1. Introduction
Overhead transmission lines (OHTL) play a crucial role in transmitting electrical power over long distances [
1]. They consist of conductors carrying alternating current (AC) from power generation sources to distribution centers and consumers. The generation of electrical and magnetic fields in OHTL is a result of the current flowing through these conductors [
2]. The EMF generated by individual conductors in the transmission line combines to form a complex pattern around the entire line [
3]. Several reasons make EMF estimation of paramount importance. The major reason is that the EMF has a significant effect on the human’s body [
4]. Many studies in recent decades have demonstrated that EMF caused by OHTLs in residential areas is one of the main reasons for the increased incidence of cancer, especially childhood leukemia [
5]. Additionally, some studies have reported an increased risk of brain tumors among individuals exposed to prolonged and high-intensity EMFs [
6]. Furthermore, it has an extensive effect on the corrosion of buried metallic infrastructures including pipelines, cables, shielding conductors, and grounding systems [
7,
8,
9,
10]. Therefore, health scientists, utility companies, and governments have to set limits to prevent future health problems [
10]. Many organizations consider 0.4 µT as an acceptable level for long-term exposure to electromagnetic fields, while a few allow for a lower threshold of 0.2 µT as the critical point for leukemia risk [
11,
12,
13]. The National Institute of Environmental Health Sciences and the International Agency for Research on Cancer (IARC), a part of the World Health Organization (WHO), have jointly identified the range of 0.3–0.4 µT as a critical threshold for leukemia risk, categorizing it as Group-B level [
14,
15,
16]. In contrast, the International Non-Ionizing Radiation Committee (ICNIRP) recommends a considerably higher limit of 200 µT for public exposure, which significantly exceeds the recommendations of other reputable health organizations [
17]. Also, for the purpose of real-time monitoring of the systems, the engineers need to estimate the EMF with exceptionally low latency [
18].
There are some common methods among researchers for EMF estimation. Measuring the EMF in an experiment using sensors is one of the most accurate ways [
2]. That said, since in real-world cases the load of OHTL varies, the experiment is not under fully controlled conditions. Also, the change in temperature can affect the height of the conductor in the long-term experiments which will result in changes in the measured fields. This can be overcome by measuring seasonal datasets or annual ones and then creating large datasets and using “big data” techniques and AI to estimate the field values. Moreover, it needs precise instruments and skilled operators to avoid inaccuracies which turns it into an expensive practice [
19]. Another means of EMF evaluation is analytical equations which can estimate the EMF with some simplifications in the boundary conditions, limiting them to being merely useful for simple problems [
20]. Moreover, EMF can be estimated using numerical finite element methods (FEM). In this method, the entire analysis domain must be divided into small elements where the governing equations will be solved numerically [
21]. This means that an enormous number of equations have to be solved, leading to this method being computationally intensive, expensive, and time-consuming [
22]. Moreover, FEM accuracy heavily relies on the quantity and quality of meshes that are used to discretize the domain [
7]. Achieving optimal mesh refinement requires careful consideration and expertise to balance accuracy and computational costs. For complex real-world geometries or high-resolution simulations, the mesh size should be adapted to consider every dynamic change in the parameters, ensuring accurate results [
23,
24]. Furthermore, using FEM can be challenging for time-dependent EMF simulations, particularly when dynamic effects and transient behaviors are involved [
25]. All these reasons hinder the implementation of this method in certain EMF estimation scenarios and encourage researchers to explore or develop alternative methods.
Artificial intelligence (AI) techniques have been promoted among engineering and physics researchers since the early years of this century [
26,
27]. These techniques can be useful for EMF estimation due to their ability to handle and learn from complex datasets, provide accurate predictions, and offer several advantages over traditional approaches [
28,
29]. They also can be retrained by more datasets to update the model to enhance various conditions coverage or improve the accuracy of the model. One of the most renowned types of AI for engineering and physics problems is artificial neural networks (ANN), which has been inspired by the human brain’s learning and decision-making processes [
19]. The flexibility and adaptability of ANNs enable them to handle small and large datasets and extract meaningful insights from vast amounts of data/information [
30]. Also, ANN’s capability of handling non-linear relationships makes them particularly advantageous in addressing complex and dynamic problems [
27]. Moreover, they are able to predict the target parameters with remarkably high accuracy in a noticeably short timeframe. Feed-forward neural network (FFNN) is one of the most common ANN methods among researchers due to its simplicity and high accuracy [
13]. Cascade-forward neural network is a variant of ANN that is a more complex version of FFNN due to connecting the input and hidden layers to all preceding layers [
31,
32]. Due to its naturally complex design, in some cases, it can yield more accurate results than a simple FFNN [
33].
In the most recent decade, researchers tried to implement AI methods to predict electric or magnetic fields. In [
34], Ekonomou et al. initially provided a setup to measure the EMF to make a dataset for the AI model. Then, they developed a multilayer FFNN model to predict the EMF radiation via electrostatic discharges. The relative error between the predicted and actual value for EMF was reported between 5.437% and 23.620%. In [
13], Carlak et al. used a simple multilayer perceptron (MLPNN) and a generalized regression neural network (GRNN) model to predict electric and magnetic fields. They proposed several models for both electric and magnetic fields, each one considering only one longitude position of the conductor. The performance of MLPNN and GRNN models using the Root Mean Squared Error (RMSE) value as the index was reported as 0.030855 and 0.053084 for the electric field and 0.02719 and 0.03666 for the magnetic field, respectively. In [
35], Salam et al. implemented single and double-layer models based on FFNN to predict magnetic fields for four substations in Brunei. They trained the models for each of these substations separately and the results indicated that the R-squared value range of their models was from 70.9039% to 98.881%.
In [
20], Sivakami et al. suggested a model using a cuckoo search algorithm (CSA) and neuro-fuzzy controller (NFC). First, they used the cuckoo search algorithm as an optimizer to optimize the conductor spacing, which has a significant effect on the intensity of EMF to make an input dataset with minimum electric field intensity for the NFC. This is because they generated the training data using some base equations while considering some simplifications to be able to use those formulas. Finally, by training the NFC with that dataset, they reached a model that was able to estimate the intensity by 5–190% relative error for different data points. In [
30], Alihodzic et al. implemented two algorithms, namely the charge simulation method and Biot–Savart law, to generate target values for the electric field intensity and magnetic flux density datasets, respectively. After that, they developed a FFNN model using a Scaled Conjugate Gradient (SCG) as the training function. In another paper with the same process for data collection, Turajilic et al. implemented two FFNN models for each of the magnetic and electric fields. Their models’ accuracy was reported using RMSE and R-squared as indices. For the electric model, RMSE and R-squared were 0.6172 and 0.9121, while for the magnetic field, they were 0.3602 and 0.9471, respectively. However, since these papers used analytical models with some simplifications, the final model might not be accurate enough for real-world study cases [
36].
While there have been efforts to estimate or measure EMF near the OHTLs, a significant gap exists in the literature regarding the development of a fast, precise, and experimental-based EMF estimation approach, as opposed to relying solely on conventional analytical models. Recent research has predominantly focused on utilizing FFNN for EMF estimation, resulting in suboptimal accuracy. Consequently, there is a pressing need for a more advanced model capable of effectively handling highly non-linear data. One of the best AI models is the CFNN, renowned for its ability to provide accurate predictions for complex and non-linear problems. The CFNN’s sophistication lies in its capacity to update layer parameters based on the outputs from preceding layers, enabling the model to derive more optimal weight and bias factors, ultimately yielding higher accuracy results. This paper aims to propose the CFNN models for estimating the electric and magnetic fields of OHTLs. These models have been assessed through sensitivity analysis on the effective parameters, ensuring that they are trained with the best setup to reach the most accurate and stable results. In the following, first, the models will be introduced and explained in detail. Then, the sensitivity analysis process will be discussed. In the
Section 3, the results of each step of sensitivity analysis will be presented and the performance of both FFNN and CFNN models will be discussed. Finally, a brief conclusion will be made in the
Section 4.