Data pre-processing mainly includes filling in missing values and encoding character data.

(1) Character data encoding

Character data encoding is carried out for the load distribution L _d and abnormal line loss categories in the substation area. Table 2 shows character data encoding for the load distribution in the substation area.

(2) Missing data filling

Upon the analysis of existing data, there were some missing data such as the daily power factor and daily three-phase unbalance in some substations. To solve this problem, the candidate set is generated by the substation. Then, the k-nearest neighbor method is adopted to select the k-samples which are the most similar from the candidate set and fill in the missing values by taking the average value of k-samples.

According to the substation operation data, feature extraction is carried out on the daily power supply quantity, daily power sales quantity, daily line loss rate, daily power factor, and other data. The statistical features such as the average value, maximum value, minimum value, and variance in monthly are generated.

(1) Monthly average value

For the loss rate, lr; power factor, μ; maximum load rate, MaxL; and the three-phase voltage unbalance rate, U, the average value is calculated with the month as the statistical length, as shown in Equation 1: x i , avg = 1 N i ∑ j = 1 x i j , (1) where x _{i j} indicates the measured value of the j-th day of the i-th month; x = lr, μ, MaxL or U; x _i,avg indicates the average value of the indicator in the i-th month; and N _i indicates the total number of days in the i-th month.

(2) Monthly maximum/minimum value

For the daily line loss rate lr and daily maximum load rate MaxL, the maximum and minimum values are calculated with the monthly statistical length, which is defined by Equation 2. x i , ⁡ max = ⁡ max x i 1 , x i 2 , . . . , x i N i x i , ⁡ min = ⁡ min x i 1 , x i 2 , . . . , x i N i , (2) where x _{i j} represents the measured value of the corresponding index on the j-th day of the i-th month.

(3) Monthly fluctuation rate of daily line loss

The fluctuation of the monthly line loss rate can also reflect the abnormality of the line loss to a certain extent. Considering the difference of the average line loss rate in months, the fluctuation rate of the monthly line loss is defined as in Equation 3 in order to remove the impact of the average level of the line loss rate on the statistical results. l r i , d e v = 1 l r i , avg 1 N i ∑ j = 1 N i l r i j − l r i , avg 2 , (3) where lr _i,avg represents the average value of the line loss rate in the i-th month.

(4) Monthly abnormal rate of daily line loss

If the daily line loss rate is 0, negative, or too high, to some extent, it implies that the line loss rate may also be abnormal. Therefore, in order to reduce the influence of the accidental occurrence of the abnormal daily line loss rate, this paper defines the abnormal rate of monthly line loss, γ _lr,i , as shown in Equation 4. In this paper, the threshold of the excessive line loss rate is set as 7%. γ l r , i = 1 N i ∑ j = 1 N i 1 l r i j ≤ 0 l r i j > 7 % l r i j . (4)

XGBoost adopts the idea of boosting. The basic idea is to stack the base classifiers layer by layer. Each layer gives a higher weight to the misclassified samples of the previous layer when training. The XGBoost tree is constructed by extending a node into two branches, and the layers of the nodes continue to split until the entire tree is formed. Starting from the depth of the tree equal to 0, each node traverses all the features and sorts them according to the value of the feature gain function, as shown in Equation 6. In this way, all the features are sorted according to the contribution of the features to the objective function. Then, the feature is linearly scanned to determine the best segmentation point. G a i n = 1 2 G L 2 H L + λ + G R 2 H R + λ − G R + G L 2 H L + H R + λ − δ , (6) where G _L represents the cumulative sum of the first-order partial derivation of the objective function by the samples contained in the left subtree after the current node splitting. G _R represents the cumulative sum of the first-order partial derivation of the objective function by the samples contained in the right subtree after the current node splitting. H _L represents the cumulative sum of the second-order partial derivation of the objective function by the samples contained in the left subtree after the current node splitting. H _R represents the cumulative sum of the second-order derivation of the objective function of the samples contained in the right subtree after the current node splitting. λ is the regularization parameter, and δ is the threshold to control the minimum gain of the split.

Since there are less labeled data for abnormal line loss types, most abnormal line losses only mark whether there is an anomaly but do not mark the specific reason of the anomaly. Therefore, this paper adopts the self-training semi-supervised learning method to model the abnormal line loss category classification. It trains an initial model with labeled data and then uses the model to predict the unlabeled data. The data with high confidence are added to the labeled dataset and used to retrain the model. The final model is obtained by iterating the process until the converge condition is satisfied.

According to whether the abnormal line loss type is labeled, the dataset is divided into the labeled sample dataset D _L = {( x ₁, y ₁), ( x ₂, y ₂), …, ( x _n , y _n )}and unlabeled sample dataset D _U . The number of sample label categories is N _c . In self-training semi-supervised learning, the pseudo-label sample selection strategy is the core part of the model performance. The purpose of the pseudo-labeled sample selection strategy is to select the samples that are more likely to be correctly labeled from the unlabeled samples and add them to the labeled samples to form a new training set so as to further improve the accuracy and generalization performance of the model. If pseudo-label samples, which are falsely labeled, are added to the training set, the performance of the model may be degraded. In this paper, pseudo-label sample selection based on the Mahalanobis distance is adopted, and the process is as follows:

In the labeled sample dataset D _L , the samples are divided according to the sample category. The sample set of class m is denoted as D _L , _m = {( x _i , y _i )| y _i = m}, m = 1, … , N _c . The average value of its feature vector is calculated based on Equation 7. x ¯ m = 1 D L , m ∑ x i , y i ∈ D L , m x i . (7)

In the unlabeled sample dataset D _U , the corresponding pseudo-labeled sample set D _P is obtained after labeling. y _{p, j} is denoted as the pseudo-label of sample x _j , x _j ϵ D _U . Suppose y _{p, j} = m, the Mahalanobis distance between the pseudo-label sample ( x _j , y _{p, j} ) and x ¯ m is calculated by Equation 8. d x j , x ¯ m = x j − x ¯ m C m − 1 x j − x ¯ m T , (8) where C _m is the covariance matrix of D _L , _m , which is shown in Equation 9. C m = 1 D L , m − 1 ∑ x i , y i ∈ D L , m x i − x ¯ m ∑ x i , y i ∈ D L , m x i − x ¯ m . (9)

The detailed processing of abnormal line loss category classification based on semi-supervised learning and XGBoost is shown as follows:

In this paper, the operation data on three power supply stations in Lvliang, Shanxi Province, China, spanning half a year are used for comparison experiments. The three power stations contain 1,175 10-kV substations, which mainly include residential load, industrial load, public lighting, and commercial load. The substation operation data contain the daily active power supply, reactive power supply, line loss rate, input power, output power, power factor, maximum load rate, three-phase unbalance rate, and other data on substations spanning from May 2022 to November 2022.

In the experiment, the abnormality of the substation line loss is labeled by the month. There are a total of 7,050 samples in the dataset, including 1,503 abnormal line loss samples and 5,547 normal line loss samples. Due to the limited labor, only the abnormal causes in the part of the substation are verified, which includes 988 samples, accounting for 65.73% of the whole abnormal line loss samples. The distribution of abnormal line loss causes is shown in Figure 4. The main cause of abnormal line loss is the meter problem, including data acquisition exception and meter device fault. The electricity theft accounted for the smallest proportion. A part of the reason is that the electricity theft by users is difficult to confirm in reality due to user privacy. The detailed causes of different abnormal line loss categories are shown as follows:

• Line infrastructure problem: too long supply wire or too small wire radius and aging of the line equipment.

In the abnormal line loss recognition model, the dataset is divided as 7:3, where 70% of the data comprises the training set and 30% of the data comprises the test set. The hyperparameters of the random forest and XGBoost model used in this paper are shown in Table 3.

In this paper, the abnormal line loss detection and category classification is a two-stage classification problem. Thus, the confuse matrix is used to display the result. In stage 1, the abnormal line loss identification is a binary classification problem.

In stage 2, the abnormal line loss category classification is a multi-classification task, and the evaluation metrics include accuracy, precision, recall, and the F1-score. Considering the unbalanced sample problem, this paper utilizes the macro average value, as shown in Equations 10–13. The TP is the number of the positive samples detected as positive. The TN is the number of negative samples detected as negative. The FP is the number of negative samples detected as positive. The FN is the number of positive samples detected as negative. A c c = 1 N c ∑ i = 1 N c T P i + T N i T P i + F P i + F N i + T N i , (10) P = 1 N c ∑ i = 1 N c T P i T P i + F P i , (11) R = 1 N c ∑ i = 1 N c T P i T P i + F N i , (12) F 1 = 2 × P × R P + R . (13)

To further analyze the performance of feature engineering, the Spearman correlation analysis is first employed to quantify the relationship between the statistical features and abnormal line loss. The result is displayed in Figure 5. It is clear that the monthly abnormal rate of daily line loss, γ _lr,i , is the most important feature in abnormal line loss identification. The maximum value of line loss, the minimum value of line loss, and the average value of line loss also have a certain correlation with abnormal line loss. The three-phase unbalance rate is the least related to abnormal line loss and is not regarded as the input of the identification model.

The result of our proposed abnormal line loss identification and category classification model is shown in Table 4. It shows that a good performance is achieved in abnormal line loss identification. All the evaluation metrics obtain good results. Figure 6 displays the decision boundary of one decision tree of the random forest. It is clear that all the samples with negative line loss are recognized as abnormal. The sample with a high monthly abnormal rate and high minimal line loss rate is also identified as abnormal.

In abnormal line loss category, the classification result is not better than that of abnormal line loss identification. The small sample size and unbalanced sample distribution significantly impact the precision and recall values. The confusion matrix of the XGBoost model is presented in Figure 7. The classification result of electricity theft is the worst. The meter problem classification is the best. It is because the number of electricity theft incidents is too small and impacts the model learning. All the categories are easily misidentified as meter problems, especially electricity theft. In reality, the meter problem is the most common cause of abnormal line loss, including different abnormal line loss scenarios, such as data error, communication problem, and data collection terminal fault. Thus, other causes are easily misidentified as meter problems.

In this section, the comparison experiments are conducted from different aspects, including abnormal line loss identification with different algorithms, abnormal line loss category classification with different algorithms, and comparison of supervised learning and semi-supervised learning.

1) Comparison of abnormal line loss identification with different algorithms

In abnormal line loss identification, the decision tree (DT), XGBoost, BP, and support vector machine (SVM) are utilized as the comparison algorithms. In DT, the max depth of tree is set as 12. In XGBoost, the learning rate is 0.1 and the number of estimators is set as 100. In BP, the number of hidden layers is set as 2, with 100 neurons in each hidden layer. The kernel function of SVM is the radial basis kernel function, and the regularization parameter is 1.

The identification results of different algorithms are shown in Figure 8. Since the abnormal line loss identification problem is a relatively simple binary classification problem, all the algorithms can achieve a good performance. From the aspect of accuracy, BP achieves the best performance. The accuracy values of RF, DT, XGBoost, and SVM are close. From the aspect of all metrics, the RF performs the best. The precision, recall, and F1-score of the RF are the highest. The precision result of BP implies that the model easily launches false alarms than RF. The performance of DT and SVM is the worst. Further analyzing the result with data, it is found that the monthly line loss with the negative daily line loss rate is easily recognized as abnormal. The abnormal monthly line loss with a small and positive line loss is the most difficult to detect compared to other abnormal line loss scenarios.

2) Comparison of abnormal line loss category classification with different algorithms

In abnormal line loss category classification, random forest, DT, and BP are used as the comparison algorithms. In random forest, the number of decision trees is set as 105, and the maximum depth of the decision tree is set as 10. In DT, the maximum depth of the tree is set as 15. In BP, the number of hidden layers is set as 3, with 85 neurons in each hidden layer. All the algorithms are conducted with the semi-supervised learning.

The result of the abnormal line loss category classification is displayed in Figure 9. It is obvious that the performance of XGBoost is the best and that of DT is the worst. Due to limited samples, the accuracy of abnormal line loss category classification is not higher than that of abnormal line loss identification. In another aspect, the input feature is generated based on monthly line loss, which cannot reflect the fluctuation of the intra-day line loss rate. In particular, the theft of electricity is closely related to the intra-day line loss rate, which cannot be well-detected.

3) Supervised learning vs. semi-supervised learning

In this section, the performance of supervised learning and semi-supervised learning is compared with XGBoost in the abnormal line loss category classification task. The supervised learning directly uses 70% of the labeled samples to train XGBoost, and the rest 30% was used for the test. The evaluation metric results are displayed in Table 5. From Table 5, it is obvious that the classification results are significantly improved by semi-supervised learning, especially recall. It is implied that the phenomenon of leaking alarm is relieved. The category of theft of electricity is the most difficult to detect. It is because of the limited electricity theft samples and because electricity theft is mostly impacted by the intra-day line loss rate. The confusion matrix of supervised learning is presented in Figure 10. Compared to Figure 7, the classification accuracy of all the categories is enhanced. For the semi-supervised learning, the unlabeled samples are used, which can help the model learn to increase the classification accuracy. However, the current data cannot reflect the situation of intra-day line loss, and the category classification performance is limited. To further improve the abnormal line category classification, detailed line loss data are needed.