## 3 Applications
### Development of DRSFO-EES Model
Constraint (56) means that the amount of coke processing is not less than the coke demands, and \(Cr\{D\widetilde{M}\widetilde{C}_{I}\big{|}CKPA_{t}\geq DM\widetilde{C}_{I} \}\geq\alpha\) means the credibility of the \(CKPA_{t}\geq DM\widetilde{C}_{I}\) is higher than or equal to confidence-level \(\alpha\). Constraint (57) depicts the coal consumption amount for coke processing. \[Cr\{DM\widetilde{C}_{I}\big{|}CKPA_{t}\geq DM\widetilde{C}_{I}\}\geq\alpha \tag{56}\]  
\[CFCJ_{t}\ =\ CKPA_{t}/CTCKE_{t} \tag{57}\]  
where \(DM\widetilde{C}_{I}(DMCJ_{t}{}^{1},DMCJ_{t}{}^{2},DMCJ_{t}{}^{3})\) is the coke demand expressed as a triangular fuzzy number. \(CTCKE_{t}\) denotes the coal consumption for unit of coke processing; \(CFCJ_{t}\) is the coal consumption of coke processing. Constraints for mass balance of coal and natural gas resources. These constraints are established to calculate the consumption amounts of coal and natural gas for electricity generation. \[[(W_{1t}^{-}+rr_{1t}\cdot WC_{1t})+Y_{1 th}+EVEC_{t}]\cdot FE_{1t}=ECOALM_{t} \tag{58}\]  
\[[(W_{2t}^{-}+rr_{2t}\cdot WC_{2t})+Y_{2 th}]\cdot FE_{2t}=ENGM_{t} \tag{59}\]where \(FE_{it}\) is the coal (\(i=1\)) and natural gas (\(i=2\)) consumption for a unit of electricity generation. Constraints of electricity demand and supply balance. Constraint (60) is established to ensure the electricity demand be satisfied by domestic electricity generation and importation. And \(Cr\{\widetilde{D}_{th}|\sum\limits_{i=1}^{4}\sum\limits_{t=1}^{3}\left[\left(W_{ it}^{-}+rr_{it}WC_{it}\right)+Y_{ith}+EVEC_{t}\right]+Ed_{th}\geq\widetilde{D}_{ th}\}\geq\alpha\) means the credibility of the \(\sum\limits_{i=1}^{4}\sum\limits_{t=1}^{3}\left[\left(W_{it}^{-}+rr_{it}WC_{it} \right)+Y_{ith}+EVEC_{t}\right]+Ed_{th}\geq\widetilde{D}_{th}\) is higher than or equal to confidence-level \(\alpha\). For constraint (61), the optimized amount of electricity generated in the first-stage is given by \(W_{kt}^{\pm}=W_{kt}^{-}+rr_{kt}\cdot\Delta W\), where \(rr_{kt}\) denotes the decision variables, and \(rr_{kt}\in[0,\,1]\). \[Cr\{\widetilde{D}_{th}|\sum\limits_{i=1}^{4}\sum\limits_{t=1}^{3}\left[\left(W _{it}^{-}+rr_{it}WC_{it}\right)+Y_{ith}+EVEC_{t}\right]+Ed_{th}\geq\widetilde{ D}_{th}\}\geq\alpha \tag{60}\]  
\[0\leq rr_{it}\leq 1 \tag{61}\]  
where \(\widetilde{D}_{th}\) is the electricity demand under various electricity demand levels, which show the characteristics of stochastic and fuzzy sets. \(rr_{it}\) denotes the decision variable of first-stage electricity generation. Constraints for electricity capacities. These constraints mainly depict that the amount of generated electricity must not exceed its existing and expanded capacities, which are established to ensure that the available electricity-generation capacity is greater than the generated electricity. \[\sum\limits_{i=1}^{4}\left(W_{it}^{-}+rr_{it}\cdot WC_{it}\right)+Y_{ith}+EVEC_{ t}\leq\sum\limits_{i=1}^{2}\left[CF_{it}\cdot\left(RC_{it}+Z_{ith}\right)\right] \tag{62}\]  
\[\sum\limits_{i=1}^{4}\left(RC_{i}+Z_{ith}\right)\geq U_{t};\forall t,\,\,h \tag{63}\]  
\[\sum\limits_{i=1}^{4}\left(RC_{i}+Z_{ith}\right)\leq US_{t};\forall t,\,\,h \tag{64}\]  
\[\sum\limits_{i=1}^{4}Y_{ith}\leq\sum\limits_{i=1}^{4}CF_{it}\cdot Z_{ith} \tag{65}\]  
\[CFCJ_{t}\ =\ CKPA_{t}/CTCKE_{t} \tag{57}\]0  
\[Q_{ith}=\left\{ \begin{array}{l}=1,\text{ if capacity expansion of is undertaken }\forall i,\,\,t,\,h\\ =0,\text{ if otherwise }\end{array} \right. \tag{67}\]  
where \(CF_{kt}\) is the operating hours of electricity conversion technology \(k\); \(RC_{kt}\) is the original capacity; \(U_{t}\) is the lower bound of load demands; \(US_{t}\) is the upper bound of load demands. (6) Constraints of energy production. The constraint is established to ensure the primary energy production amount must be less than the lower bound and higher than the upper bound. \[EPAL\leq EPA_{it}\,\,\leq\,\,EPAL_{it} \tag{68}\]  
where \(EPAL_{it}\) is the upper bound of amount of primary energy production.