# A Small-Sample Scenario Optimization Scheduling Method Based on Multidimensional Data Expansion
## 3 XGBoost PV Power Prediction Method Considering Data Imbalance in Small-Sample Scenarios
### Optimized Scheduling Objective Functions for Integrated Energy Systems  
The optimal energy system dispatch problem can be modeled using the MINLP model given in Equations (16)-(27). The objective function in Equation (16) aims to minimize the cost of operating the entire time horizon \(\mathcal{T}\), including the operating cost of the DG unit shown in Equation (17) and the cost of buying/selling power to the main grid in Equation (18). Given the output power \(P_{i,t}^{G}\) of the DG unit P, the operating cost can be estimated using the quadratic model in Equation (17). The transaction costs between the energy system and the main grid are settled on a time-of-use tariff, where the sale price is assumed to be lower than the purchase price. In Equation (18), \(p_{t}\) is the time-sharing tariff for time period \(t\), and \(P_{t}^{N}\) is the power transaction of output/input to the primary grid. \[\min\sum_{t\in\mathcal{T}}\sum_{i\in\mathcal{G}}(C_{i,t}^{G}+C_{t}^{E})\Delta t \tag{16}\]  
\[C_{i,t}^{G}=a_{i}\cdot\left(P_{i,t}^{G}\right)^{2}+b_{i}\cdot P_{i,t}^{G}+c_{i },\quad i\in\mathcal{G}. \tag{17}\]  
\[C_{t}^{E}= \begin{cases}\rho_{t}P_{t}^{N}&P_{t}^{N}>0,\\ \beta\rho_{t}P_{t}^{N}&P_{t}^{N}<0.\end{cases}  \tag{18}\]  
\[\sum_{i\in\mathcal{G}}P_{i,t}^{G}+\sum_{m\in\mathcal{V}}P_{m,t}^{V}+P_{t}^{N}+ \sum_{j\in\mathcal{G}}P_{j,t}^{B}=\sum_{k\in\mathcal{L}}P_{k,t}^{L},\quad \forall t\in\mathcal{T} \tag{19}\]\[\underline{E}_{l}^{G}\cdot u_{i,t}\leq P_{i,t}^{G}\leq\overline{P}_{i}^{G}\cdot u _{i,t}\quad\forall i\in\mathcal{G},\forall t\in\mathcal{T} \tag{20}\]  
\[P_{i,t}^{G}-P_{i,t-1}^{G}\leq RU_{i}\quad\forall i\in\mathcal{G},\forall t\in \mathcal{T} \tag{21}\]  
\[P_{i,t}^{G}-P_{i,t+1}^{G}\leq RD_{i}\quad\forall i\in\mathcal{G},\forall t\in \mathcal{T} \tag{22}\]  
\[-\underline{B}_{j}^{B}\leq P_{j,t}^{B}\leq\overline{P}_{j}^{B}\quad\forall j \in\mathcal{B},\forall t\in\mathcal{T} \tag{23}\]  
\[SOC_{j,t}^{B}=SOC_{j,t-1}+\eta B_{j,t}^{B}\Delta t\quad\forall j\in\mathcal{ B},\forall t\in\mathcal{T} \tag{24}\]  
\[\underline{E}_{j}^{B}\leq SOC_{j,t}^{B}\leq\overline{E}_{j}^{B}\quad\forall j \in\mathcal{B},\forall t\in\mathcal{T} \tag{25}\]  
\[-\overline{P}^{C}\leq P_{t}^{N}\leq\overline{P}^{C}\quad\forall t\in\mathcal{T} \tag{26}\]  
\[u_{i,t}\in\{0,1\},\quad\forall i\in\mathcal{G},\forall t\in\mathcal{T} \tag{27}\]  
The power balance constraint is formulated in Equation (19). The commitment state of the DG units is modeled using a binary variable, i.e., \(u_{i,t}=1\) indicates that the \(i\)-th DG unit is in operation. The generation capacity limits of DG units are specified in Equation (20), while Equations (21) and (22) impose ramp-up and ramp-down constraints, respectively. For the energy storage system (ESS), Equations (23)-(25) define its operational model. It is worth noting that this formula does not take into account the operating costs of the energy storage system, and it can also pre-plan charging and discharging operations. Equation (23) defines the charging and discharging power limits, and Equation (24) models the state of charge (SOC) as a function of the charging and discharging power. The expression in Equation (25) limits the energy stored in the cell and avoids the effects of overcharge and over-discharge. Finally, the main grid output/import power limit is modeled by the expression in Equation (26). The constraint in Equations (20)-(25) is achieved through hard constraints, while the power balance in Equation (19) and Equation (26) is constrained by adding penalty terms.