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Probabilistic power flow (PPF) plays a critical role in the analysis of power systems. However, its high computational burden makes practical implementations challenging. This paper proposes a model-based deep learning approach to overcome these computational challenges. A deep neural network (DNN) is used to approximate the power flow calculation process, and is trained according to the physical power flow equations to improve its learning ability. The training process consists of several steps: 1) the branch flows are added into the objective function of the DNN as a penalty term, which improves the generalization ability of the DNN; 2) the gradients used in back propagation are simplified according to the physical characteristics of the transmission grid, which accelerates the training speed while maintaining effective guidance of the physical model; and 3) an improved initialization method for the DNN parameters is proposed to improve the convergence speed. The simulation results demonstrate the accuracy and efficiency of the proposed method in standard IEEE and utility benchmark systems. Index Terms-Model-based deep learning, initialization methods, deep neural networks, probabilistic power flow.

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Random forest (RF) has obtained great success in hyperspectral image (HSI) classification. However, RF cannot leverage its full potential in the case of limited labeled samples. To address this issue, we propose a unified framework that embeds active learning (AL) and semi-supervised learning (SSL) into RF (ASSRF). Our aim is to utilize AL and SSL simultaneously to improve the performance of RF. The objective of the proposed method is to use a small number of manually labeled samples to train classifiers with relative high classification accuracy. To achieve this goal, a new query function is designed to query the most informative samples for manual labeling, and a new pseudolabeling strategy is introduced to select some samples for pseudolabeling. Compared with other AL-and SSL-based methods, the proposed method has several advantages. First, ASSRF utilizes the spatial information to construct a query function for AL, which can select more informative samples. Second, in addition to providing more labeled samples for SSL, the proposed pseudolabeling method avoids bias caused by AL-labeled samples. Finally, the proposed model retains the advantages of RF. To demonstrate the effectiveness of ASSRF, we conducted experiments on three real hyperspectral data sets. The experimental results have shown that our proposed method outperforms other state-of-the-art methods.Random forest, proposed by Breiman [16], is one of the most important machine learning methods. Compared with other machine learning methods, RF has the following advantages [9][10][11][17][18][19][20][21][22][23][24]. First, RF is suitable for high-dimensional data, which can alleviate the influence of curse of dimensionality [20]. Second, RF is very fast because it is implemented in parallel [9]. Third, RF is effective to handle data with imbalanced class labels or missing values [21]. Fourth, RF is not easy to fall into overfitting compared with the Boosting method [18]. Moreover, RF measures the importance of variables automatically [25]. Finally, RF can obtain a higher classification accuracy compared to other well-known classifiers such as SVM [5,6] and maximum likelihood (ML) [17,23], with fewer parameters.Active learning is a kind of iterative method that queries the most informative samples for manual labeling at each iteration [14]. Initially, the data set consists of two parts: a small number of labeled samples and a large number of unlabeled samples. At each iteration, a query function is adopted to investigate the unlabeled samples and select the most informative samples for manual labeling. After multiple iterations, the performance of the classifier improves along with the increase in the number of manually labeled samples. The query function plays an important role in determining the samples selected for manual labeling, which directly affects the classification performance [15].Semi-supervised learning, another way to tackle the issue of limited samples, trains a classifier with the labeled and unlabeled samples together, which does not r...