## 4 Multi-Objective Scheduling Optimization Model
### Multi-Objective Scheduling Model
#### 4.1.2 Constraint Conditions  
To ensure the safe and stable operation of the MECS, some necessary constraint conditions should be considered, for example, the load supply and demand balance constraints, the TPP operation constraints, the ESD operation constraints, and the system reserve constraints. A detailed description of the MECS operation constraints is as follows. (1) Supply and demand balance constraint  
\[\underbrace{\left[g_{WPP,t}(1-\rho_{WPP})+g_{PV,t}(1-\rho_{PV})+\left(g^{ dis}_{ISD,t}-g^{ch}_{ISD,t}\right)+g_{HS,t}(1-\varphi_{HS})+g_{TPP,t}(1-\varphi_{TPP}) \right]}_{\text{{MECS}}\mathit{output}}+g_{GC,t}=L_{t} \tag{19}\]  
where \(\varphi_{WPP}\), \(\varphi_{PV}\), \(\varphi_{HS}\), and \(\varphi_{TPP}\) are the power loss rates of the WPP, PV, HS, and TPP, respectively. \(g_{GC,t}\) is the exchange power from the GC at time \(t\). When \(g_{GC,t}\) is positive, the MECS will purchase power from the GC to satisfy its power demand. When the \(g_{GC,t}\) is negative, the MECS will transfer surplus power to the utility grid. (2) HS power output constraint  
\[g_{HS,t}=\rho_{HS}^{0}+\rho_{HS}^{1}Q_{HS,t}+\rho_{HS}^{2}V_{HS,t}+\rho_{HS}^{3}Q_{ HS,t}V_{HS,t}+\rho_{HS}^{4}Q_{HS,t}^{2}+\rho_{HS}^{5}V_{HS,t}^{2}+\varepsilon \tag{20}\]  
where \(\varepsilon\) is an error term calculated by history data, \(\rho_{HS}^{0}\), \(\rho_{HS}^{1}\), \(\rho_{HS}^{2}\), \(\rho_{HS}^{3}\), \(\rho_{HS}^{4}\), and \(\rho_{HS}^{5}\) are the power output coefficients of the HS. \(Q_{HS,t}\) and \(V_{HS,t}\) are the output capacity and the reservoir capacity of HS at time \(t\). \[Q_{HS,t}^{\min}\leq Q_{HS,t}\leq Q_{HS,t}^{\max} \tag{21}\]  
\[V_{HS,t}^{\min}\leq V_{HS,t}\leq V_{HS,t}^{\max} \tag{22}\]  
where \(Q_{HS}^{\min}\) is the minimum water flow for the HS power output. \(Q_{HS}^{\max}\) is the maximum water flow for the HS power output. \(V_{HS}^{\min}\) is the lower bound of the reservoir capacity. \(V_{HS,t}^{\max}\) is the upper bound of the reservoir capacity. The reservoir capacity of the HS at the initial and end time points of the scheduling period should meet the following constraint:  
\[V_{HS}^{T}=V_{HS}^{0}+\sum_{t=1}^{T}\left[\widetilde{Q}_{t}-\underset{t}{Q}- \underset{t}{Q}-S_{HS,t}+\sum_{h=1}^{H}\left(Q_{h,t-\tau_{h,HS}}+S_{h,t-\tau_{ h,HS}}\right)\right] \tag{23}\]  
where \(\widetilde{Q}_{t}\) is the water inflow at time \(t\), \(\underset{t}{Q}\) is the water outflow at time \(t\). \(S_{HS,t}\) is the amount of wastewater at time \(t\), and \(H\) is the number of upstream hydropower stations. \(\tau_{h}\) is the delayed water flow between the HS and the upstream hydropower station \(h\). \(V_{HS}^{T}\) is the water storage capacity at time \(T\). \(V_{HS}^{0}\) is the water storage capacity at the initial time point. Water runoff into the reservoir comprises the water supply and the outflow from the upstream reservoir, as follows:  
\[V_{HS,t}=V_{HS,t}^{0}+\Delta t\sum_{\tau=1}^{t-1}\left(\widetilde{Q}_{HS,\tau} -Q_{HS,\tau}-S_{HS,\tau}+Q_{HS-1}^{\tau-lup}+S_{HS-1}^{\tau-lup}\right) \tag{24}\]  
where \(Q_{HS-1}^{\tau-lup}\) and \(S_{HS-1}^{\tau-lup}\) are the power generation flow and wastewater flow, respectively, of the upstream hydropower station. \(t_{\text{up}}\) is the time lag corresponding to the flow from the upstream hydropower station to the downstream hydropower station. (3) ESD operation constraints  
The ESD can complementarily operate with the WPP and the PV by utilizing its discharge-charge cycle. Two operation modes exist, namely, the longest life cycle (LLC) mode and the optimum economic efficiency (OEE) mode. Thus, the main constraints are as follows:  
\[\sum_{t=1}^{T}\left(Q_{0}+g_{ESD,t}^{ch}-Q_{t}\right)(1-\rho_{ESD})=\sum_{t=1}^ {T}g_{ESD,t}^{dis} \tag{25}\]  
\[0\leq g_{ESD,t}^{ch}\leq\overline{g}_{ESD,t}^{ch} \tag{26}\]  
\[0\leq g_{ESD,t}^{dis}\leq\overline{g}_{ESD,t}^{dis} \tag{27}\]  
where \(\overline{g}_{ESS,t}^{ch}\) and \(\overline{g}_{ESS,t}^{dis}\) are the upper bounds for ESS charging-discharging at time \(t\), respectively. \[Q_{t+1}=\left\{