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Power system flexibility refers to the ability to respond to changing net loads within a predefined timeframe. The main causes of flexible demand are resource and load uncertainties. The joint interaction of resource and load uncertainty makes flexible demand more complex. Accurately assessing the power system flexibility and promoting flexible supply are important for maintaining grid safety and stability. In this study, a joint probability distribution for intermittent renewable energy sources (IRES) output and load forecast deviation is proposed. Subsequently, an optimal dispatch model that considers flexible demand and hydropower flexible supply is built to improve the operation scheme of flexible resources. A hydropower dispatch model that does not consider the probability of IRES output and load forecast deviation is used as the contrast model. A comparison of the results shows that the optimal dispatch model is more beneficial for maintaining grid safety and stability. The results show that the maximum upward adjustment flexible (UAF) margin of the optimal dispatch model of the Yalong River Downstream is 376 MW, whereas the maximum downward adjustment flexible (DAF) margin is 197 MW. This implies that a greater UAF supply is required in the case study to satisfy the power system flexible demand. • A quantitative method for power system flexibility is proposed. • A joint probability distribution considering resource and load uncertainties is presented. • An optimal dispatch model considering flexible margins is proposed. [ABSTRACT FROM AUTHOR]

• The effects of fairness and efficiency on water allocation are explored by constructing different of optimization models. • Welfare economics theory is introduced to balance the efficiency and fairness of water-resource regulations. • An integrated model of water-resource scheduling and allocation is constructed based on a proxy model, which significantly improves computational efficiency. Water is an important natural resource with economic and social attributes. The most important aspects of water resource regulations are fairness and efficiency. Achieving a balance between fairness and efficiency is a challenging but popular topic in related research. In this study, three optimal water-resource allocation models were constructed: efficiency priority (E-P), fairness priority (F-P), and stratified water supply (SWS). The three models were applied to the Yellow River Basin (YRB), and the results show that, compared with the E-P and F-P models, the SWS model is a more equilibrium model with significant advantages. Considering the impact of reservoirs on water allocation, an optimal operation model for cascade reservoirs groups, considering fairness and efficiency, was constructed based on the principle of welfare economics and coupled with the SWS model. Proxy modeling was used in the coupling process to enhance the computational efficiency of the integrated model. Compared with the SWS model, the integrated model can better satisfy the water supply demand outside a river while ensuring the ecological environment and sand flushing demand inside the river; its fairness and efficiency are also significantly improved. In addition, integrated models can effectively address hydrological uncertainty and reduce losses. The model proposed in this study has significant flexibility and application potential and can provide a reference for other governments and water-resource management institutions worldwide. [ABSTRACT FROM AUTHOR]