State evaluation of zinc oxide arresters based on the initial probabilistic self-learning Bayes algorithm

Improving the accuracy of state evaluation for Zinc Oxide (ZnO) surge arresters is essential for grid safety. This paper proposes a comprehensive state evaluation method based on an initial probabilistic self-learning Bayesian algorithm. The method firstly utilizes association rules to mine the correlations of state parameters under various fault modes. A five-level hierarchical criterion is established to quantify the health state, where thresholds and parameter scores are determined via fuzzy membership functions. By integrating historical, current, and predicted state information, the method performs self-learning on conditional probability tables to construct a Bayesian network assessment model. The conditional probability tables and transaction matrix heat maps for all state levels were obtained. Practical case verification indicates that this method achieves a condition assessment accuracy of 93.33%. Comparative analysis confirms that this performance is significantly superior to traditional single-parameter methods and clustering algorithms, validating the model's effectiveness in assessing equipment operation status.

1 Introduction

Installing zinc oxide arresters (ZnO arresters) is a key measure for enhancing the lightning protection performance of distribution networks. However, prolonged operation can lead to performance degradation due to aging, potentially creating safety hazards [1, 2]. Therefore, accurately assessing the operational status of ZnO arresters is crucial for ensuring grid safety and power supply reliability [3].

Currently, most ZnO arrester condition assessment methods rely solely on single-parameter judgments. Reference [4] proposed a technique utilizing combined high-frequency and ultra-high-frequency signal localization to identify internal faults, particularly targeting internal discharge sources in ZnO arresters. Reference [5] evaluated the aging condition of ZnO arresters by analyzing the decomposition of frequency-domain dielectric spectrum curves. Reference [6] analyzed leakage currents in ZnO arresters and further investigated the distribution of harmonic components. Although the single-feature analysis methods mentioned above are relatively straightforward, they are susceptible to interference and struggle to comprehensively reflect the overall condition of ZnO arresters, leading to certain errors in the evaluation results.

In recent years, with the continuous development of artificial intelligence (AI) technologies and machine learning (ML) algorithms, many researchers have applied them to assess the condition of zinc oxide arresters. The research results in the engineering field indicate that the optimized hybrid intelligent algorithm can significantly improve the accuracy, robustness, and generalization ability of the prediction and has high transferability for the condition assessment of zinc oxide arresters. Reference [7] proposed an enhanced beluga whale optimization algorithm, which achieved robust estimation of the state of lithium batteries under extreme conditions. For example, Reference [8] constructed a BWO-TCN-BiGRU-Attention hybrid neural network, which significantly improved the generalization ability of transformer oil temperature prediction. Reference [9] utilized the fish eagle optimization algorithm to optimize the bidirectional LSTM network, effectively suppressing the interference of high-frequency noise on the prediction accuracy. Reference [10] proposed an integrated NRBO-TXAD model, solving the lag problem in wind speed prediction in renewable energy systems. Reference [11] established a state assessment model based on reliability center maintenance, achieving the transition from single-parameter monitoring to full lifecycle risk management. Reference [12] designed and implemented the corresponding online health assessment hardware system for zinc oxide arresters, verifying the effectiveness of the algorithm in the actual engineering environment through fast Fourier transform and harmonic analysis techniques. These algorithms, by integrating multiple parameters and improving model performance, have fully verified their effectiveness in equipment condition assessment. When the algorithms exhibit strong applicability and the models are adequately trained, their prediction accuracy is relatively high. However, due to the lack of real defect samples in the actual engineering application involving zinc oxide arresters, these artificial intelligence algorithms encounter challenges when dealing with large amounts of data.

In contrast, Bayesian networks can improve and optimize the model by integrating fault mechanisms and on-site operational experience and can efficiently extract valuable information from limited data, making them highly suitable for solving the data scarcity problem in the condition assessment of zinc oxide arresters. Current research on the application of Bayesian networks in cross-domain equipment condition assessment further highlights their potential in compensating for the shortcomings of pure data-driven models. Reference [13] proposed a sequential and adaptive probability integral method, effectively solving the computational efficiency problem of edge possibility estimation in Bayesian model updates; this improvement in computational efficiency is of crucial significance for applying Bayesian models to real-time arrester state monitoring scenarios. Reference [14] implemented a Bayesian physical information neural network to solve the problem of poor physical consistency of pure data-driven models in inverter-dominated power systems; its integration of physical mechanisms meets the requirements for interpretable state assessment of zinc oxide arresters as these arresters rely on clear electrical fault mechanisms. Reference [5] proposed a state estimation method based on Bayesian optimization, using Gaussian process regression to adaptively compensate for the temperature drift of lithium-ion batteries. Reference [15] constructed a Transformer–LSTM hybrid model based on Bayesian optimization, achieving high-precision prediction of battery health status. However, these algorithms overly rely on threshold comparison and surface features and fail to better utilize the deep information from the data. Furthermore, they did not analyze the correlations among multiple characteristic indicators.

In view of this, this paper first uses association rule mining to explore the correlation of multiple state parameters such as leakage current, resistive leakage current, and infrared temperature under various fault modes before the comprehensive condition assessment of ZnO arresters. Then a five-level state hierarchical criterion is established for ZnO arresters to evaluate their historical, current, and predicted state information. Finally, the data characteristics of these three types of information are comprehensively considered to perform self-learning of the conditional probability table (CPT) and complete the condition assessment of ZnO arresters based on the self-learning Bayesian algorithm with initial probability, thus effectively improving the accuracy of the condition assessment of ZnO arresters.

Compared with the existing probability models, this paper first optimizes the prior probability using association rule mining and uses the Apriori algorithm to mine the association rules between state parameters, providing data-driven initial values for the Bayesian Network conditional probability table instead of being set subjectively [16]. At the same time, multi-timescale information fusion is carried out. The BN model simultaneously inputs historical, current, and predicted state scores, enabling dynamic comprehensive evaluation. In addition, this paper introduces a self-learning mechanism, using Bayesian parameter estimation to enable the CPT to be updated with new data.

2 Principle of association rule mining and the hierarchical assessment criteria for the condition of ZnO arresters

2.1 Association rule mining

To establish quantitative correlations between the state parameters and fault modes of ZnO arresters and to provide data-driven prior knowledge for the subsequent Bayesian network-based condition assessment, this study uses the Apriori algorithm for association rule mining. This algorithm can effectively discover strong association rules from historical data among leakage current exceeding thresholds, infrared temperature abnormalities, and aging faults. This algorithm generates candidate item sets through iteration and verifies their frequency, effectively mining potential valuable association patterns within the dataset [17]. Based on the association characteristics among multi-source features, a mapping relationship between state indicators and fault modes is constructed. Feature weight analysis is conducted to determine the influence degree distribution of each parameter, and the initialization process of prior probability is optimized accordingly. In the Apriori algorithm, binary representation is utilized, indicating that data are encoded as 0 or 1. Each row corresponds to the test data of each indicator obtained from one detection, and each column corresponds to the test data of a certain indicator obtained from multiple detections. If the data exceed the corresponding threshold, the value is 1, indicating that the ZnO arrester has failed for that indicator; otherwise, it is 0 [18, 19].

To describe the fault types of ZnO arresters in detail, they are classified into four categories, as shown in Table 1.

Table 1

Table 1. Classification of the failure modes.

In particular, historical fault data from the ZnO arresters are treated as a transaction database $D$. Each transaction, that is, a fault record, consists of an item set composed of $n$ specific independent fault modes $T$. Mathematically, a transaction $D$ consisting of specific independent fault modes is represented as Equation 1.

$D=\left\{T_1,T_2,...,T_n\right\}.(1)$

Among them, $T_k\left(k=1,...,n\right)$ represents the set of various fault items included when each piece of equipment malfunctions.

For each $T_k$, a set containing $m$ items is called an $m$-item set. In the framework of fault feature analysis, the set of these condition assessment indicators constitutes an $m$-dimensional feature space, and these indicators jointly determine the occurrence mechanism of the fault. The $m$-dimensional feature space constituted by these evaluation indicators is defined as Equation 2.

$T_k=\left\{A_1,A_2,...,A_m\right\}.(2)$

Among them, $A_p\left(p=1,...,m\right)$ represents an evaluation indicator. For a specific fault instance $X$, the corresponding support degree, denoted as $\mathit{Sup}\left(X\right)$, is calculated as Equation 3.

$\mathit{Sup}\left(X\right)=\frac{N\left(X\subseteq T_k\right)}{\left|D\right|}.(3)$

The set of fault items when another type of fault different from $X$ occurs is denoted as $Y$. The expression of the shape of the association rule referring to fault association rules is $X\to Y$ and $X\cap Y$. Consequently, the support degree for such fault association rules is

$\mathit{Sup}\left(X\to Y\right)=\frac{N\left(X\cup T_k\right)}{\left|D\right|}.(4)$

The support parameter in Equation 4 reflects the co-occurrence characteristic of fault modes $X$ and $Y$. Its calculation follows the support measurement criterion for frequent item sets. Based on the support metric, the confidence of an association rule is determined by the ratio:

$Conf\left(X\to Y\right)=\frac{\mathit{Sup}\left(X\to Y\right)}{\mathit{Sup}\left(X\right)}.(5)$

The confidence described in Equation 5 reflects the probability that item set $X$ contains $Y$ when it contains $X$. Therefore, association rules with high support and confidence are mainly discovered by defining the values of $\mathit{min}\_support$ and $\mathit{min}\_confidence$. The discovery of association rules is to find these rules that satisfy $s\ge \mathit{min}\_support\_limit$ and $c\ge \mathit{min}\_confidence\_limit$ for a given $T$, where $\mathit{min}\_support\_limit$ and $\mathit{min}\_confidence\_limit$ define the minimum boundary conditions of support and confidence, respectively, during the association rule mining process.

2.2 Criteria for the layered evaluation of the state of ZnO arresters

Based on the framework of multi-source information fusion for equipment condition assessment, the status information is divided into three dimensions: historical, current, and predictive. A comprehensive condition assessment model is constructed. The traditional binary status classification method divides the status of electrical equipment into only two states: qualified or unqualified, which is intuitive but difficult to meet the requirements of refined condition assessment. This paper no longer adopts a single yes–no classification method. Instead, the comprehensive status of ZnO arrester equipment is divided into five grades: normal (A), attention (B), slight (C), abnormal (D), and emergency (E), denoted as F = {normal (A), attention (B), slight (C), abnormal (D), emergency (E)} [20, 21]. The specific description is shown in Table 2. On this basis, this paper establishes the Bayesian network criterion for the comprehensive assessment of the status of ZnO arresters.

Table 2

Table 2. Equipment operation and maintenance requirements at different levels.

The criteria for hierarchical assessment of the status of ZnO arresters are shown in Table 3. This figure shows parent layers and their corresponding parallel sub-layers. The score of each parent layer is calculated by integrating the scores of the corresponding sub-layers. If the scores of all sub-layers are not less than 0.7, the arithmetic mean of the sub-layers is taken as the final score of the parent layer; otherwise, the minimum score among the sub-layers is taken as the final score of the parent layer.

Table 3

Table 3. State evaluation model and characteristic quantity selection of ZnO arresters.

3 Evaluation of the comprehensive condition of ZnO arresters

3.1 Self-learning of initial probabilities

Based on the theory of probabilistic graphical models, Bayesian networks represent the conditional independence relationships among variables using directed acyclic graphs [22]. Each node in the Bayesian network $I_i$ is independent of other non-descendant nodes given its parent node $Pa\left(I_i\right)$. The conditional probability distribution of the node is represented by the CPT, denoted as $P\left(I_i|Pa\left(I_i\right)\right)$. The joint probability distribution of the entire network can be decomposed into the product of conditional probabilities Equation 6.

$P\left(I_1,I_2,...,I_n\right)={\displaystyle \prod _{i=1}^n}P\left(I_i|Pa\left(I_i\right)\right).(6)$

The Bayesian network modeling framework comprises three core elements: variable selection and feature engineering, network structure learning, and conditional probability parameter estimation [23, 24]. By applying the association rule mining method, the potential correlation characteristics among fault state parameters are revealed. Based on these characteristics, the initial probability parameters of the Bayesian network model are optimized. Then starting from the Bayesian statistical learning theory, the maximum a posteriori probability estimation method is adopted to optimize the parameters of the network’s conditional probability distribution.

Let $r_i$ be the number of possible states for $I_i$ and $q_i$ be the number of possible state configurations for the parent set $Pa\left(I_i\right)$. The conditional probability that the child node $I_i$ is in the $k$th state given that its parent set is in the $j$th configuration is denoted by

${\theta }_{ijk}=P\left(I_i=k|Pa\left(I_i\right)=j\right).(7)$

In Equation 7, $k\in \left\{1,...,r_i\right\}$ represents the state index of the child node and $j\in \left\{1,...,q_i\right\}$ represents the configuration index of the parent nodes. According to the axiom of probability normalization, for any fixed parent configuration $j$, the summation of probabilities over all possible states $k$ satisfies

${\displaystyle \sum _{k=1}^{r_i}}{\theta }_{ijk}=1.(8)$

Based on Bayesian statistical learning theory, the parameters of the CPT are updated using maximum a posteriori (MAP) estimation. To address the issue of data sparsity, the confidence obtained from association rule mining is utilized to construct an informative prior. The update formula, which combines the prior knowledge with observed data, is constructed as

$P\left(I_i=k|Pa\left(I_i\right)=j\right)=\frac{N_{ijk}+M·{\delta }_k}{N_{ij}+M}.(9)$

In Equation 9, $N_{ijk}$ represents the number of samples in the observed dataset, where node $I_i$ is in state $k$ and its parent set $Pa\left(I_i\right)$ is in configuration $j$. $N_{ij}$ represents the total number of samples where the parent set $Pa\left(I_i\right)$ is in configuration $j$, satisfying $N_{ij}={\sum }_{k=1}^{r_i}{N}_{ijk}$. ${\delta }_k$ represents the initial probability derived from the association rules for state $k$, serving as the prior probability distribution. $M$ represents the equivalent sample size, which represents the weight of the prior knowledge relative to the observed data.

In summary, the core of the proposed method lies in the initial probability self-learning mechanism of the Bayesian network. This process consists of two steps:

1. Initialization through association rules: The confidence index of the mined association rules as Formula 5 is used to provide information for the initial condition probability table of the Bayesian network, thereby providing a data-based starting point instead of random or uniform initialization.

2. Iterative update through Bayesian estimation: Subsequently, the initial CPT is refined using the maximum a posteriori estimation as Formulas 7–9, while using the collected state evaluation scores as training data, enabling the model to self-learn and adjust its parameters based on evidence.

The historical, current, and future health status of the ZnO arrester determines its overall condition. The former can be determined by the hierarchical assessment criteria established in Section 2.2. Based on the previous analysis, this paper has established a Bayesian network-based comprehensive assessment model for the state of the ZnO arrester as shown in Figure 1. The lowest-level node of the model is the overall state of the ZnO arrester, and this node has five states (Table 2).

Figure 1

Figure 1. Condition assessment model of ZnO arresters based on Bayesian networks.

3.2 Evaluation of the operating condition of ZnO arresters

The equipment operation condition assessment model integrates historical operation data, real-time monitoring information, and trend prediction results. These three types of data have different timescale characteristics and jointly reflect the influence mechanism of faults on the equipment operation status.

In the condition assessment of ZnO arresters, a semi-trapezoidal membership function is introduced for the quantitative scoring of state information. Through reasonable simplification of this engineering approach, a monotonic and interpretable mapping from physical parameter values to health scores is achieved. According to the characteristics of the indicators, the upward semi-trapezoidal function is adopted for the forward indicators; the downward semi-trapezoidal function is adopted for the backward indicators. The upward semi-trapezoidal function for forward indicators is expressed as Equation 10.

$R_1\left(x\right)=\left\{\begin{array}{l}0,0\le x<\mathrm{a},\\ \frac{x-\mathrm{a}}{\mathrm{b}-\mathrm{a}},\mathrm{a}\le x<\mathrm{b},\\ 1,x\ge \mathrm{b}.\end{array}\right.(10)$

Conversely, the downward semi-trapezoidal function for backward indicators is defined by Equation 11.

$R_2\left(x\right)=\left\{\begin{array}{l}1,0\le x<\mathrm{a},\\ 1-\frac{x-\mathrm{a}}{\mathrm{b}-\mathrm{a}},\mathrm{a}\le x<\mathrm{b},\\ 0,x\ge \mathrm{b}.\end{array}\right.(11)$

For the assessment of the current status of the equipment, assuming that the fault state and the operating state are linearly related, a trapezoidal distribution function is adopted for modeling to achieve precise quantification of the status information.

Here, a and b are both model thresholds and x is the parameter value of the score. According to GB/T 11032-2020 [25], taking YH5WZ-17/45, a 10-kV distribution-type ZnO arrester, as an example, the model thresholds of typical scoring items are presented (Table 4).

Table 4

Table 4. ZnO arrester scoring model threshold table.

However, when x approaches the threshold, the value at this point makes it difficult to precisely describe the specific state of the equipment at that moment. Therefore, a fuzzy mathematics membership function is introduced here to soften the hard boundary conditions. In this paper, a trapezoidal membership function is selected, as shown in Table 5. The thresholds in question were all derived from GB/T 11032-2020 [18].

Table 5

Table 5. Membership degree function table.

3.3 Algorithm summary and workflow

The pseudocode of the Bayes algorithm is shown in Table 6. The algorithm presents an integrated framework combining adaptive association rule mining with Bayesian network modeling. Initially, historical monitoring data—in particular, leakage current, resistive leakage current, and infrared temperature—are cleansed and transformed into binary transaction sets based on an 80% quantile threshold. Subsequently, an adaptive iterative strategy is implemented for rule generation: if no valid rules are identified under current initialization, the algorithm dynamically reduces the support and confidence thresholds until a non-empty rule set is extracted and visualized. Finally, the process integrates multidimensional score features (HScore, NScore, and FScore) to train a naive Bayes model, calculating and preserving the conditional probability table for diagnostic purposes.

Table 6

Table 6. Pseudocode of the Apriori algorithm.

The pseudocode of the Bayes algorithm is shown in Table 7. The algorithm proceeds through three core phases: data scoring, state prediction, and Bayesian modeling. Initially, the process normalizes both historical and current datasets by applying half-ladder membership functions to calculate individual scores for current and temperature, which are aggregated into comprehensive grades (stored as HScore and NScore). Subsequently, a state prediction mechanism is implemented using a weighted fusion strategy, where future state scores (FScore) are derived by averaging current and historical scores with equal weights. Finally, the algorithm integrates these multi-temporal score features to train a naive Bayes classifier, generating a CPT for comprehensive state assessment.

Table 7

Table 7. Pseudocode of the Bayes algorithm.

The overall flowchart for conducting comprehensive condition assessment on ZnO arresters is shown in Figure 2. Figure 2a shows the condition assessment flowchart based on the Bayesian algorithm. Figure 2b presents the algorithm flowchart obtained through association rule mining. Figure 2c depicts the state data scoring flowchart.

Figure 2

Figure 2. Condition assessment flowchart. (a) Condition assessment flowchart based on the Bayesian algorithm. (b) Flowchart of the association rule mining algorithm. (c) Status rating flowchart.

4 Analysis of examples

From the perspective of theoretical verification, the proposed Bayesian network method was experimentally analyzed using the state data of ZnO arresters. The results proved the scientificity and effectiveness of this method. Based on the condition assessment results, the overall operating status of the equipment can be understood. When the equipment is in an abnormal state, faults can be diagnosed and located according to the above method, or possible faults can be further excluded.

This experiment was conducted on a computer equipped with an Intel Core i5-1135G7 processor and 8 GB of memory. The operating system used was Windows 11. The programming language and version adopted was MATLAB R2025a. First, association rule mining was carried out. This paper selected 870 operational status data points from YH5WZ-17/45, a 10-kV distributed ZnO arrester, that failed due to aging at a substation of a power supply company in Jiangsu Province, China. The time range of these data points was within 5 years up to 1 September 2024. Subsequently, the dataset was expanded to 10,000 entries using linear interpolation, thereby forming a fault sample set. The analysis focused on three key indicators: leakage current, resistive leakage current, and relative infrared temperature. Given the diversity of state parameters, it was necessary to set appropriate support thresholds. At the same time, to ensure the reliability of the association rules, the confidence threshold should be maintained at a relatively high level. The experimental analysis identifies the optimal minimum support and confidence thresholds, respectively, as Equations 12, 13.

$\mathit{min}\_support=0.2,(12)$

$\mathit{min}\_confidence=0.6.(13)$

However, given the characteristics of our ZnO arrester status dataset, these initial thresholds were too strict, resulting in an insufficient number of rules. Therefore, an iterative adjustment strategy was adopted. During each iteration, if the generated rules fail to meet these thresholds, the algorithm will gradually lower the thresholds and try different support and confidence values until the minimum limit is reached. To ensure sufficient rule generation, the thresholds for the faulty item set are adjusted to Equations 14, 15.

$\mathit{min}\_support\_limit=0.05,(14)$

$\mathit{min}\_confidence\_limit=0.3.(15)$

To assess the robustness of the final selected thresholds and validate their selection rationale, we conducted a sensitivity analysis. We performed association rule mining under various combinations of support and confidence thresholds, observing changes in the quantity and quality of generated rules, as illustrated in Figure 3.

Figure 3

Figure 3. Heatmap of association rule counts at different thresholds.

When thresholds were overly stringent, the sparse number of rules failed to form an effective knowledge network. When thresholds are too lenient, the number of rules surges, but many are redundant or inconsistent with physical mechanisms, introducing noise. The ultimately selected threshold combination resides in a transitional zone, achieving an optimal balance between generating a sufficient number of meaningful rules and avoiding rule explosion. This demonstrates the effectiveness of our threshold selection method.

The association rules obtained through mining are shown in Table 8.

Table 8

Table 8. Partial synthetic state conditional probability.

In addition, the visualization output of the association rules was obtained. Among them, Figure 4 illustrates the corresponding frequency chart of fault items, showing the occurrence frequency of each fault item in all transactions. Figure 5 shows the corresponding heatmap of the transaction matrix, indicating whether each transaction contains a specific fault item. The darker the color, the more frequently the fault item appears.

Figure 4

Figure 4. Frequency diagram of failure items.

Figure 5

Figure 5. Transaction matrix heatmap.

Then conditional probability self-learning was carried out. Taking 10,000 state data of a 10-kV distribution-type ZnO arrester of model YH5WZ-17/45 as an example, the state of this equipment was evaluated. According to the scoring method in Section 2.2, the state information of the ZnO arrester was scored as the training sample of the Bayesian network condition assessment model. Based on the Bayesian network learning framework proposed in Section 2.1, the final conditional probability distribution of the model was obtained through parameter estimation. Table 7 lists some conditional probability tables of the comprehensive state nodes of the ZnO arrester.

5 Model performance verification

5.1 Uncertainty analysis

As a probabilistic graphical model, the Bayesian network constructed in this study has its core advantage in its inherent ability to handle uncertainty and to provide quantified outputs of uncertainty. Unlike traditional models that output a single-state label, this model’s output is a complete probability distribution regarding the states. Taking the example of outputting p(A) = 0.75, p(B) = 0.2, p(C) = 0.05, p(D) = 0, and p(E) = 0 for analysis, this distribution directly quantifies the uncertainty of the evaluation results. The most likely state is A, but its probability of 75% does not indicate complete certainty. At the same time, there is a 20% probability of being B. Based on this probability distribution, the expected value $\mathrm{\mu }$ and standard deviation $\mathrm{\sigma }$ are calculated to yield the prediction interval as Equations 16, 17.

$\mu =0.75*1+0.20*2+0.05*3+0.00*4+0.00*5=1.3,(16)$

$\sigma =\sqrt{0.75*{\left(1-1.3\right)}^2+0.20*{\left(2-1.3\right)}^2+0.05*{\left(3-1.3\right)}^2}\approx 0.552.(17)$

Therefore, the prediction interval for one standard deviation is [0.748, 1.852]. Consequently, the uncertainty quantification capability, as shown in Table 9, enables the operation and maintenance strategy to upgrade from binary decision-making to refined and differentiated control based on different risk probabilities.

Table 9

Table 9. Link gauge status under the aging mode.

5.2 Model statistical indicators

To demonstrate the superiority of this method, a portion of the original sample data was selected for analysis. The performance accuracy of this method was compared with other condition assessment methods, such as the assessment method based on leakage current as a single indicator, clustering algorithms, support vector machines (SVMs), and random forests (RFs). The SVM model uses the radial basis function (RBF) as the kernel function, and its hyperparameters (the penalty factor C and the kernel function coefficient gamma) are optimized through grid search. In the RF model, the number of decision trees is set to 100, and the other parameters are set to their default values. The results show that the condition assessment method for ZnO arresters based on the initial probability self-learning Bayesian algorithm has an accuracy rate of up to 93.33%. Compared with other assessment methods, its overall performance is the highest, as shown in Figure 6. This accuracy rate is based on actual data samples and represents the probability of deviation between the overall state obtained through the algorithm’s assessment and the current actual state.

Figure 6

Figure 6. Accuracy of different condition assessment methods.

To ensure the fairness of the experimental comparison and the credibility of the results, all the comparison models involving adjustable parameters have undergone systematic hyperparameter tuning. The tuning process adopts the industry-standard grid search combined with a fivefold cross-validation method. The ratio of the training set to the validation set is 8:2 to prevent data overfitting. For SVM, the key parameters to focus on are the penalty factor C, with the search range [0.1, 1, 10, 100, 1000], and the kernel parameter γ, with the search range [0.001, 0.01, 0.1, 1, 10]. For RF, the two key hyperparameters are decision tree quantity, with the search range [50, 100, 150, 200], and the maximum depth of the decision tree, with the search range [5, 10, 15, none]. Through this tuning process, all the comparison models achieve the optimal performance on the current dataset, providing a fair and objective experimental basis for the subsequent accuracy comparison.

Considering the time and space complexities of each method, as shown in Table 10, n denotes the number of samples, d denotes the number of features, k denotes the number of clusters, i denotes the number of iterations, $n_{sv}$ denotes the number of support vectors, m denotes the number of trees, and c denotes the number of categories. This Bayesian method exhibits a simple linear relationship with the dataset size in both time and space complexities. It ensures high temporal efficiency while achieving low spatial consumption.

Table 10

Table 10. Comparison of time and space complexity among different methods.

The proposed ZnO arrester state evaluation method is deployed on the State Grid Jiangsu Electric Power’s Integrated Equipment Condition Monitoring Platform, adopting a Java Spring Boot-Vue.js/Element UI-separated microservice architecture; its core algorithm runs independently in MATLAB Runtime (decoupled from business logic), with structured data stored in PostgreSQL and real-time data cached in Redis. The system performs time-consuming training offline, with online P95 response time ≤500 ms, recommended for 8-core CPU, 32 GB RAM, and SSD servers; the model’s linear computational complexity balances 93.33% high accuracy with efficient, low-consumption operation.

6 Conclusion

In view of the deficiencies of the current methods for evaluating the status of ZnO arresters, this paper proposes a method for evaluating the status of ZnO arresters based on the initial probability self-learning Bayesian algorithm:

1. The Apriori algorithm is used to mine association rules among state parameters under various fault modes, thereby establishing a hierarchical condition assessment criterion for ZnO arresters.

2. The hierarchical criteria are applied to evaluate historical, current, and future states. A Bayesian network is then employed to derive the overall state of the ZnO arrester and generate corresponding conditional probability tables.

3. Theoretical validation using empirical data demonstrates the feasibility and effectiveness of the proposed method for assessing the condition of ZnO arresters.

Test results validated the superiority of this method, achieving an evaluation accuracy of 93.33% on the test dataset. Furthermore, this model effectively balances accuracy and computational efficiency. This paper separates offline complex learning from online efficient inference: offline training utilizes historical data and high-performance computing resources, while online state evaluation requires only forward probability inference on Bayesian networks. The compact Bayesian network constructed in this paper features a limited number of nodes, enabling faster execution speeds.

However, this study has limitations. On the one hand, the model’s sensitivity to extreme environmental variables such as humidity and pollution levels has not been systematically incorporated into the analytical framework. Instead, the influence of environmental humidity and pollution levels on parameters including leakage current and resistance current is implicitly considered. Second, experimental validation primarily relies on data from specific ZnO arrester models. Although this facilitates focused verification of the methodological framework under controlled conditions, differences in design parameters, material properties, and rated values may lead to variations in normal threshold values and aging trajectories among different ZnO arrester models. Therefore, future research will focus on the following directions: explicitly incorporating key environmental factors into Bayesian networks as input nodes for explicit modeling while simultaneously exploring model transferability to adapt this methodological framework to diverse equipment types and enhance model robustness.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

XH: Methodology, Writing – original draft, Writing – review and editing. HH: Validation, Writing – original draft. XZ: Validation, Writing – original draft. YW: Writing – review and editing. HD: Writing – review and editing.

Funding

The author(s) declared that financial support was received for this work and/or its publication.

Conflict of interest

Authors XH, HH, XZ, YW, and HD were employed by State Grid Jiangsu Electric Power Co., Ltd., Yancheng Dafeng District Power Supply Branch.

The authors declare that this research was supported by the Science and Technology Funding Project (J2024068) of State Grid Jiangsu Electric Power Co., Ltd. The involvement of the funding entity in the research is as follows: During the research design phase, the funding entity provided the necessary space for experimental equipment; during the manuscript preparation phase, the funding entity provided essential revision suggestions.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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References

1. Xie J, Liu M, Fang K, Tao Z, Zheng Y, Xiang Z. Influence of damp status on leakage current of ZnO lightning arrester. In: 2024 International Conference on Smart Grid and Energy (ICSGE); Singapore, Singapore (2024). p. 7–10. doi:10.1109/icsge61115.2024.00008

CrossRef Full Text | Google Scholar

2. Li H, Hua Y, Jiang Y. MOA resistive current measurement method based on multi-frequency point interpolation and three-dimensional surface modeling. High Volt Eng (2019) 45(11):3512–20. doi:10.13336/j.1003-6520.hve.20191031015

CrossRef Full Text | Google Scholar

3. Han Y, Li Z, Zheng H, Guo W. A decomposition method for the total leakage current of MOA based on multiple linear regression. IEEE Trans Power Del (2016) 31(4):1422–8. doi:10.1109/tpwrd.2015.2462071

CrossRef Full Text | Google Scholar

4. Ma Y, Shu H, Qian G. Evaluation of aging state of zinc oxide ZnO arresters based on decomposition of frequency-domain dielectric spectra curve. Trans China Electrotech Soc (2024) 39(3):901–13. doi:10.19595/j.cnki.1000-6753.tces.231244

CrossRef Full Text | Google Scholar

5. Fan Han R, Zheng Y, Jiang P, Wang W, Dong X. Reliability centered condition-based maintenance of zinc oxide lightning arresters: concept, process and case study. Front Energy Res. (2024) 12:1359849. doi:10.3389/fenrg.2024.1359849

CrossRef Full Text | Google Scholar

6. Zhang Wang R, Yu H. Online monitoring system for metal oxide ZnO arresters based on multi-layer support vector machine. Insulators and ZnO Arresters (2020) 1:59–70. doi:10.16188/j.isa.1003-8337.2020.01.010

CrossRef Full Text | Google Scholar

7. Kitahara M, Kitahara T. Sequential and adaptive probabilistic integration for Bayesian model updating. Mech Syst Signal Process (2025) 223:111825. doi:10.1016/j.ymssp.2024.111825

CrossRef Full Text | Google Scholar

8. Stock S, Babazadeh D, Becker C, Chatzivasileiadis S. Bayesian Physics-informed neural networks for system identification of inverter-dominated power systems. Electric Power Syst Res (2024) 235:110860. doi:10.1016/j.epsr.2024.110860

CrossRef Full Text | Google Scholar

9. Yuan Z-R, Huang K-F, Xu C-H, Zou J-C, Yan J. Bayesian optimization-based state-of-charge estimation with temperature drift compensation for lithium-ion batteries. Batteries (2025) 11(7):243. doi:10.3390/batteries11070243

CrossRef Full Text | Google Scholar

10. Li S, Ou D, Chen Z, Li B. A bayesian optimization-based Transformer-LSTM method for lithium-ion battery state of health estimation. In: Proceedings of the International Conference on Computer Science and Artificial Intelligence. Springer (2025). doi:10.1007/978-981-96-1395-3_53

CrossRef Full Text | Google Scholar

11. Liu J, Hou Z, Xu Y, Wang B. Enhanced beluga whale optimization meets GRU and adaptive cubature kalman filter: a novel approach for state of charge estimation in lithium-ion batteries. Ionics (2025) 31:10643–69. doi:10.1007/s11581-025-06578-6

CrossRef Full Text | Google Scholar

12. Liu J, Hou Z, Liu B, Zhou X. Mathematical and machine learning innovations for power systems: predicting transformer oil temperature with beluga whale optimization-based hybrid neural networks. Mathematics (2025) 13(11):1785. doi:10.3390/math13111785

CrossRef Full Text | Google Scholar

13. Liu J, Hou Z, Yin T. Short-term power load forecast using OOA optimized bidirectional long short-term memory network with spectral attention for the frequency domain. Energy Rep (2024) 12:4891–908. doi:10.1016/j.egyr.2024.10.050

CrossRef Full Text | Google Scholar

14. Hou Z, Liu J, Shao Z, Ma Q, Liu W. Machine learning innovations in renewable energy systems with integrated NRBO-TXAD for enhanced wind speed forecasting accuracy. Electronics (2025) 14(12):2329. doi:10.3390/electronics14122329

CrossRef Full Text | Google Scholar

15. Zhao S, Yang M. Zinc oxide arrester health status online evaluation system. In: 2024 4th International Conference on Energy, Power and Electrical Engineering (EPEE); Wuhan, China (2024). p. 381–5. doi:10.1109/epee63731.2024.10875455

CrossRef Full Text | Google Scholar

16. Yu X, Rong H, Sun Y, Zhang D. Method for assessing the operational state of transformers based on association rule mining and bayesian network. In: 2024 3rd International Conference on Energy and Electrical Power Systems (ICEEPS); Guangzhou, China (2024). p. 1373–6. doi:10.1109/iceeps62542.2024.10693108

CrossRef Full Text | Google Scholar

17. Song W, Wu G, Zhu L, Chi S. The study of model for power grid diagnosis based on naive bayes theory. In: 2016 IEEE/WIC/ACM International Conference on Web Intelligence Workshops (WIW), Omaha, NE, USA (2016). p. 41–4. doi:10.1109/wiw.2016.023

CrossRef Full Text | Google Scholar

18. Lin Q, Gao C. Discovering categorical main and interaction effects based on association rule mining. IEEE Trans Knowl Data Eng (2023) 35(2):1379–90. doi:10.1109/tkde.2021.3087343

CrossRef Full Text | Google Scholar

19. Bashkari MS, Sami A, Rastegar M. Outage cause detection in power distribution systems based on data mining. IEEE Trans Ind Informat (2021) 17(1):640–9. doi:10.1109/tii.2020.2966505

CrossRef Full Text | Google Scholar

20. Zhu Y, Wu L, Li X, Yuan J. A transformer condition assessment framework based on data mining. IEEE Power Eng Soc Gen Meet (2005) 2:1875–80. doi:10.1109/PES.2005.1489207

CrossRef Full Text | Google Scholar

21. Wu L, Zhu Y, Li X, Yuan J. Application of multi-agent and data mining techniques in condition assessment of transformers. In: 2004 International Conference on Power System Technology, 2004. PowerCon 2004; Singapore, 1 (2004). p. 823–7.

Google Scholar

22. Hong J, Li J, Qiu X. Numerical correlation analysis of power grid construction project based on apriori algorithm. In: 2021 6th Asia Conference on Power and Electrical Engineering; Chongqing, China. Chongqing, China: IEEE (2021). p. 361–4.

CrossRef Full Text | Google Scholar

23. Chen Q, Cao F, Xing Y, Liang J. Evaluating classification model against bayes error rate. IEEE Trans Pattern Anal Mach Intell (2023) 45(8):9639–53. doi:10.1109/TPAMI.2023.3240194

PubMed Abstract | CrossRef Full Text | Google Scholar

24. Tang B, He H, Baggenstoss PM, Kay S. A Bayesian classification approach using class-specific features for text categorization. IEEE Trans Knowl Data Eng (2016) 28(6):1602–6. doi:10.1109/tkde.2016.2522427

CrossRef Full Text | Google Scholar

25. GB/T 11032-2020. Metal oxide ZnO arresters without gaps for a.c. systems[S]. Beijing: Standards Press of China (2020).

Google Scholar