## 4 Multi-Objective Scheduling Optimization Model
### Multi-Objective Scheduling Model
#### 4.1.1 Objective Functions  
MECSs can utilize the complementarity of different power sources, make full use of the adjustable power sources' peaking capacities, and promote the grid connection of clean energy sources, such as wind and photovoltaic power. Therefore, the MECS operation revenue comprises the economic revenue, environmental revenue, and peak shaving revenue. \[\left(\text{P1}\right)\ \max F_{1}=\sum\limits_{t=1}^{T}\sum\limits_{k=1}^{K} \gamma_{k}\left(R_{WPP,t}+R_{PV,t}+R_{ESD,t}+R_{HS,t}+R_{TPP,t}\right)_{k} \tag{11}\]  
where \(t\) is the index for time. \(k\) is the index for the scenario. \(\gamma_{k}\) is the probability of scenario \(k\). \(R_{WPP,t}\), \(R_{PS,t}\), \(R_{HS,t}\), and \(R_{G,t}\) denote the operation revenue for the WPP, the PV, the ESD, the HS, and the TPP at time \(t\), respectively. The operation revenue for the WPP, the PV, and the ESD have already been studied in Reference [10]. As is the case with the WPP and the PV, the cost of the initial investment, operation, and maintenance of the HS is classified as the sunk cost, whereas the energy consumption cost of the HS can also be ignored. Therefore, the revenue of the HS operation equals the product of the power price and power output. Thus, the revenue of the TPP operation comprises three parts: the power generation cost, the startup-shutdown cost, and the pollutant emission cost. The detailed calculations are as follows:  
\[R_{TPP,t}=\rho_{TPP,t}\delta_{TPP,t}-\rho_{\text{coal}}\left(\pi_{TPP,t}^{pg}+ \pi_{TPP,t}^{ss}\right)-\pi_{TPP,t}^{pe} \tag{12}\]  
\[\pi_{TPP,t}^{pg}=a_{TPP}+b_{TPP}\delta_{TPP}+c_{TPP}(g_{TPP,t})^{2} \tag{13}\]  
\[\pi_{TPP,t}^{ss}=\left[u_{TPP,t}(1-u_{TPP,t})\right]\cdot\left\{ \begin{array} []{l}N_{CGT}^{hot},T_{CGT}^{\min}<T_{CGT,t}^{off}\leq T_{CGT}^{\min}+T_{CGT}^{ cold}\\ N_{CGT}^{cold},T_{CGT,t}^{off}>T_{CGT}^{min}+T_{CGT}^{cold}\end{array} \right. \tag{14}\]  
\[\pi_{TPP,t}^{pe}=\sum\limits_{c=1}^{E}p_{c}\left[\widetilde{a}_{TPP}^{s}+ \widetilde{b}_{TPP}^{c}\delta_{TPP}+\widetilde{c}_{TPP}^{r}(g_{TPP,t})^{2}\right] \tag{15}\]  
Figure 3: The uncertainty simulation flowchart. PV: photovoltaic power plant; WPP: wind power plant. where \(\pi^{pg}_{TPP,t}\), \(\pi^{ss}_{TPP,t}\), and \(\pi^{pe}_{TPP,t}\) denote the power generation cost, startup-shutdown cost, and pollutant emissions cost of the TPP at time \(t\), respectively. \(a_{TPP}\), \(b_{TPP}\), and \(c_{TPP}\) are the cost coefficients of the TPP power generation. \(\rho_{\text{coal}}\) is the coal price. \(u_{TPP,t}\) and \(D_{TPP,t}\) are the status and cost of the TPP startup-shutdown at time \(t\). \(N^{\text{{{NoId}}}}_{TPP}\) and \(N^{\text{{{NoId}}}}_{TPP}\) are startup costs of the TPP operation in the cold mode and hot mode. \(T^{\text{{{min}}}}_{TPP}\) and \(T^{\text{{{off}}}}_{CGT,t}\) are the minimum allowable and continuous downtime of TPP at time \(t\), respectively. \(T^{\text{{{cold}}}}_{TPP}\) is the required time for TPP startup in cold-start mode. \(a_{TPP}\), \(b_{TPP}\), and \(c_{TPP}\) are the cost coefficients of TPP power generation. \[\text{(P2)}\ \text{min}\ F_{2}=\sum\limits_{l=1}^{T}\sum\limits_{k=1}^{K} \gamma_{k}\left[\rho_{WP,t}\left(g^{*}_{WPP,t}-g_{WPP,t}\right)+\rho_{PV,t} \left(g^{*}_{PV,t}-g_{PV,t}\right)+\rho_{HS,t}\left(g^{*}_{HS,t}-g_{HS,t} \right)\right]_{k} \tag{16}\]  
where, \(F_{2}\) is the abandoned energy cost of WPP, PV, and HS during the scheduling period. For MECS, determining how to use the complementary characteristics of different power sources is the key to reducing the amount of abandoned energy. In fact, if the abandoned energy could be absorbed, more economic revenue could be gained. Correspondingly, this part of the economic revenue can be regarded as the opportunity cost. \(\rho_{WPP,t}\), \(\rho_{PV,t}\), and \(\rho_{HS,t}\) are the power prices of WPP, PV, and HS, respectively, at time \(t\). \(g^{*}_{WPP}\), \(g^{*}_{PV}\), and \(g^{*}_{HS,t}\) are the available outputs of WPP, PV, and HS at time \(t\), respectively.