## 5 Probabilistic Analysis of the Electricity-Gas-Heat IES  
When considering uncertainties in such electricity/gas/heat loads and renewable energy sources, the DEF problem of the EGH-IES becomes a PEF problem. Any output variable of the EGH-IES, such as bus voltage, line power, nodal pressure, gas flow within gas pipeline, mass flow rate within heat pipe or supply temperature of heat load, can be expressed by  
\[\mathbf{y}=\mathbf{f}(\mathbf{x}) \tag{14}\]  
where \(\mathbf{x}\) is a vector of random input variables (e.g., electricity/gas/heat loads, wind speed and solar power, etc. ), and \(\mathbf{y}\) is the output variable. In this paper, uncertainties in wind speed, photovoltaic power and electricity/gas/heat loads are considered, and the most widely used distributions are adopted to model these uncertainties (i.e., Weibull distribution for wind speed [18], Beta distribution for photovoltaic power [20] and normal distributions for loads [20,21]). Correlations among these uncertainties are also considered in this paper and are represented by a correlation matrix. Therefore, probabilistic analysis of an EGH-IES or the task of PEF problem (14) is to obtain the probabilistic characteristics (e.g., mean and standard deviation (SD)) and probabilistic distributions  
Figure 3: Flowchart for solving the DEF problem of an EGH-IES. (e.g., probabilistic density function (PDF) and cumulative distribution function (CDF)) of any output variable from the given distributions of the correlated uncertain wind speed, photovoltaic power and electricity/gas/heat loads. The PEF problem (14) can be solved by many methods, and the MCS-LN method is developed to solve it in this paper. The MCS-LN method is based on LHS, Nataf transformation and repetitive solving of the DEF problem, where LHS is used to improve the sampling efficiency of Monte Carlo simulation and Nataf transformation is used to deal with correlations between uncertainties, especially correlations between uncertainties following non-normal distributions. Both the procedure and the flowchart of the MCS-LN method for solving PEF problem 14 shown in Figure 4 can be summarized as follows. **Step 1**: Calculate the correlation matrix of a standard normal distribution vector from the given correlation matrix of input random vector with Nataf transformation. Let \(\mathbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{\mathrm{T}}\) be the \(n\)-dimensional input random vector with correlation matrix \(\mathbf{C}_{\mathbf{X}}\). By applying the inverse transformation method, the random vector \(\mathbf{x}\) can be described by a standard normal distribution vector \(\mathbf{Z}\) composed of \(Z_{k}\) (\(k=1,2,\ldots,n\)) with a different correlation matrix \(\mathbf{C}_{\mathbf{Z}}\). \[\mathbf{x}_{k}=\mathbf{F_{k}}^{-1}(\mathbf{\Phi}(\mathbf{Z}_{k})),\mathbf{k}=1,2,\cdots,n \tag{15}\]  
where \(F_{k}\) is the CDF of \(x_{k}\), and \(\mathbf{\Phi}(\cdot)\) denotes the single-variable standard normal CDF of \(Z_{k}\). The principal problem of Nataf transformation is to obtain the correlation matrix \(\mathbf{C}_{\mathbf{Z}}\) in normal space from the known correlation matrix \(\mathbf{C}_{\mathbf{X}}\) of input random vector \(\mathbf{x}\).