## 7 Conclusions  
A PEM based on the Nataf transformation for computing PEF in an IES was proposed in this paper. This method is appropriate for large-scale electricity, gas, and thermal power interconnected systems because the energy flow calculations for such systems are complicated and time consuming; therefore, it is essential to reduce the cost of computation. From the results of the case studies, the following conclusions can be drawn:  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Original Method** & **Power Transformation** &
\begin{tabular}{c} **Equality Constraint** \\ **Transformation** \\ \end{tabular} \\ \hline
2-PEM **(m/s)** & 10.939 & 9.1205 & 9.0384 \\  & 3.351 & 5.3445 & 5.2664 \\ \hline
3-PEM **(m/s)** & 14.3745 & & 10.285 \\  & 6.7123 & invalid & 4.0934 \\  & 1.6625 & & 3.5 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of estimation points for different methods.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Relative Error** & **3-PEM(\%)** & **2-PEM(\%)** \\ \hline mean & / & 0.8347321 \\ variance & / & 16.034543 \\  & (a) **Power transformation method** & \\ \hline
**Relative Error** & **3-PEM(\%)** & **2-PEM(\%)** \\ \hline mean & 0.8222351 & 0.8123421 \\ variance & 14.983132 & 16.854671 \\  & (b) **Equal constraint transformation method** & \\ \hline \hline \end{tabular}
\end{table}
Table 7: Average relative error of gas network after constraining estimation points.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Relative Error** & **3-PEM(\%)** & **2-PEM(\%)** \\ \hline mean & / & 0.2566954 \\ variance & / & 7.1256984 \\  & (a) **Power transformation method** & \\ \hline
**Relative Error** & **3-PEM(\%)** & **2-PEM(\%)** \\ \hline mean & 0.8888845 & 0.9123241 \\ variance & 7.9845636 & 7.16879958 \\  & (b) **Equal constraint transformation method** & \\ \hline \hline \end{tabular}
\end{table}
Table 8: Average relative error of power network after constraining estimation points.  
1. The proposed modified MPEM is effective in situations where the numbers of estimation points and variables are relatively large. The combination of the Nataf transformation with MPEM can obtain good accuracy. The proposed method makes the computational cost of m-PEM equal to that of \((\text{m}-1)\)-PEM, thus improving the computational efficiency.
2. The power transformation and equality constraint transformation methods are used to reduce the large errors when the estimation points are outside of the boundary constraints. Simulation results show the effectiveness of the proposed methods.  
Note that these methods are not only suitable for IESs but also for other scientific applications, such as the reliability analysis of structures and geotechnical reliability analysis.  
Conceptualization, B.Q.; methodology, B.Q., C.F.; software, C.F., B.Q.; validation, C.F., K.M.; investigation, C.F., J.L.; writing--original draft preparation, B.Q., C.F.; writing--review and editing, K.M., B.Q., and J.L.; supervision, B.Q., K.M.  
This work was funded by the National Natural Science Foundation of China (51707147, 21703153), Key Research and Development Program of Shaanxi (grant number 2017 ZDCXL-GY-02-03), China Postdoctoral Science Foundation (grant number 2017M623171), Shaanxi Postdoctoral Science Foundation (grant number 2017 BSHEDZZ27).  
The authors declare no conflict of interest.