## 2 Distributed Energy Storage Aggregation Model Based on Inner Approximation of Minkowski Sum
### Day-Ahead Optimal Scheduling Model
#### 3.1.1 Objective Function
 \tag{27}\]  
where \(f_{1}\) represents the objective function of the day-ahead scheduling optimization model, which denotes the system operating cost; \(f_{\text{G},t}\), \(f_{\text{ESS},t}\), \(f_{\text{DG},t}\), \(f_{\text{load},t}\), and \(f_{\text{des},t}\) represent the cost functions for GTs, ESS, PV, user load shedding, and the regulation of DES clusters, respectively; \(N_{S}\) denotes the number of scenarios; \(p_{s}\) represents the probability coefficient of the \(s\)-th scenario; \(N_{G}\) refers to the number of GTs; and \(P_{\text{G}i,t,s}\) represents the power output of the \(i\)-th GT at time \(t\) under the \(s\)-th scenario; \(a_{i}\), \(b_{i}\), and \(c_{i}\) represent the power generation cost coefficients of the \(i\)-th GT; \(S_{i}\) is the start-stop cost coefficient of the \(i\)-th GT; \(u_{\text{G}i,t}\) is the start-stop state of the \(i\)-th GT at time \(t\), with 1 being the start and 0 being the stop; \(N_{\text{ESS}}\) is the number of ESS; \(P_{\text{ESS},i,t,s}\) is the amount of power generated by the ESS \(i\) under the \(s\)-scenario at time \(t\); \(C_{\text{ESS},i}\) and \(W_{\text{ESS},i}\) represent the operating cost coefficient and maintenance cost coefficient of the ESS \(i\), respectively; \(\pi_{\text{ESS}}\) represents the depreciation cost coefficient per unit of time of the ESS; \(u_{\text{ESS},i,t,s}\) represents the on/off status of ESS \(i\) at time \(t\); \(N_{\text{DG}}\) is the number of DG; \(P_{\text{DG},i,t,s}\) is the output of the \(i\)-th DG in scenario \(s\) at time \(t\); \(C_{\text{DG},i}\) is the operating cost coefficient of the DG \(i\); \(k_{c,\text{DG}}\) is the penalty cost coefficient for wind abandonment; \(P_{\text{DG},i,t,s}^{\text{pre}}\) is the predicted output of DG in scenario \(s\) at time \(t\); \(k_{c,\text{load}}\) is the penalty coefficient for loss of power of the loads; \(P_{\text{loss},t,s}\) is the amount of power lost by loads in scenario \(s\) at time \(t\); \(c_{\text{buy},t}\) is the electricity price at moment \(t\); \(f_{\text{buy},t}\) is the amount of electricity purchased from the superior grid at moment \(t\); \(N_{\text{des}}\) represents the number of nodes containing DES clusters; \(k_{c,\text{des}}\) represents the cost of regulating the DES clusters; \(P_{\text{des},n,t}\) represents the power of the DES clusters at node \(n\) at moment \(t\); \(u_{\text{des},n,t}\) represents the state of the DES clusters at node \(n\) participating in the regulation, with 1 representing the participation in the regulation, and 0 representing the non-participation in the regulation; \(c_{\text{des},n}\) represents the cost of participating in regulation of the DES clusters at node \(n\). To quantify the economic and environmental value of the operational regulation of DES, a carbon emission cost model is established by dividing carbon emissions into three segments [34]. The baseline method is adopted to allocate carbon emission allowances, and it is approximately assumed that the system carbon emission allowances are proportional to the output of generation units. The calculation formula for the system carbon emission allowance is as follows:  
\[E_{L}=\sum_{t=1}^{T}\varepsilon P_{t} \tag{28}\]  
\[P_{t}=\sum_{i=1}^{N_{\text{G}}}P_{\text{G},i,t}+P_{\text{DG},t}+P_{\text{buy },t} \tag{29}\]  
where \(E_{L}\) is the total carbon emission allowance of the system; \(T\) is the length of an operation cycle; \(\varepsilon\) is the emission allowance per unit of electricity; \(P_{t}\) is the total power generation of all generating units in the system at the moment \(t\). PV generation, as a clean and environmentally friendly energy source, does not produce carbon emissions. Therefore, the system's carbon emissions primarily originate from GTs and power purchased from the upstream grid.