A Novel Detection Algorithm for the Icing Status of Transmission Lines

Abstract

As more and more transmission lines need to pass through areas with heavy icing, the problem of transmission line faults caused by ice and snow disasters frequently occurs. Existing ice coverage monitoring methods have defects such as the use of a single monitoring type, low accuracy of monitoring results, and an inability to obtain ice coverage data over time. Therefore, this study proposes a new algorithm for detecting the icing status of transmission lines. The algorithm uses two-dimensional multifractal detrended fluctuation analysis (2D MF-DFA) to determine the optimal sliding-window size and wave function and accurately segment and extract local feature areas. Based on the local Hurst exponent ($L_h\left(z\right)$) and the power-law relationship between the fluctuation function and the scale at multiple continuous scales, the ice-covered area of a transmission conductor was accurately detected. By analyzing and calculating the key target pixels, the icing thickness was accurately measured, achieving accurate detection of the icing status of the transmission lines. The experimental results show that this method can accurately detect ice-covered areas and the icing thickness of transmission lines under various working conditions, providing a strong guarantee for the safe and reliable operation of transmission lines under severe weather conditions.

1. Introduction

As the world experiences the onset of a 300-year cycle known as the ‘30-Year Little Ice Age’ alongside the exacerbation of global warming, the frequency and severity of extreme weather events leading to disasters are on the rise. The increasing occurrence of icing disasters in local power grids indicates a worsening trend, making power grid icing the new normal. This presents significant challenges for the safe operation and reliability of power grids. At present, the most common types of icing on overhead transmission lines can be roughly divided into the following four categories: snow glaze, rime, glaze, and mixed glaze (Figure 1). Glaze is formed by snow accumulating on a wire; it has poor adhesion abilities and easily falls off due to wind. Rime is composed of small ice particles with pores between gaps, and the cohesion between adjacent ice particles is poor, so it is also a kind of icing that easily and naturally falls of off due to the effect of wind. Rime is attached to transmission wires and towers in the form of transparent or translucent hard ice. The ice particles are hard and dense, which means that it is easy for the wires to break. This is a relatively bad type of icing. Mixed glaze involves glaze and rime at a temperature below the freezing point; due to wind, this mixed formation is frozen, resulting in strong adhesion and high density. The formation speed is particularly fast, and it causes great harm to the wire.

Transmission lines are an important part of a power grid; maintaining their safe operation is part of the stable operation of the power system to guarantee social and economic development [1]. Transmission lines are very widely distributed, and many lines need to pass through high-humidity and low-temperature areas. Micro-terrain and micro-meteorological factors often lead to disasters due to conductors that are covered with snow and ice, and many areas of domestic and foreign power systems have been deeply affected in this way [2,3]. Hence, methods for detecting conductor icing have garnered significant attention from scholars worldwide. Once transmission lines are covered with thick ice, their structure and performance may be severely affected, potentially leading to power supply accidents, such as wire overstretching, breakage, or even tower collapse [4,5]. In the research by Xu et al. [6], it was found that due to China’s complex and diverse terrain, many regions’ transmission lines are frequently affected by extreme weather conditions, such as ice and snow. Hence, precise monitoring and analysis of transmission line conditions across various weather conditions are crucial for bolstering the safety and reliability of transmission lines. This not only aids in averting ice and snow calamities but also effectively ensures the safe and steady functioning of power systems [7,8]. The expanding magnitude, capacity, and complexity of power grids, along with their widening geographical coverage, subject power production to increasingly severe meteorological and weather conditions, consequently amplifying the degree and intensity of their impact. In recent years, the use of image segmentation techniques to study images of transmission line icing caused by snowfall, freezing rain, snow, and frost has become a research hotspot. Research in this field not only helps us gain a deeper understanding of the impact of icing phenomena on transmission lines but also provides powerful technical support for the maintenance and management of power systems, demonstrating broad application prospects and value in the field of electricity.

Image segmentation technology, as a core branch of computer vision, aims to divide images into non-overlapping regions to accurately extract and analyze target information [9,10,11]. In the monitoring and management of power transmission lines, image segmentation technology plays a crucial role. Through this technology, we can accurately segment and identify different targets or regions in transmission line images, thereby achieving meticulous monitoring and in-depth analysis of transmission line status. By analyzing images of transmission line icing and related data, we can promptly detect icing conditions, effectively prevent safety hazards caused by icing, and provide important reference and technical support for the operation and management of transmission lines and subsequent de-icing work [12,13,14].

To date, numerous scholars have conducted in-depth explorations of the use of image segmentation technology for images of transmission line icing and weather-affected transmission lines. They have employed various image processing methods and algorithms, such as convolutional neural networks (CNNs) based on deep learning [15,16], genetic algorithms [17,18], and optimal threshold methods [19,20], to achieve precise segmentation and identification of transmission line images. These studies not only provide important technical support to enhance the safety and reliability of transmission lines but also lay a solid theoretical foundation for subsequent research. For example, Hu et al. [21] proposed a method for recognizing the thickness of ice on transmission lines based on visual sensor data. They segmented the iced areas of transmission lines in images, extracted the edge features of iced areas, and calculated the ice thickness based on the pixels of those areas. Zhong et al. [22] designed an algorithm that can automatically detect icing conditions on transmission lines and can adapt to different outdoor icing environments. As machine learning and neural network technologies continue to advance in the realm of image segmentation, an increasing array of innovative methods are being deployed for the analysis of transmission line images. Yue et al. [23] determined the presence of ice by studying the shapes of edge components and classified ice and non-ice images by inputting feature matrices into a multilayer perceptron neural network. Hu et al. [24] proposed an optimized network, SGAN-UNet, composed of generative adversarial networks (GANs) and S_Unet; it significantly improved the segmentation performance according to a comparison of ground-truth-labeled images of transmission line icing with predicted images. In addition, Yang et al. [25] extracted texture features of six typical types of images of transmission line icing using uniform local binary patterns (ULBPs) and improved uniform local binary patterns (IULBPs), providing an effective means for identifying icing types. Huang and Wei [26] designed a master–slave monitoring structure that effectively addresses the shortcomings of existing online monitoring technologies for transmission line icing and provides more effective guidance for de-icing work. These studies not only promote the application of image segmentation technology in the field of power but also provide powerful technical support for the safe operation and maintenance of transmission lines. In this study, the multifractal spectrum in the Multifractal Detrended Fluctuation analysis (MF-DFA) method that we used essentially embodied features of symmetry at different scales, reflecting the scaling invariance of complex structures across local regions. At the same time, the sliding window method that we used also exhibited symmetry in two dimensions, which helped to uniformly capture local features in the images.

As more transmission lines need to traverse areas prone to severe icing, the problem of transmission line faults caused by ice and snow disasters has become increasingly frequent. Traditional icing monitoring methods have several shortcomings, such as a limited number of monitoring types, low accuracy of monitoring results, and the inability to promptly obtain ice coverage data. These issues pose significant threats to the safety of power grids and increase maintenance costs and operational difficulties. Therefore, developing a new algorithm capable of effectively and accurately monitoring the icing status of transmission lines are of critical importance. Existing icing monitoring methods still have some immediate monitoring types and cannot comprehensively cover different icing scenarios. In addition, the accuracy of monitoring results is insufficient, making it difficult to provide reliable data support. Additionally, existing methods are not precise enough in detecting icing thickness to meet the demand for the accurate measurement of transmission line icing thickness. These shortcomings limit the grid’s response speed and protective capabilities under adverse weather conditions. Therefore, a new detection algorithm is urgently needed to overcome these limitations and provide more accurate information on icing conditions.

In order to address the challenging issue of ice detection on transmission line wires, this study proposes a method for detecting the state of ice coverage on transmission wires based on 2D MF-DFA. By converting images of ice-covered wires to grayscale, we determined the optimal sliding-window size and wave function. Building upon this foundation, we divided the ice-covered images and accurately extracted local feature areas. By combining 2D MF-DFA, we could precisely identify both the icing area and thickness of the wire, enabling accurate detection of its icing state and ensuring the safe operation of the power system. This approach holds significant practical value in engineering applications.

The main contributions of this study are summarized as follows:

• Novel Algorithm: We developed a new detection algorithm for the icing status of transmission lines using 2D MF-DFA.
• Accurate Detection: The algorithm accurately segments and extracts local feature areas, detecting ice-covered areas and measuring icing thickness with high precision.
• Versatility: We demonstrated the algorithm’s adaptability across various environmental conditions and image types, showing potential for broader applications in power grid monitoring.
• Practical Implementation: We provide the detailed MATLAB code in the Appendix A, facilitating the application and further development of the proposed method.

The structure of the remaining sections of this article can be outlined as follows. In Section 2, we present the MF-DFA method and provide a comprehensive description of the edge detection algorithm. Section 3 is dedicated to empirical experiments, where we showcase the image detection results and engage in discussions. Finally, in Section 4, we draw conclusions based on our findings.

2. Detection Principle of the State of Ice Coverage on Transmission Lines Based on 2D MF-DFA

2.1. Preliminary Concepts

Multifractals are structures that exhibit complex scaling behaviors across different regions. To define multifractals formally, suppose that for every point ($\theta$) in a set ($\mathcal{X}$), there exists a function ($\gamma \left(\theta \right)$) such that the measure ($\rho \left(B_s\left(\theta \right)\right)$) of a ball ($B_s\left(\theta \right)$) (a sphere centered at $\theta$ with a radius of s) scales as follows:

$$\rho \left(B_s\left(\theta \right)\right)\propto s^{\gamma \left(\theta \right)},$$

where $\rho$ is a measure defined on a subset of $\mathcal{X}$. This set ($\mathcal{X}$) is referred to as a multifractal set. The function $\gamma \left(\theta \right)$ is called the local Holder exponent or local scaling exponent, which characterizes how the measure scales locally around point $\theta$.

To further understand the distribution of the scaling exponents ($\gamma \left(\theta \right)$) within the set, we define the level sets as follows:

$$E_{\gamma }=\{\theta \in \mathcal{X}:\gamma \left(\theta \right)=\theta \}.$$

The singularity spectrum ($g\left(\gamma \right)$) is then defined as the fractal dimension of the $E_{\gamma }$ set:

$$g\left(\gamma \right)=\mathrm{FD}\left(E_{\gamma }\right),$$

where $\mathrm{FD}(·)$ denotes the fractal dimension. The pair of $(\gamma ,g(\gamma \left)\right)$ forms the singularity spectrum, which provides insights into both the local properties (through $\gamma$) and the overall distribution of these local properties (through $g\left(\gamma \right)$).

The principle of MF-DFA begins with a core step. First, a ”profile” is constructed from the data, which is then divided into segments. Each segment is fitted with a local trend, which helps interpret the behavior of fluctuations. By calculating the variance of each segment, we can evaluate fluctuations across different time scales. In that case, the $\gamma -g\left(\gamma \right)$ profile reflects how frequently different scaling exponents ($\gamma$) occur across the set, giving a comprehensive picture of the underlying complexity.

Multifractal detrended fluctuation analysis (MF-DFA) is a method that is widely used to analyze multifractal properties in various types of data, including time series and images. An intriguing aspect is that MF-DFA also helps identify the source of multifractality. By comparing the original time series with its shuffled version, we can reveal whether multifractality arises from a broad probability distribution or from long-range correlations. If the shuffled series still exhibits multifractality, this indicates that the cause is the probability distribution rather than correlations [27]. Initially, a 1D time series was introduced by Kantelhardt [27], and we extended the method to 2D images in this study to analyze the multifractal properties of images. The steps of the 2D MF-DFA method are outlined as follows:

Step 1: Image Discretization: We discretized the 2D image into an $A\times B$ matrix $Y_{p,q}$, where $p=1,2,\dots ,A$ and $q=1,2,\dots ,B$. The image was divided into non-overlapping sub-regions of equal length (r), denoted as $A_r\times B_r$. Each sub-region is expressed as follows:

$$Y_{x,y}(p,q)=Y((x-1)r+p,(y-1)r+q)\mathrm{for}1\le p,q\le r.$$

Next, the cumulative sum of surface pixels within each sub-region is calculated as follows:

$$V_{x,y}(p,q)=\sum _{m=1}^p\sum _{n=1}^q{Y}_{x,y}(m,n).$$

Step 2: Detrending the Sub-Regions: To remove trends within each sub-region, we fit a plane (${\overline{V}}_{x,y}(p,q)$) using the following simple function:

$${\overline{V}}_{x,y}(p,q)=ap+bq+c,$$

where a, b, and c are fitting parameters derived through least squares regression. The residual matrix, representing the deviation from the trend, is given by

$$v_{x,y}(p,q)=V_{x,y}(p,q)-{\overline{V}}_{x,y}(p,q).$$

Step 3: Detrended Fluctuation Function: The detrended fluctuation function ($G^2(x,y,r)$) for each sub-region is then computed as

$$G^2(x,y,r)={\displaystyle \frac{1}{r^2}}\sum _{p=1}^r\sum _{q=1}^r{\left(v_{x,y}(p,q)\right)}^2.$$

Step 4: Wave Function Calculation: For each sub-region, the z-order fluctuation function is derived as follows:

$$G_z\left(r\right)={\left\{{\displaystyle \frac{1}{A_r{B}_r}}\sum _{x=1}^{A_r}\sum _{y=1}^{B_r}{\left[G(x,y,r)\right]}^z\right\}}^{{\displaystyle \frac{1}{z}}},$$

with the following special case for $z=0$:

$$G_z\left(r\right)=exp\left({\displaystyle \frac{1}{A_r{B}_r}}\sum _{x=1}^{A_r}\sum _{y=1}^{B_r}ln\left[G(x,y,r)\right]\right).$$

Step 5: Scale Dependence: By adjusting the sub-region size (r), the following power-law relationship is obtained between $G_z\left(r\right)$ and r:

$$G_z\left(r\right)\propto r^{h\left(z\right)},$$

where $h\left(z\right)$ is the generalized Hurst exponent. The key lies in how the fluctuation function ($G_z\left(r\right)$) scales with the time scale (r), which is characterized by the generalized Hurst exponent ($h\left(z\right)$). If $h\left(z\right)$ varies with z, the time series exhibits multifractality. If $h\left(z\right)$ remains constant, the time series is monofractal [27].

Step 6: Sliding Window for Local Analysis: To capture local multifractal properties, a sliding window of size $w\times w$ is used to compute $h\left(z\right)$ for each pixel $(p,q)$, resulting in the local Hurst exponent ($L_{h\left(z\right)}$) for each pixel.

The multifractal spectrum can be derived using the Legendre transform of $h\left(z\right)$ as follows:

$$\alpha =h\left(z\right)+z{h}^{\prime }\left(z\right),f\left(\alpha \right)=z[\alpha -h\left(z\right)]+1,$$

where $\alpha$ represents the singularity strength and $f\left(\alpha \right)$ denotes the singularity spectrum, which captures the distribution of singularities in the image.

Figure 2 shows the sliding window technique used to compute the local Hurst exponents across the image. Through this analytical approach, MF-DFA can uncover complex hidden structures in both time series and spatial data, providing researchers with a powerful tool for understanding these multifractal phenomena.

2.2. Proposed Algorithm for Icing Status Detection

Multifractal analysis has emerged as a powerful tool in image processing, particularly in the context of texture analysis and segmentation. Traditional fractal analysis is often limited by its inability to capture the heterogeneity and complex scaling behaviors found in real-world images. Multifractal techniques, however, allow for a more nuanced characterization by describing variations in local singularities across an image. This is especially valuable in fields such as medical imaging [28], remote sensing [29], and material science [30], where images often exhibit complex structures that require advanced segmentation methods.

The algorithm is grounded in a multifractal segmentation methodology, where each image is depicted as a two-dimensional matrix featuring 256 gray levels. By employing the described algorithm, we compute $L_{h\left(z\right)}$ for these images. Initially, we traverse the entire image using a $v\times v$ window, computing the generalized $L_{h\left(z\right)}$ for every pixel $(p,q)$. By identifying the maximum and minimum $L_{h\left(z\right)}$ values across all pixels, we establish an interval of $[{\left(L_{h\left(z\right)}\right)}_{min},{\left(L_{h\left(z\right)}\right)}_{max}]$.

Following this, we partition the interval into L equidistant segments. Within each segment, we employ the box-counting method to calculate the fractal dimension of a sub-image constructed from particular $L_{h\left(z\right)}$ values within that segment. This process yields a series of fractal dimensions, denoted as $d_1\left(L_{h\left(q\right)}\right)$, $d_2\left(L_{h\left(z\right)}\right)$, …, $d_n\left(L_{h\left(z\right)}\right)$. We set $L=30$ for subsequent evaluations.

Fractal and multifractal dimensions provide critical insights into the self-similarity and scaling properties of images. These dimensions have been successfully applied in image texture classification, image compression, and biomedical image analysis. One key advantage of multifractal analysis is its ability to capture both smooth and rough regions within an image, making it a versatile tool for segmentation tasks in complex images. In practice, applying multifractal-based segmentation to medical [31] or natural images [32] can enhance accuracy in identifying structures such as tumors, textures in natural scenes, or even geological formations in satellite imagery.

The choice of window size significantly impacts the algorithm’s precision. A large window may overlook details, while a small one might lack sufficient data points for curve fitting. Hence, we adopt distinct window sizes ($w\times w$) and scales (r) within each window to accommodate varying pixel densities in different images.

Within each segment, we employ a box dimension to compute an image segment consisting of $L_{h\left(z\right)}$ values within the interval of $[{\left(L_{h\left(z\right)}\right)}_{min},{\left(L_{h\left(z\right)}\right)}_{max}]$. Subsequently, we utilize a box with dimensions of $\gamma \times \gamma$ to encompass a region within the sub-image, counting the instances where $L_{h\left(z\right)}$ falls within the interval. As the box traverses the entire sub-image, we tally the recorded box count as $N\left(\gamma \right)$. This process yields a series of $N\left(\gamma \right)$ values computed according to the following power-law relationship:

$$N\left(\gamma \right)={\left({\displaystyle \frac{1}{\gamma }}\right)}^{-d}.$$

(1)

The spectral function of $L_{h\left(z\right)}$, denoted as $d\left(L_{h\left(z\right)}\right)$, in MF-DFA is expressed as

$$d\left(L_{h\left(z\right)}\right)=lim{\displaystyle \frac{lnN\left(\gamma \right)}{ln(1/\gamma )}}.$$

(2)

where we consider $\gamma =2,4,8,16$, and 32. $d\left(L_{h\left(z\right)}\right)$ serves as a holistic measure of singularity within the image. Singularity assessment relies on $d\left(L_{h\left(z\right)}\right)$, which guides image segmentation. If $d\left(L_{h\left(z\right)}\right)$ is approximately 1, the pixels denote smooth boundary points; value near 2 indicates a smooth surface. By analyzing the numerical outcomes, the image’s singularity is ascertained. The range of $d\left(L_{h\left(z\right)}\right)$ assists in identifying both singular regions ($1\le d\left(L_{h\left(z\right)}\right)\le C$) and boundaries ($C\le d\left(L_{h\left(z\right)}\right)\le 2$), where C acts as a threshold.

To demonstrate the scientific rigor and validity of the method proposed in this study, we outline the implementation steps of the algorithm for image segmentation as follows.

Step 1 

We begin by reading the image and converting it to grayscale, as shown in Figure 3. The image is processed in blocks by determining the optimal sliding-window size and the order of the fluctuation function, as shown in Figure 4a.

Step 2 

Taking values of s ranging from 2 to 4, each sub-image can be divided into $5\times 5$, $3\times 3$, and $2\times 2$ sub-regions for $r=2,3$, and 4, respectively, as shown in Figure 5.

Step 3 

Then, we can obtain the log–log plots of the fluctuation function ($F_q\left(s\right)$) versus s. The  line fit using the least squares method is shown in Figure 6.

Step 4 

Blockwise cumulative plane fitting is performed for each extracted local area. The algorithm iterates through the image, extracting a local area of size $w\times w$, with each pixel as its center, as shown in Figure 4b. The local generalized Hurst exponent ($L_h\left(z\right)$) for each pixel is calculated within each sub-block. The stored results are depicted in Figure 7.

Step 5 

Matrix J is filtered to obtain matrix C. The minimum and maximum values of C are used to create an arithmetic progression ($A_n$), dividing it into f equal parts. Matrix D is created with the same size as C and filled with 255. C is traversed, and  the corresponding positions in D are set to 0 if the value at $C(x,y)$ falls within $(A\left(k_1\right),A\left(k_2\right))$, where $0\le k_1\le k_2\le n$; otherwise, it is left unchanged. Finally, the segmentation results are displayed using the imshow and contourf functions in MATLAB to visualize matrix D.

Figure 3. Original grayscale icing image.

Figure 3. Original grayscale icing image.

Figure 4. (a) Initial state of a sliding window with a size of $11\times 11$ in an icing image; (b) route passing points of the sliding window in the retinal icing image.

Figure 4. (a) Initial state of a sliding window with a size of $11\times 11$ in an icing image; (b) route passing points of the sliding window in the retinal icing image.

Figure 5. Sub-regions of the sub-images of (a) $r=2$, (b) $r=3$, and (c) $r=4$.

Figure 5. Sub-regions of the sub-images of (a) $r=2$, (b) $r=3$, and (c) $r=4$.

Figure 6. Double-log plots of the sub-image.

Figure 6. Double-log plots of the sub-image.

Figure 7. Values of the local Hurst exponent as the sliding window moves.

Figure 7. Values of the local Hurst exponent as the sliding window moves.

To improve the readability, we provide the pseudocode in Algorithm 1.

Algorithm 1 Multiscale Fractal Dimension Calculation for Image Segmentation.

Input: Color image $I_{RGB}$ ▹ Step 1: Load and preprocess the image Convert $I_{RGB}$ into grayscale: $I_{gray}=rgb2gray\left(I_{RGB}\right)$ Convert the grayscale image into double precision: $I_{double}=double\left(I_{gray}\right)$ Get image dimensions $(M,N)=size\left(I_{double}\right)$ Set the window size w, calculate the half-window size: $w1=(w-1)/2$ ▹ Step 2: Initialize the variables for the calculation of the fractal dimension Initialize matrices J and $DE$ for storing local fractal dimensions and errors for q in the range (e.g., $q=-10$) do       for each window position $(k1,k2)$ inside the image do             Extract sub-image $I=I_{double}(k1-w1:k1+w1,k2-w1:k2+w1)$             for each scale s from 2 to $w1$ do                    Compute the number of sub-regions: $Ms=\lfloor m/s\rfloor$, $Ns=\lfloor n/s\rfloor$                    Initialize matrix $F1$ to store local fluctuations                    for each sub-region $(u,v)$ do                          Extract sub-region $X1=I\left(\right(u-1)\ast s+1:u\ast s,(v-1)\ast s+1:v\ast s)$                          Compute cumulative sum: $z=cumsum{\left(cumsum{\left(X1\right)}^{\prime }\right)}^{\prime }$                          Fit a plane to z and compute residual error $rr$                          Compute fluctuation: $F1(u,v)=\sum \sum r{r}^2/s^2$                    end for                    Calculate fluctuation function: $F\left(s\right)={\left(\sum \sum F{1}^{q/2}/(Ms·Ns)\right)}^{1/q}$             end for             ▹ Step 3: Fractal dimension estimation via the log-log fit             Fit a line to the log-log plot: $p=polyfit(log(s),log(F\left(s\right)),1)$             Extract the slope as a local fractal dimension: $J(k1-w1,k2-w1)=p\left(1\right)$             Compute the fitting error: $DE(k1-w1,k2-w1)=\sqrt{\sum {(yy1-yy)}^2}/{(1+p\left(1\right))}^2$       end for       Compute the mean fractal dimension: $MH(q+11)=mean\left(mean\right(J\left)\right)$       Compute the mean fitting error: $MDE(q+11)=mean\left(mean\right(DE\left)\right)$ end for Output: Local fractal dimension matrix J, fitting error matrix $DE$ ▹ Step 4: Post-process the fractal dimension matrix Apply the median filter: $C=medfilt2(J,[2,2\left]\right)$ Crop the matrix: $C=C(2:end-1,2:end-1)$ Set the segmentation levels $n=30$, compute the range $a\left(1\right)=min\left(C\right)$, $a\left(n\right)=max\left(C\right)$, $dd=\left(a\right(n)-a(1\left)\right)/(n-1)$ for $i=2:n-1$do       Calculate the segmentation threshold: $a\left(i\right)=a\left(1\right)+(i-1)·dd$ end for ▹ Step 5: Segment the image based on the fractal dimensions Initialize a binary image $D=255·ones\left(size\right(C\left)\right)$ for each pixel $(i,j)$ in C do       if $a\left(6\right)<C(i,j)<a\left(30\right)$ then             Set $D(i,j)=0$       end if end for Display the segmented image D Output: Segmented binary image D

3. Experimental Results

This section presents a series of experiments conducted on both artificially synthesized images and real-world images of icing on transmission lines to demonstrate the effectiveness and reliability of the proposed method. The dataset of line icing was taken from a large-scale line icing event that occurred in November 2020 in Siping, Jilin Province, China. The images were taken on site by maintenance personnel from the Siping Power Supply Company of the State Grid. All experiments in this study were performed using MATLAB R2021b in an environment equipped with an AMD Ryzen 7 7840H CPU clocked at 3.80 GHz.

3.1. Experimental Results of Icing Detection

3.1.1. Analysis of Synthetic Images

In the first set of experiments, we artificially synthesized an image containing regions with diverse features, as shown in Figure 8a. By setting the sliding-window parameter (w) to 7 and performing edge segmentation based on the local Hurst exponent ($Lh\left(z\right)=Lh\left(6\right)$), we successively adjusted the constraint parameters ($(k_1,k_2)$ to $(13,16)$, $(3,6)$, and $(9,26)$), resulting in the segmented regions depicted in Figure 8b–d, respectively.

It can be observed that by adjusting the parameters, we achieved the desired segmentation regions. This simple image segmentation process serves to illustrate how our algorithm operates and demonstrates the feasibility of obtaining target segmentation regions by adjusting various parameters, highlighting the method’s versatility.

3.1.2. Detection Results for Line Icing without Background Interference

First, we performed multifractal analysis on the icing image shown in Figure 9a to ensure that it possessed multifractal features. The size of the image was $256\times 256$. Here, we set the r parameter to vary from 2 to 127 with a step of 1. In addition, the z parameter changed from $-10$ to 10. We calculated the double-logarithmic curve between $ln\left(r\right)$ and $ln\left(G_z\left(r\right)\right)$, as shown in Figure 9b. Based on the slope of the curve, we calculated the generalized Hurst exponent, as shown in Figure 9c. We can see that the fractal exponent changed with the variation in z and gradually decreased, indicating that the image exhibited multifractal characteristics, which could also be indicated in the form of the multifractal spectrum shown in Figure 9d.

After determining that the ice-covered image had multifractal features, we next used the local Hurst exponents to detect and segment the ice-covered areas in the image. To visually demonstrate the operation process of our algorithm, we first used a synthesized image with a size of $20\times 20$ for illustration. The original synthesized image is shown in Figure 10a, and the corresponding pixel values are shown in Figure 10b. We utilized a $5\times 5$ sliding window to calculate the local Hurst value of the image, resulting in a $16\times 16$ pixel image, as shown in Figure 10c. We arranged the feature values in Figure 10c in ascending order and set them as an arithmetic sequence with $n=60$. We also set $(k_1,k_2)=[8,60]$ and defined the feature values located in this interval as 0 and 255 otherwise. Then, the newly obtained matrix was point-multiplied with the original image to obtain a new matrix, which was the segmented image. We present the contour segmentation state image and segmentation results in Figure 10e.

We also applied the proposed algorithm to other real-world scenarios. Given that various electrical and mechanical faults caused by icing on transmission lines significantly impact the safety and stability of power grids, we conducted segmentation experiments on different images of icing on transmission lines. As shown in Figure 11, with parameters set to $z=-10$ and $w=5$, we segmented a portion of the transmission lines in an image of icing at $(k_1,k_2)=(12,23)$ and isolated the icing at $(k_1,k_2)=(15,30)$.

These results were achieved while effectively retaining the detailed features of the target segmentation regions, demonstrating the robustness of our proposed algorithm.

3.1.3. Detection Results for Line Icing against a Chaotic Background

Furthermore, we applied the algorithm to the icing image shown in Figure 12a. In this experiment, we set z to −10 while keeping the sliding-window parameter (w) at 5. Despite the complex background of the icing image, which included numerous branches and weeds, we obtained the desired icing segmentation region by adjusting the parameters from $(25,30)$ to $(6,22)$, then to $(6,24)$. The segmentation was unaffected by the cluttered background, showcasing the strong segmentation performance and robustness of the proposed algorithm.

Using the characteristics of the algorithm proposed in this study, we were able to calculate the ice thickness in different images of icing at the same camera position. Taking the camera perspective of the icing image in Figure 12a as an example, we show the segmentation results in Figure 12d on a grid, as shown in Figure 13. Figure 13a indicates the segmented grayscale image result. In Figure 13b, we display the segmentation results when processed using a two-dimensional grid, and in Figure 13c, we enlarge the details on the grid. We denote the distance between the left segmentation edge and the right segmentation edge in Figure 13d as T (see Figure 13d) and the distance of the manually measured icing edge as K. Then, we define $U=K/T$ as the icing thickness coefficient.

Through coefficient conversion calculation, we calculated 20 distances of icing on the left side of the transmission line, as shown in Figure 14; the calculated average value was $1.7485$ cm.

In the subsequent experiments, by obtaining the segmentation results for icing images at this camera position, we were able to multiply the distance ($T_0$) in the image by the icing thickness coefficient (U) to obtain the real icing thickness. This method alleviates the difficulty and high intensity of manual icing measurement, as well as the pain point of being unable to achieve it on-site due to the difficulty of personnel reaching the scene.

In addition, we selected another image of ice coverage to verify the robustness and effectiveness of the algorithm for ice-covered area segmentation, as shown in Figure 15. As shown in Figure 15c, our algorithm was able to clearly capture the ice-covered area.

3.2. Multiple Evaluation Indicators

In this study, we evaluated the performance of the segmentation algorithm using the following four values from the confusion matrix: true positives (TPs), true negatives (TNs), false positives (FPs), and false negatives (FNs). Based on these values, we calculated several key performance metrics. Accuracyrepresents the proportion of correctly classified pixels and is computed as $\mathrm{Accuracy}={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}}$. Precision measures the proportion of true positives among the predicted positives, given by $\mathrm{Precision}={\displaystyle \frac{TP}{TP+FP}}$. Recall evaluates the proportion of actual positives that are correctly predicted, expressed as $\mathrm{Recall}={\displaystyle \frac{TP}{TP+FN}}$. To balance the precision and recall, we used the F1 score, which is the harmonic mean of precision and recall and is defined as $F1=2\times {\displaystyle \frac{\mathrm{Precision}\times \mathrm{Recall}}{\mathrm{Precision}+\mathrm{Recall}}}$. Additionally, to measure the overlap between the segmentation results and the ground truth, we computed the intersection over union (IoU), given by $\mathrm{IoU}={\displaystyle \frac{TP}{TP+FP+FN}}$. These metrics provide a comprehensive evaluation of the algorithm’s performance across various dimensions, helping us understand its effectiveness from multiple perspectives.

In evaluating the segmentation results of the images of ice coverage, it was evident that the algorithm demonstrated outstanding performance across various metrics. As shown in

Table 1. Results of the image segmentation metrics for ice coverage segmentation.

Image	Accuracy (%)	Precision (%)	Recall (%)	F1 Score (%)	IoU (%)
Figure 11d	99.09	99.57	91.60	95.42	91.24
Figure 12d	97.17	89.65	93.68	91.62	84.54

, the accuracy of Figure 11d reached 99.09%, indicating a very high rate of correctly classified pixels, meaning that the majority of the pixels were classified accurately. The precision was also notably high, at 99.57%, reflecting the model’s strong ability to distinguish true positives. The recall was 91.60%, showing that the model identified most of the actual positive pixels. Moreover, the F1 score was 95.42%, highlighting the algorithm’s excellent balance between precision and recall, ensuring the overall robustness of the segmentation results. Finally, the IoU stood at 91.24%, further verifying the high overlap between the segmentation results and the ground truth, which confirmed the accuracy of the algorithm.

In comparison, Figure 12d exhibits an accuracy of 97.17%, which is slightly lower than that in Figure 11d but still at a high level. The precision was 89.65%, the recall was 93.68%, the F1 score was 91.62%, and the IoU was 84.54%. Although these values are slightly lower than those in Figure 11d, the results still reflect the robustness of the algorithm in handling different conditions for the task of segmenting images of ice coverage.

Overall, these metric values suggest that the proposed algorithm performs stably and achieves balanced performance across multiple dimensions in the task of segmenting images of ice coverage.

3.3. Applicability Analysis

In order to further verify the applicability and advancement of the proposed method in the detection of icing on actual transmission lines, a sample database of transmission line icing images in a certain province of China was constructed for experimental testing and verification. There were 2924 images of ice coverage in the sample database. The images of ice coverage in the sample database were divided into the four levels according to the degree of ice coverage and ice coverage level classification standards: I, II, III, and IV. The specific details of this database of sample images of ice coverage are provided in

Table 2. A detailed description of the ice image database.

Ice Grade	Ice Thickness (mm)	Ice Cover	Training Set	Test Set	Total
I	0	Ice-free	920	426	1346
II	0–5	Light icing	764	345	1109
III	5–10	Moderate icing	243	107	350
IV	>10	Heavy icing	80	39	119
Total	/	/	2007	917	2924

.

In order to verify the detection accuracy of the proposed method at different icing levels, the methods proposed in [24,33] were selected for comparison. The comparative experimental results are shown in

Table 3. Comparison of the icing detection results of different detection methods.

Detection Method	Icing Test Results
	I	II	III	IV	mAP
[24]	83.8	79.1	75.7	89.7	82.1
[33]	71.1	64.1	61.7	64.1	65.2
This article	92.1	89.4	88.3	92.6	90.6

, where the mAP (mean average precision) is the average recognition accuracy.

As can be seen in Table 3, in comparison with references [24,33], the proposed method had the highest thickness identification accuracy, and the mean average precision (mAP) for different levels of ice coverage in images in the whole sample database reached 90.6%.

4. Conclusions

This study introduced a 2D multifractal detrended fluctuation analysis (MF-DFA)-based segmentation algorithm to effectively tackle ice accumulation on transmission lines, enhancing both the monitoring and maintenance of power systems. Our experiments with synthetic and real-world images confirmed the algorithm’s precision in detecting and assessing ice thickness, which is crucial for grid reliability and structural integrity. The algorithm’s adaptability across various environmental conditions and image types highlights its potential for broader application and underscores its role in proactive grid maintenance and disaster prevention. In future work, we intend to explore the use of image binarization, either globally or with a local adaptive threshold, as a segmentation method. As demands on power infrastructure increase, further development of such advanced techniques is critical, with future efforts being aimed at refining performance under diverse conditions and integrating real-time dynamic analysis to ensure grid safety and efficiency amidst escalating environmental challenges.