## 1 Introduction
Others focus on improving the calculation efficiency using methods such as parallel computing [19]. The sampling sequence can also be changed to reduce the number of samples, such as in the quasi-Monte Carlo method [20]. This method obtains random numbers using the Sobol sequence, which generates more uniformity than random numbers generated by the pseudo random number technique. The analytical method is used to solve the PPF problem with a linearized alternating current model and convolution. The number of calculations in this method increases exponentially when the system is relatively large and several components need to be considered. Therefore, this method is not as widely used as the other three methods. Stochastic spectral analysis includes the Karhunen-Loeve expansion method (KLEM) [21], polynomial chaos expansion [22], and surrogate modelling [23]. KLEM expands the system response into a deterministic part and a stochastic part. The stochastic part consists of a series of unrelated random variables and deterministic coefficients (eigenvalues and eigenfunctions). The disadvantage of KLEM is that it is not applicable to the covariance function of unknown stochastic processes. Polynomial chaos expansion expands the structural probability using the orthogonal polynomial of the basic variable and uses the chaotic polynomial as the basis of the expansion. Therefore, it can be applied to both Gaussian and non-Gaussian outputs. However, when the number of uncertain parameters increases, the number of polynomial coefficients also increase, which greatly increases the cost of computation. Hence, stochastic spectral analysis method is a new method developed recently and remains an on-going research. Currently, the approximation method is an important approach to solving the PPF problem. This method mainly includes the first-order second-moment method (FOSMM) [24] and point estimate method (PEM) [25]. The FOSMM is fast and precise. Because it adopts the Taylor series expansion method, only the linear term is considered and the higher-order term is ignored; therefore, when the system is highly nonlinear or the random variables are very asymmetrical, a large error will be caused. PEM is now widely used for solving the PPF problem; through Gaussian interpolation, the estimated points can be easily obtained, resulting in fewer power flow calculations. PEM is very fast. However, the original PEM cannot deal with correlated variables [26] and the estimated points might exceed the estimator boundary constraints, which results in a less accurate result [27]. Therefore, PEM is restricted to specific applications. However, studies examining cases wherein a great number of probability variables with different marginal distributions co-exist in IESs are scarce. In practice, the probability variables, such as those of RESs, are highly correlated with each other; for example, wind speeds and solar radiation at different sites within the same geographical area are typically dependent owing to similar weather conditions. Thus, large errors will occur if we ignore the correlations among different probability variables. The commonly used approaches to deal with the correlation of random variables are transformation methods such as the third-order polynomial normal transformation (TPNT) [28], rotational transformation and orthogonal transformation (RTOT) [29], Rosenblatt transformation [30], unscented transformation [26], and copula function method (CFM) [20]. TPNT transfers variables into the third-order polynomial normal space, which causes mapping errors and low accuracy in some cases. RTOT transfers the correlation variables from the linear correlation coefficient matrix into the generated samples. Unlike the unscented transformation, which does not require a linear transformation and is suitable for any arbitrary nonlinear function, RTOT is essentially a linear transformation method.