB2 = 3.000, and λB3 = 3.000, respectively. Finally, the weight coefficients of each matrix were obtained, as shown in Table 4.
After inspection, the random CR of each matrix was found to be less than 0.1, indicating that the matrix exhibited satisfactory consistency.
1.2. Solution to total regional weight
The entire hierarchical ordering satisfied the consistency requirements, as observed from the results obtained on calculating Eq. (9), with CR = 0.048 < 0.10. Notably, the total weight of each region was derived, as indicated in Table 5.
The target areas in the considered example were residences, schools, and factories. Points were taken evenly in the span direction in each region to calculate the average electric field intensity in each of the areas, represented by E1, E2, and E3, respectively. Consequently, Em can be expressed as follows:
2. Masked Optimization Solution Set
This section uses an example to investigate the process of optimizing the field strength of a multi-target area by deploying three shielding wires to arrive at an economical shielding solution. The length s of a single shielded wire was set to 400, and the value of the shielded wire n was considered 3. Eqs. (1) and (2) can then be changed to: Em = f(h, d), MinC = a · h + 1200b. Usually, shielding wire erection cost is solely correlated with height. However, the field strength in a region is closely connected to both the height and horizontal spacing of the wires. Therefore, it is assumed that the cost of erecting a shielded wire is 1,000 ¥/m for height and 10 ¥/m for length, with C = 1000h + 12000 being the cost per unit. Table 6 provides the shielding scheme parameters (h, d) pertaining to the example considered in this section.
2.1. Validity verification
To verify the effectiveness of the improved algorithm, the initial value wmax of the inertia weight factor was set to 0.9, and the terminal value wmin was set to 0.4 for the improved particle swarm algorithm. Notably, in the traditional particle swarm algorithm, the inertia factor is set to 1. The learning factors c1 and c2 of both algorithms were set to 2, the species size of 100, and the maximum number of iterations was 200. As shown in Figs. 8 and 9, a convergence curve of the electric field strength and erection cost was achieved by verifying the calculation operation.
Figs. 8 and 9 demonstrate that the enhanced particle swarm method is able to carry out more iterations than the original particle swarm algorithm, suggesting that it is capable of conducting better global searches. This further supports the idea that the algorithm described in this article is superior to its counterparts.
2.2. Comparison of algorithms
The effectiveness of the improved algorithm was verified using Eq. (1), which confirmed that the AHP-IPSO algorithm proposed in this paper was able to successfully address the multi-target area electric field shielding optimization problem under transmission lines. To conduct an in-depth horizontal comparison, the AHP-IPSO algorithm, non-dominated sorting genetic algorithms (NSGA), firefly algorithm (FA), simulated annealing (SA), and optimal compromise solution for PSO were considered.
Table 7 shows that the electric field intensity values of the NSGA, FA, SA, and PSO optimization algorithms are 1,908 V/m, 1,913 V/m, 1,919 V/m, and 1,897 V/m, respectively, while their installation costs are 24,423 ¥, 24,409 ¥, 24,362 ¥, and 24,410 ¥, respectively. In contrast, the Pareto optimal compromise solution obtained for the AHP-IPSO algorithm estimated an electric field intensity value of 1,891 V/m and an erection cost of 24,350 ¥, indicating that the proposed algorithm effectively reduces both erection cost and electric field intensity. Therefore, the effectiveness of the AHP-IPSO algorithm in the case is demonstrated.
2.3. Multi-objective optimization scheduling
The inertia weight coefficient attained an initial value of 0.9 and a terminal value of 0.4 in the multi-objective particle swarm algorithm. Notably, a population of 200 and a maximum of 200 iterations were considered. The learning factors c1 and c2 were both set to 2. Fig. 10 displays the Pareto optimal solution set obtained for the multi-target shielding strategy by encoding multi-target particle swarms.
Fig. 10 presents the variations in shielding effects in response to varying erection heights and line spacing when three shielded lines are erected. Notably, the cost of erection and the shielding effect are inversely correlated—the greater the shielding effect, the higher the erection cost. The shielding optimization solution is represented in terms of each point on the Pareto optimal solution set curve. We used the slope approach to carefully weigh the erection cost and shielding effect to ultimately identify the best erection solution for the given scenario, representing the optimal compromise in the Pareto optimal solution set. In Fig. 10, considering that the target area’s electric field intensity is 1,891 V/m, the shielding cost that represents the best compromise solution is 24,350 ¥. Notably, at this point, the height of the shielding wire was considered to be roughly 13 m, and its horizontal spacing was 5.5 m. Overall, this installation plan guarantees economic rationality and the lowest electric field intensity value for several target regions.
VI. Conclusion
In this study, the shielding effect of shielding wires on the electric field intensity beneath transmission lines was analyzed using the finite element method. Moreover, the effects of various numbers, heights, and spacings of shielding wires on electric field intensity were thoroughly examined. In addition, to optimize the electric field intensity under the transmission line in a multi-target area, the AHP-IPSO algorithm was adopted. Its results were compared with those of the NSGA, FA, SA, and PSO algorithms to confirm that the algorithm employed in this paper offers better adaptability and achieves better optimization performance. The AHP-IPSO algorithm was specifically implemented to study two significant aspects: the shielding effect and erection investment. Ultimately, the following conclusions were derived from the findings of this study:
• The field strength below the transmission line can be effectively inhibited by installing shielding wires. The maximum field strength value decreased with an increasing number of shielding wires. In this study, the maximum field strength decreased to 3.84 kV/m after one shielded wire was erected, which was 13.3% less than it would have been without using a shielded wire. The maximum field strength value decreased by 9.2% and 17.4% on erecting two and three shielding wires, respectively, in comparison to erecting only one shielded wire. Furthermore, when examining the target point and target region, it was discovered that altering the height and the spacing between shielding lines might modify the field strength value.
• The enhanced particle swarm algorithm yielded the optimal Pareto solution considering multi-objective conditions, which led to the identification of the optimal shielding line erection position. The AHP-IPSO algorithm efficiently optimized the electric field intensity in multi-objective areas under the transmission line. Furthermore, the AHP was utilized to obtain a simplified objective function of the weight of each area by accounting for various factors, such as shielding line erection height, horizontal spacing, and erection cost.
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