A Gaussian copula is widely used to define correlated random variables. To obtain a prescribed Pearson correlation coefficient of ρ_x between two random variables with given marginal distributions, the correlation coefficient ρ_z between two standard normal variables in the copula must take a specific value which satisfies an integral equation that links ρ_x to ρ_z. In a few cases, this equation has an explicit solution, but in other cases it must be solved numerically. This paper attempts to address this issue. If two continuous random variables are involved, the marginal transformation is approximated by a weighted sum of Hermite polynomials; via Mehler’s formula, a polynomial of ρ_z is derived to approximate the function relationship between ρ_x and ρ_z. If a discrete variable is involved, the marginal transformation is decomposed into piecewise continuous ones, and ρ_x is expressed as a polynomial of ρ_z by Taylor expansion. For a given ρ_x, ρ_z can be efficiently determined by solving a polynomial equation.

For the issue of generating correlated random vector containing discrete variables, one major obstacle is to determine a suitable correlation coefficient ρ_z in normal space for a specified correlation coefficient ρ_x. This paper develops a method to solve this problem. First, the double integral evaluated for ρ_x is transformed into independent standard uniform space, then, a Quasi Monte Carlo method is introduced to calculate the double integral. For a given ρ_x, an appropriate ρ_z is determined by a false position method. Compared with existing methodologies, the proposed method is less efficient, but it is relatively easy to implement.

In this study, the probabilistic power flow (PPF) problem is modelled as a multiple integral with an implicit integrand. A detailed discussion is given about the dimension reduction (DR) method for solving PPF. To accommodate the dependency of the input random variables, Nataf transformation is introduced to map PPF problems in the independent standard normal space. Emphasis is put on the computational burden of DR method. Based on Taylor expansion, a deep sight is given in the residual error of univariate DR (UDR) method. Using the first four standardised central moments from UDR method, the generalised lambda distribution is employed to represent the probability distribution of the output variable. Finally, an IEEE‐118 case is performed to check the proposed method.

The probabilistic power flow (PPF) computation involves quantifying and propagating uncertainty over hundreds of variables. Zhao's point estimate method (PEM) has been widely used for PPF computation, and the computational burden increases linearly with respect to the number of PPF inputs. In this paper, Zhao's PEM is reformulated as the Kronecker product of Gauss-Hermite quadrature and an identity matrix. Following this formulation, a new PEM termed Nested PEM is developed based on Hadamard matrix, which can reduce the computational burden nearly by half relative to Zhao's PEM. In order to consider the dependence among PPF inputs, the extended generalized lambda distribution (EGLD) is employed to model marginal distributions of correlated PPF inputs, and the $t$-copula is introduced to map PPF problem to the independent standard normal space. Finally, case studies are conducted on a modified IEEE 118-bus system to illustrate the proposed PEM.

In this article, the Latin hypercube sampling (LHS) based probabilistic optimal power flow (P‐OPF) technique is employed to assess the performance of power systems under large wind power penetration. In the case where only wind speed samples are available, the kernel tuned polynomial expansion series are developed to recover marginal distributions of wind speeds. Because the dependence among wind speeds has a considerable impact on P‐OPF solutions, 13 Archimedean copulae are derived to represent the dependence structure of correlated wind speeds at multiple sites. The rejection sampling method and Metropolis‐Hastings algorithm are deployed to generate samples of copula models, and a midpoint LHS technique is proposed to produce correlated low discrepancy sequences for P‐OPF computation. Finally, a case study is performed on an IEEE 118‐bus system to illustrate the proposed methods.