## 5 Case Study
### Case 1: Estimation Points within the Limit Range  
In this case, the mean of wind speed is 8.5 m/s, and each wind power plant follows the distribution given in Equation (9), where \(k_{p}=2\), \(c_{p}=10\), \(x_{ci}=3.5\)\(m/s\), \(x_{co}=18m/s\), \(P=0.315 MW\). Moreover, each photovoltaic power plant follows the distribution given in Equation (10), where \(x\) is the standard light intensity value, \(x=G_{stc}(t)/G_{stcmax}\), \(\alpha=3\), \(\beta=2\), \(\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)+\Gamma(\beta)}=20\), and \(p^{stc}=0.5 MW\). The mean of the gas load 3 is \(\mu=204m^{3}/h\), its standard deviation is \(\delta=2\), We then have  
\[f(v)=\frac{2}{10}(\frac{v}{10})e^{-(\frac{v}{10})^{2}}\quad\ v\geq 0\enspace, \tag{53}\]  
\[h(v)= \begin{cases}0&0\leq v<3.5\\ \frac{1}{46}(v-3.5)&3.5\leq v<18\\ 0.315&18\leq x\end{cases}  \tag{54}\]  
\[f(x)=20(x)^{2}(1-x)\quad\ 0\leq x\leq\ 1. \tag{55}\]  
Figure 3 shows the probability density of the sampling points and probability distribution of the power plants and photovoltaic power plants after the Nataf transformation. The appearance of this figure noisy is because the pseudo random variables produced by the computer do not strictly satisfy the probability density function of samples. Figure 3c,d show that the results agree well the theoretical sampling points, indicating that the Nataf transformation has a very good precision. Figure 3: Probability density of sampling points in (**a**) wind power plants and (**b**) photovoltaic power plants. Cumulative probability distribution of (**c**) wind power plants and (**d**) photovoltaic power plants. Figure 4 shows the correlation between each wind power plant and photovoltaic power plant after the Nataf transformation. The distribution of the sampling points is all concentrated in an ellipse. A flatter ellipse indicates a stronger correlation. It can be seen that the Nataf transformation controls the correlation well. Next, these points are used to calculate the PEF. Figure 5 shows the results of m-PEM combined with the Nataf transformation. To show specific errors, Tables 2 and 3 show the errors in the variance and mean. The calculation results show that this method has a very good accuracy when calculating the mean value, especially when calculating the mean value of the grid voltage. When calculating variance, it performs worse, which may be related to the number of Monte Carlo iterations. Because the PEF solution to an IES is very time-consuming, the number of Monte Carlo iterations must be as low as possible (3000). For 2-PEM, 3-PEM, 5-PEM, and 7-PEM, the computational quantities are 12, 18, 25, and 37, respectively. Figure 4: Results for point estimate method with the Nataf transformation for Case 1. Correlations in the wind speeds of (**a**) wind power plants 1 and 2, (**b**) wind power plants 1 and 3, and (**c**) wind power plants 2 and 3. (**d**) Correlation in the light intensity for the photovoltaic power plants. To compare the results of PEF with and without the Nataf transformation, we used a common pseudo-random sample method for wind and photovoltaic farms. Figure 6 shows the correlation of common pseudo-random sampling points. It can be seen that the correlation of samples is very small: the correlation graph of the sample is approximately distributed in a circle in the plane. If the random error of sampling is ignored, the samples are independent of each other. \begin{table}
\begin{tabular}{c c c c c} \hline
**Relative Error** & **7-PEM (\%)** & **5-PEM (\%)** & **3-PEM (\%)** & **2-PEM (\%)** \\ \hline mean & 0.1164472 & 0.1725141 & 0.1930306 & 0.240743 \\ variance & 14.783333 & 15.873716 & 21.8975776 & 25.6881084 \\ \hline \end{tabular}
\end{table} 
Table 2: Average relative error of the gas network. Figure 5: Case 1 results. (**a**) Mean pressure for the natural gas network. (**b**) Variance in pressure for the natural gas network. (**c**) Mean voltage for the power network. (**d**) Variance in voltage for the power network. Next, the results of the two sampling methods were compared using MCS.