## 4 Multi-Energy Microgrid Reliability Assessment
### Reliability Evaluation Indexes  
This study selected the expectation of energy supplied (EES), loss of energy expectation (LOEE), and system average interruption duration index (SAIDI) as the indices to evaluate the reliability of different energy types in the multi-energy microgrid. The multi-energy reliability economic indicator was built by combining energy prices. (1) EES  
This index refers to the probability that a certain energy type reaches the user demand side in a certain period. A higher EES suggests that the energy can meet the demand in the sampling period with high reliability. The specific solution is as follows:  
The energy conversion matrix [2] in the energy hub model is extended to the energy transmission power matrix \(C\). Matrix \(C\) includes the unit supply matrix \(C_{\mathbf{M}}\) and adjustable resource matrix \(C_{\mathbf{S}}\). The matrix expression forms are shown in Equations (20) and (21). The matrix rows represent different run-time scenarios (including normal operation and failure scenarios); the matrix columns represent the supplies of power, heat, and cold energy types. The unit supply matrix \(\mathbf{C_{M}}\) refers to the energy output of the unit following the different energy demands in a certain period; the adjustable resource matrix \(\mathbf{C_{S}}\) refers to the energy provided by backup resources or a utility grid to the microgrid after a certain period. Specific values are associated with specific operation strategies. The energy transmission matrix of the \(i\)th scenario in the sampling period is: \(C_{i}=C_{\text{M},i}+C_{\text{S},i}\). \[C_{\text{M}}=\left[ \begin{array}{c}P_{\text{output}}^{\text{CCHP}}+P_{\text {output}}^{\text{PV}}&P_{\text{output}}^{\text{CCHP}}+P_{\text{output}}^{\text{ GHHP}}&P_{\text{output}}^{\text{CCHP}}+P_{\text{output}}^{\text{CATC}}\\ P_{\text{output}}^{\text{PV}}&P_{\text{output}}^{\text{GFHP}}&P_{\text{output }}^{\text{CCATC}}\\ P_{\text{output}}^{\text{CCHP}}+P_{\text{output}}^{\text{PV}}&P_{\text{output }}^{\text{CCHP}}+P_{\text{output}}^{\text{GHHP}}&P_{\text{output}}^{\text{CCHP }}\\ P_{\text{output}}^{\text{CCHP}}+P_{\text{output}}^{\text{PV}}&P_{\text{output }}^{\text{CCHP}}+P_{\text{output}}^{\text{CGHP}}&P_{\text{output}}^{\text{CCHP }}+P_{\text{output}}^{\text{CCATC}}\\ P_{\text{output}}^{\text{PV}}&0&P_{\text{output}}^{\text{CCHP}}\\ P_{\text{output}}^{\text{PCATP}}&P_{\text{output}}^{\text{CCHP}}+P_{\text{output }}^{\text{GGHP}}&P_{\text{output}}^{\text{CCHP}}+P_{\text{output}}^{\text{CCATC}}\\ P_{\text{output}}^{\text{PCATP}}+P_{\text{output}}^{\text{PV}}&P_{\text{output }}^{\text{CCHP}}+P_{\text{output}}^{\text{CCHP}}&P_{\text{output}}^{\text{CCHP }}+P_{\text{output}}^{\text{CCATC}}\\ P_{\text{output}}^{\text{PCATP}}+P_{\text{output}}^{\text{PV}}&P_{\text{output }}^{\text{CCHP}}+P_{\text{output}}^{\text{CCHP}}&P_{\text{output}}^{\text{CCHP }}+P_{\text{output}}^{\text{CCATC}}\\ \end{array} \right] \tag{20}\]  
\[C_{\text{S}}=\left[ \begin{array}{c}P_{\text{output}}^{\text{PG}}+P_{\text {output}}^{\text{ES}}&P_{\text{output}}^{\text{HS}}&0\\ P_{\text{output}}^{\text{PG}}+P_{\text{output}}^{\text{ES}}&P_{\text{output }}^{\text{HS}}&0\\ P_{\text{output}}^{\text{PG}}+P_{\text{output}}^{\text{ES}}&P_{\text{output }}^{\text{HS}}&0\\ P_{\text{output}}^{\text{PG}}+P_{\text{output}}^{\text{ES}}&P_{\text{output }}^{\text{HS}}&0\\ P_{\text{output}}^{\text{PG}}+P_{\text{output}}^{\text{ES}}&P_{\text{output }}^{\text{HS}}&0\\ P_{\text{output}}^{\text{PG}}+P_{\text{output}}^{\text{ES}}&P_{\text{output }}^{\text{HS}}&0\\ P_{\text{output}}^{\text{PG}}+P_{\text{output}}^{\text{ES}}&P_{\text{output }}^{\text{HS}}&0\\ P_{\text{output}}^{\text{PG}}+P_{\text{output}}^{\text{ES}}&P_{\text{output }}^{\text{HS}}&0\\ P_{\text{output}}^{\text{PG}}&P_{\text{output}}^{\text{HS}}&0\\ \end{array} \right] \tag{21}\]  
The load energy demand matrix \(L=[P_{\text{Load}}^{\text{elec}},P_{\text{Load}}^{\text{heat}},P_{\text{Load }}^{\text{cold}}]\) reflects the demand for power, heat, and cold energies in the sampling period.