## 6 Case Studies
### Effects of Different Correlation Levels among Loads  
The previous subsection studied the effects of different conversion levels of power to gas/heat on the operation of the EGH-IES and the corresponding uncertainty transmission among the subnetworks. In this subsection, we study further the effects of different correlation levels among uncertainty factors, especially the effects of different correlation levels among electricity, gas and heat loads. Effects of correlation levels between wind speeds and solar power, etc. can be studied in a similar way. The following three testing scenarios with different correlation levels are considered:  
Scenario 1: \(\rho_{\rm{EG}}=\rho_{\rm{EH}}=\rho_{\rm{GE}}=0\);  
Scenario 2: \(\rho_{\rm{EG}}=\rho_{\rm{EH}}=0.1\), \(\rho_{\rm{GH}}=0.3\);  
Scenario 3: \(\rho_{\rm{EG}}=\rho_{\rm{EH}}=0.3\), \(\rho_{\rm{GH}}=0.5\). Obviously, the loads are assumed to be uncorrelated in Scenario 1, and the correlation levels of Scenario 3 are higher than those of Scenario 2. Note that the other correlation coefficients (i.e., \(\rho_{\rm{WW}}\) and \(\rho_{\rm{PP}}\)) are the same as those in Section 6.2, and both \(\lambda_{\rm{P2G}}\) and \(\lambda_{\rm{P2H}}\) are set to be 0.1. The PEF analysis proposed is conducted in this subsection to study the effects of correlation level on both the mean values and SDs of any output variable of the EGH-IES. Simulation results indicate that different correlation levels among loads have negligible influences on the mean values and only some results related to the SDs (i.e., SDs of the voltage magnitudes and nodal pressures) are presented in Figures 19 and 20 due to the limited space. Figure 19 shows that the SDs of the voltage magnitudes at buses 1, 2, and 26-33 are slightly affected by variations in correlation of loads, while the SDs of other buses (especially buses 6-8 and 17) increase notably as the correlation levels increase from Scenario 1 to Scenario 3. This is because buses 1, 2, and 26-33 are near the slack bus (bus 33) of the electricity network, and buses 6-8 and 17 are the buses coupling to the gas and heating networks. Similar observations can be made from Figure 20. In the gas network, the SDs of pressures of nodes 1 and 2 vary slightly and the SDs of other nodes increase notably from Scenario 1 to Scenario 3. The reason is that node 1 is the gas slack node and node 2 is close to node 1, which is similar to the explanation of Figure 19.  
Theoretically, a greater correlation level among the electricity, gas and heat loads implies a higher possibility that electricity, gas, and heat loads increase or  
Based on the simulation results, the following conclusions can be obtained:  
1. The utilization of P2G/P2H has a double-edged effect on the operation of the EGH-IES. On the one hand, it is helpful in alleviating the overvoltage or overloading risks of the EGH-IES, but on the other hand, it may aggravate and complicate the uncertainty of the EGH-IES by transmitting uncertainties or variations in wind/solar power from the electricity network to the gas and heating networks. 2. Correlations among the electricity, gas and heat loads have non-negligible effects on the operation of the EGH-IES, and therefore they should be considered carefully in the PEF analysis of an EGH-IES. 3. Compared with existing studies on individual electricity, gas and heating systems, the electricity-gas IES, and electricity-heat IES, there are more complicated interactions and uncertainty transmissions among the subnetworks in an EGH-IES, which highlights the necessity of applying a probabilistic tool to the operation analysis of an EGH-IES. 4. The results obtained from the proposed probabilistic analysis framework for an EGS-IES can disclose the effects of uncertainties and interactions on the operation of EGH-IES, and thus provide an insight into the potential risks for the operators of EGH-IES. The results will also be a reference for the planning of EGH-IES. This work was supported in part by the National Science Foundation of China (51307186).