## 3 Applications
### Development of DRSFO-EES Model
The cost represents the operation cost of first-stage electricity generation facilities (i.e., coal-based power, gas-based power, wind power and solar power) during the planning periods. It is calculated in terms of the operation costs and the amount of electricity generation for each of the electricity generation facilities. Furthermore, the first-stage is given by \(W_{kt}^{\pm}=W_{kt}^{-}+rr_{kt}\cdot\Delta W\), where \(rr_{kt}\) denotes the decision variables, \(\Delta W=W_{kt}^{+}-W_{kt}^{-}\). \[ \begin{split}&\widetilde{f}_{7}=\sum\limits_{k=1}^{4}\sum \limits_{t=1}^{3}\left[0.25(PV_{kt}{}^{1}+2\cdot PV_{kt}{}^{2}+PV_{kt}{}^{3}) \right]\cdot\left(W_{kt}{}^{-}+rr_{kt}\cdot\Delta W_{kt}\right)\\ &+\lambda_{1}\sum\limits_{k=1}^{4}\sum\limits_{t=1}^{3}\left[PV_{ it}{}^{3}-0.25(PV_{kt}{}^{1}+2\cdot PV_{kt}{}^{2}+PV_{kt}{}^{3})\right]\cdot\left(W_{ kt}{}^{-}+rr_{kt}\cdot\Delta W_{kt}\right)\\ &+\lambda_{2}\sum\limits_{k=1}^{4}\sum\limits_{t=1}^{3}\left[0.25( PV_{kt}{}^{1}+2\cdot PV_{kt}{}^{2}+PV_{kt}{}^{3})\right.\left.-PV_{kt}{}^{1} \right]\cdot\left(W_{kt}{}^{-}+rr_{kt}\cdot\Delta W_{kt}\right)\end{split}  \tag{29}\]where \(P\widetilde{V}_{kt}(PV_{kt}{}^{1},\ PV_{kt}{}^{2},\ PV_{kt}{}^{3})\) denotes the operation cost of different power conversion technologies; \(W_{kt}{}^{\pm}=W_{kt}{}^{-}+rr_{kt}\cdot\Delta W_{kt}\) is the first-stage electricity generation amount, which is determined by decision variables \(rr_{kt}\) (\(\Delta W=W_{kt}^{+}-W_{kt}^{-}\) and \(rr_{kt}\in[0,\ 1]\)). (8) Penalty cost for second-stage electricity generation amounts (i.e., shortage electricity amount of first-stage). It is calculated in terms of the penalty costs and the amount of second-stage electricity generation by each electricity generation technology. \[ \begin{array}{l}\widetilde{f}_{8}=\sum\limits_{k=1}^{4}\sum\limits_{l=1}^{3 }\sum\limits_{h=1}^{3}[0.25(PP_{kt}{}^{1}+2\cdot PP_{kt}{}^{2}+PP_{kt}{}^{3})] \cdot p_{th}\cdot Y_{kth}\\ +\lambda_{1}\cdot\sum\limits_{k=1}^{4}\sum\limits_{l=1}^{3}\sum\limits_{h=1}^{3 }[PP_{kt}{}^{3}-0.25(PP_{kt}{}^{1}+2\cdot PP_{kt}{}^{2}+PP_{kt}{}^{3})]\cdot p _{th}\cdot Y_{kth}\\ +\lambda_{2}\cdot\sum\limits_{k=1}^{4}\sum\limits_{l=1}^{3}\sum\limits_{h=1}^{3 }[0.25(PP_{kt}{}^{1}+2\cdot PP_{kt}{}^{2}+PP_{kt}{}^{3})-PP_{kt}{}^{1}]\cdot p _{th}\cdot Y_{kth}\end{array}  \tag{30}\]  
where \(P\widetilde{P}_{kt}(PP_{kt}{}^{1},\ PP_{kt}{}^{2},\ PP_{kt}{}^{3})\) is the operating cost for the second-stage electricity generation amount (\(Y_{kth}\)). \(Y_{kth}\) is the second-stage electricity generation amount. (9) Fixed and variable costs for electric capacity expansion. These costs include the fixed and variable electric capacity expansion costs of four electricity generation technologies in the planning horizon. \[ \begin{array}{l}\widetilde{f}_{9}=\sum\limits_{i=1}^{4}\sum\limits_{l=1}^{3 }\sum\limits_{h=1}^{3}p_{th}[0.25(A_{kt}{}^{1}+2\cdot A_{kt}{}^{2}+A_{kt}{}^{3 })]\cdot Q_{kth}\\ +\lambda_{1}\cdot\sum\limits_{i=1}^{4}\sum\limits_{l=1}^{3}\sum\limits_{h=1}^{3 }p_{th}[A_{kt}{}^{3}-0.25(A_{kt}{}^{1}+2\cdot A_{kt}{}^{2}+A_{kt}{}^{3})]\cdot Q _{kth}\\ +\lambda_{2}\cdot\sum\limits_{i=1}^{4}\sum\limits_{l=1}^{3}\sum\limits_{h=1}^{3 }p_{th}[0.25(A_{kt}{}^{1}+2\cdot A_{kt}{}^{2}+A_{kt}{}^{3})-A_{kt}{}^{1}]\cdot Q _{kth}\\ +\sum\limits_{i=1}^{4}\sum\limits_{l=1}^{3}\sum\limits_{h=1}^{3 }p_{th}[0.25(B_{kt}{}^{1}+2\cdot B_{kt}{}^{2}+B_{kt}{}^{3})]\cdot Z_{kth}\\ +\lambda_{1}\cdot\sum\limits_{i=1}^{4}\sum\limits_{l=1}^{3}\sum\limits_{h=1}^{3 }p_{th}[B_{kt}{}^{3}-0.25(B_{kt}{}^{1}+2\cdot B_{kt}{}^{2}+B_{kt}{}^{3})]\cdot Z _{kth}\\ +\lambda_{2}\cdot\sum\limits_{i=1}^{4}\sum\limits_{l=1}^{3}\sum\limits_{h=1}^{3 }p_{th}[0.25(B_{kt}{}^{1}+2\cdot B_{kt}{}^{2}+B_{kt}{}^{3})-B_{kt}{}^{1}]\cdot Z _{kth}\end{array}  \tag{31}\]  
where \(\widetilde{A}_{kt}(A_{kt}{}^{1},A_{kt}{}^{2},A_{kt}{}^{3})\) denote fixed-charge cost for different electric capacity expansion. \(Q_{kth}\) are binary variables for identifying whether or not the capacity expansion needs to be undertaken by power conversion technology \(k\).