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The generalized polynomial chaos (gPC) method recently advocated in the literature, exhibits impressive efficiency and accuracy in probabilistic power flow (PPF) calculations of small-scale power systems. However, it suffers from the "curse of dimensionality" and can only be applied to systems with input random variables that follow a set of standard probability distributions. This paper overcome these weaknesses by developing a hierarchical polynomial chaos analysis of variance (ANOVA) method that shows excellent performances in terms of accuracy, efficiency, rationality, and adaptability, for small- to large-scale power systems. By proving the equivalence between the polynomial chaos expansion and the ANOVA decomposition, which is executed at no extra computational cost, the dimensionality of the polynomial chaos expansion can be adaptively reduced to improve the computational efficiency of the generalized polynomial chaos method in high-dimensional problems. Furthermore, by resorting to the Stieltjes procedure, it is extended to any assumed probability distributions of the input random variables. Simulation results carried out on the IEEE 118-bus system and the 1354-bus European high voltage system with correlated renewable energy generations reveal that the developed method outperforms the generalized polynomial chaos method and the Monte Carlo (MC) method while being compatible with real-time applications in power systems.