## 1 Introduction
Finally, supervised methods utilize machine learning models to solve classification [16; 17] and regression problems [18; 19], which are designed for outlier detecting and benchmark calculating tasks, respectively. Classification models learn from labeled samples to distinguish normal and abnormal data. However, line loss rate samples are usually unlabeled, as it is unable to recognize whether a collected value of line loss rates is normal or not. According to the relevant references, data-driven methods have been widely applied to estimate line loss values and loss rates, where clustering and regression methods are the two major ones [20]. In [21], FCM is adopted to select a reference value for each type of feeders, in order to calculate the limit line loss rates of feeders for distribution networks. Similarly, another cluster method, namely Gaussian mixture model (GMM), is utilized to calculate line loss rates for low-voltage transformer regions in [22]. The above two methods both calculate a fixed benchmark value for each cluster, where they are not designed for a single feeder or region. Considering this fact, regression methods are proposed to calculate a benchmark with certain inputs. Those inputs are the influencing factors of line losses from only one feeder or region. Reviewing the state-of-art methods, decision tree and its derived boosting models are the most frequent-used ones in line loss computation. In [23; 24], gradient boosting decision tree (GBDT) is used to predict and estimate line losses for distribution networks and transmission lines, respectively. Among those, power flow and weather information are taken as input factors. In [25], extreme gradient boosting (XGBT) model is proposed to estimate line losses for distribution feeders, based on the characteristics of feeders. Regression models can obtain benchmark values for line loss rate samples based on different influencing factors, where a value greatly away from benchmarks can be treated as an outlier. Nevertheless, a regression model may perform poor robustness and reliability when directly trained on samples with massive outliers, causing it critical to develop a highly robust method. In addition to the boosting models mentioned before, \(k\)- nearest neighbors (KNN) is one of the most common-used machine learning models with great robustness [26; 27; 28]. It analyzes the similarity between the predicted sample and original training ones, in order to calculate an averaged value based on those nearest training samples. This calculating method can relieve certain impacts from outliers, but it increases the computational burden at the application stage and can hardly deal with high-dimensional inputs. Besides, support vector machine (SVM) is also frequently utilized for robust regression [29]. As the regression results of SVM are strongly correlated with support vectors, the influences of outliers are decreased. However, SVM shows low training efficiency on a large amount of samples. Due to the great number of regions in practical application, a method with both high efficiency and robustness is proposed in this study, called robust neural network (RNN). A neural network utilizes error back-propagation (BP) and mini-batch gradient descent algorithms to update its parameters, suitable for big data samples training. Besides, RNN modifies the structure of conventional neural networks, further increasing its robustness against outliers. The main contributions of this study can be summarized in detail as follows. 1. As the number of researches that focus on benchmarking daily line loss rates is limited, a supervised regression method is proposed in this study to obtain benchmark values of daily line loss rates in different transformer regions. In the proposed supervised method, various influencing factors of line loss rates are considered, where a high computation accuracy can be thus ensured. 2. A novel RNN model is proposed in this study. It possess a multi-path architecture with denoising auto-encoders (DAEs).