Linear power flow models are widely used in power systems to simplify the nonlinear power flow equations. The DC power flow model is one of the representatives. There are many other linear power flow models that improve the DC power flow model with the inclusion of Q and $v$ . However, existing linear models are derived based on empirical mathematical approximation without a general methodology guidance. In this paper, we found that the fundamental difference among different linear power flow models lies in the formulation of “independent variables”. Based on this finding, a general formulation of linear power flow models is proposed. The linearization error is theoretically analyzed. In particular, the case when $\theta$ and $v^k$ are regarded as independent variables is thoroughly investigated. Method for finding the linear power flow with the minimum error is presented. The formulation of the independent variables associated with the minimum linearization error is determined by the distribution of state variables $v$ and $\theta$ . It is shown that the linearization error when $v^2$ is regarded as an independent variable is normally smaller than that for v because of the special properties of the distribution of $v$ in power grids.

This paper proposes a data-driven approach for optimal power flow (OPF) based on the stacked extreme learning machine (SELM) framework. SELM has a fast training speed and does not require the time-consuming parameter tuning process compared with the deep learning algorithms. However, the direct application of SELM for OPF is not tractable due to the complicated relationship between the system operating status and the OPF solutions. To this end, a data-driven OPF regression framework is developed that decomposes the OPF model features into three stages. This not only reduces the learning complexity but also helps correct the learning bias. A sample pre-classification strategy based on active constraint identification is also developed to achieve enhanced feature attractions. Numerical results carried out on IEEE and Polish benchmark systems demonstrate that the proposed method outperforms other alternatives. It is also shown that the proposed method can be easily extended to address different test systems by adjusting only a few hyperparameters.

The data-driven method with strong approximation capabilities and high computational efficiency provides a promising tool for optimal power flow (OPF) calculation with stochastic renewable energy. However, the topology change dramatically increases the learning difficulties and the demand for learning samples. In this work, we propose a physics-guided graph convolution neural network (GCNN) for OPF calculation with consideration of varying topologies, including the physics-guided graph convolution kernel, feature construction, and loss function formulation. Specifically, a physics-embedded graph convolution kernel is derived by aggregating the features from local neighborhoods utilizing the nodal OPF model formulation. An iterative feature construction method is also developed that encodes both the physical feature and practical constraints into the node vector. Finally, a correlative learning loss function to optimize the unbalanced power injection is developed. Extensive numerical results carried out on various IEEE test systems show that the prediction accuracy of OPF using the proposed method under varying topology changes can be improved by an average of 13.30% and up to 32.63% compared with state-of-the-art methods.

Probabilistic analysis tool is important to quantify the impacts of the uncertainties on power system operations. However, the repetitive calculations of power flow are time-consuming. To address this issue, data-driven approaches are proposed but they are not robust to the uncertain injections and varying topology. This article proposes a model-driven graph convolution neural network (MD-GCN) for power flow calculation with high-computational efficiency and good robustness to topology changes. Compared with the basic graph convolution neural network (GCN), the construction of MD-GCN considers the physical connection relationships among different nodes. This is achieved by embedding the linearized power flow model into the layer-wise propagation. Such a structure enhances the interpretability of the network forward propagation. To ensure that enough features are extracted in MD-GCN, a new input feature construction method with multiple neighborhood aggregations and a global pooling layer are developed. This allows us to integrate both global features and neighborhood features, yielding the complete features representation of the system-wide impacts on every single node. Numerical results on the IEEE 30-bus, 57-bus, 118-bus, and 1354-bus systems demonstrate that the proposed method achieves much better performance as compared to other approaches in the presence of uncertain power injections and system topology.

A tie-line security region provides all possible tie-line power injections with which constraints in a regional network will not be violated. With the high penetration of renewables, a sharp increase or decrease of renewable power requires a good depiction of the tie-line security region considering time coupling. However, the tie-line power security region with time coupling is a high-dimension polytope, which leads to a considerable computational burden. This paper presents a fast algorithm to approximate the original high-dimension tie-line security region considering time coupling. A few lower-dimension polytopes associated with combinations of different periods are calculated for the approximation. An evaluation method with clear physical explanations is established to measure the accuracy of the approximation. The method converts the evaluation to two linear programming problems based on the mathematical relationship between optimal points and polytope vertices. Under a Monte Carlo framework, the statistical analysis is employed to measure the accuracy using five presented indices. The performance of the proposed method is verified by the case studies on the IEEE 118-bus test system and a 661-bus utility system.