3.4.1. The Cumulant

The cumulant [32,33] is an alternative moment of a continuous probability distribution, and the relationship between the cumulant κ and moment *m* is as follows:

$$\begin{aligned} \kappa\_1 &= m\_1\\ \kappa\_2 &= m\_2 - m\_1^2\\ \kappa\_3 &= m\_3 - 3m\_1 m\_2 + 2m\_1^3\\ \kappa\_4 &= m\_4 - 2m\_2^2 - 4m\_1 m\_3 + 12m\_1^2 m\_2 - 6m\_1^4\\ \vdots \end{aligned} \tag{18}$$

Cumulants have two important characteristics. One is homogeneity. For a random variable *x*, the *n*th cumulant is homogeneous of order *r*:

.

$$
\kappa\_r(a\mathbf{x}) = a^r \kappa\_r(\mathbf{x}) \tag{19}
$$

The other is additivity; for two independent random variables *x* and *y*, and *z* = *x* + *y*, then:

$$
\kappa\_{\mathbf{r}}(z) = \kappa\_{\mathbf{r}}(\mathbf{x}) + \kappa\_{\mathbf{r}}(y) \tag{20}
$$

The uncertainty part of line flow under a single system topology is the linear combination of δ*Ri* and δ*Di*. As we know the first four order moments of δ*Ri* and δ*Di* in advance, using Equation (18) and the characteristics of the cumulant, the first four order moments of the uncertainty part of the line flow under a single system topology can be obtained.

## 3.4.2. The Johnson System

Previous works [34,35] have shown that the Johnson system is a reliable and accurate method for obtaining CDF compared to the commonly used Gram-Charlier series [32] or Cornish-Fisher series [33]; therefore, it is used here to obtain the CDF of the uncertainty part of the line flow under a single system topology.

The Johnson system is a 4-parameter transformation system that uses the following function to transform the standard Gaussian variable *u* into a variable *x* that follows an unknown arbitrary probability distribution:

$$\mathbf{x} = \mathbf{c} + d \ltimes f^{-1}(\frac{\mu - a}{b}) \tag{21}$$

where *a* and *b* are the shape parameters, *c* is the position parameter, and *d* is the scale parameter. The function *f* <sup>−</sup><sup>1</sup>(·) takes 4 forms to distinguish di fferent distribution families:

$$\begin{cases} S\_L: f^{-1}(\frac{\mathbf{u}-\mathbf{a}}{b}) = e^{(\mathbf{u}-\mathbf{a})/b} \\ S\_{ll}: f^{-1}(\frac{\mathbf{u}-\mathbf{a}}{b}) = (e^{(\mathbf{u}-\mathbf{a})/b} - e^{-(\mathbf{u}-\mathbf{a})/b})/2 \\ S\_B: f^{-1}(\frac{\mathbf{u}-\mathbf{a}}{b}) = 1/(1 + e^{-(\mathbf{u}-\mathbf{a})/b}) \\ S\_N: f^{-1}(\frac{\mathbf{u}-\mathbf{a}}{b}) = \frac{\mathbf{u}-\mathbf{a}}{b} \end{cases} \tag{22}$$

where *SL* is the family of lognormal distributions, *SU* is the family of unbounded distributions, which means the range of variables is unlimited, *SB* is the family of bounded distributions, and *SN* is the family of Gaussian distributions.

If the first four order moments of random variable *x* (the uncertainty part of the line flow in this paper) are available, the moment-based algorithm [36] can be used to obtain the parameters of the Johnson system, and the CDF of *x* is a function of the CDF of the standard Gaussian variable, which can be obtained easily.

The CDF of the uncertainty part of the line flow under each system topology is obtained. Based on the law of total probability, *CDFl*,*uncertainty* is calculated through the weighted average of each CDF of the uncertainty part of the line flow, and its inverse function *CDF*−<sup>1</sup> *l*,*uncertainty*can be e fficiently calculated.
