To address the uncertain renewable energy in the day-ahead op-timal dispatch of energy and reserve, a multi-stage stochastic programming model is established in this paper to characterize the uncertainties over the multiple stages by a scenario tree. Most importantly, to deal with the “Curse of Dimensionality” of sto-chastic programming, stochastic dual dynamic programming (SDDP) is employed, which decomposes the original problem into several sub-problems according to the stages. Specifically, the SDDP algorithm performs forward pass and backward pass repeatedly until the convergence criterion is satisfied. At each iteration, the original problem is approximated by creating a lin-ear piecewise function. The results on the IEEE 118-bus system and real-life provincial power grid show the effectiveness of the proposed model and method.

A virtual energy storage (VES) model is proposed in this paper to accommodate renewable energy under a special market regulation. Such VESs can provide or consume electricity to the main power grid under the premise that the daily net electricity energy is balanced. Furthermore, a multi-stage distributionally robust optimization (MSDRO) model is set up in this paper to address the temporal uncertainties in the day-ahead economic dispatch model. Compared with the traditional two-stage distributionally robust optimization, the proposed multi-stage approach provides more flexibilities so that the decision variables can be adjusted at each time period, leading to a complex nested formulation. To efficiently solve the MSDRO model, a stochastic dual dynamic programming method is employed to decompose the original large-scale optimization model into several sub-problems in the stages, as two steps: forward pass and backward pass. In the forward pass, the expected cost-to-go function is approximated by piecewise-linear functions and then several samples are used to generate a lower bound; the backward pass will generate Benders’ cuts at each stage from the solution of the forward pass. The forward and backward passes are performed iteratively until the convergence is reached. Numerical results on an IEEE 118-bus system and a practical power system in China verify the proposed method.

Capacity market is a long-term clearing model to coordinate the traditional thermal generation and the renewable energy generation, which minimizes the total capacity cost of traditional generation, while satisfying the operational constraints and reliability requirement. Furthermore, to address the uncertainties in the long-term optimal decision, min-max regret is employed to find the optimal solution under the worst regret, generating several representable scenarios that can help generation companies to understand how these scenarios would impact on the future generation planning. Finally, a decomposition method is proposed to solve this reliability based min-max regret stochastic optimization model by bisection. The test results on the practical NW grid in China show the effectiveness of the proposed model.

Under the background of clean and low-carbon energy transformation, renewable distributed generation is connected to the distribution system on a large scale. This study proposes a probabilistic assessment method of hosting capacity considering wind–photovoltaic–load temporal characteristics in distribution networks. First, based on time series of wind, photovoltaic, and load demands, a discretization–aggregation technique is introduced to generate and filter extreme combinations. The method can effectively reduce the scenarios that need to be evaluated. Then a holomorphic embedding method considering generation and load scaling directions is proposed. The holomorphic function of voltage about an embedding variable is established, and it is analytically expanded in the form of series. The hosting capacity restrained by the voltage violation problem is calculated quickly and accurately. Finally, the proposed stochastic framework is implemented to evaluate hosting capacity involving renewable energy types, penetration levels, and locations. The hosting capacity of single energy and hybrid wind–solar renewable energy systems is evaluated from the perspective of probability analysis. The results verify the outstanding performance of the hybrid wind–solar energy system in improving the hosting capacity.

This paper presents an exact bijection between the branch flow model (BFM) and bus injection model (BIM) in radial systems. Moreover, the equivalence and the corresponding condition are investigated and rigorously proved. The exploration reveals that the bijection exists if and only if the network is connected and there is no zero-impedance branch.