# Research on Interval Optimal Scheduling Strategy of Virtual Power Plants with Electric Vehicles
## 3 Optimized Scheduling Model
### Objective Function  
(1) Economic objectives [38]  
\[\max\left[f_{1}\right]=[C_{sell}^{energy}]+[C_{buy}^{energy}]-[C_{op}^{DGs}] \tag{20}\]  
\[[C_{sell}^{energy}]=[C_{load}]+[C_{gas}]+C_{ev} \tag{21}\]  
\[[C_{buy}^{energy}]=[C_{gas}^{ex}]+[C_{gird}^{ex}] \tag{22}\]  
\[[C_{op}^{DGs}]=[C_{WPP}]+[C_{PV}]+C_{PMT}+C_{cs}+C_{gs}+C_{P2G} \tag{23}\]  
where \([C_{sell}^{energy}]\), \([C_{buy}^{energy}]\), and \([C_{op}^{DGs}]\) are the virtual plant revenue, acquisition cost, and maintenance cost of each unit, respectively.  
The specific calculations for the latter are as follows:  
\[[C_{sell}^{energy}]=\sum_{t=1}^{24}\left(P_{ev,t}+[P_{load,t}]\right)\cdot c_{ cloud,t}+[V_{gas,t}]\cdot c_{gas,t}) \tag{24}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Description** \\ \hline \(P_{PMT,t}\) & Gas turbine output electric power \\ \(P_{P2G,t}\) & Electric to gas equipment power \\ \(P_{ev,t}\) & Electric vehicle charging power \\ \(P_{es,t}\) & Power of power storage device (both positive and negative) \\ \(V_{gs,t}\) & Gas storage equipment storage capacity (both positive and negative) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Decision variables.  
\[[C^{energy}_{buy}]=\sum_{t=1}^{24}\begin{cases}[V^{ex}_{gas,t}]\cdot c^{ sell}_{gas,t}+[P^{ex}_{grid,t}]\cdot c^{sel}_{grid,t}&V^{ex}_{gas,t}\leq 0,P^{ex}_{grid,t}\leq 0\\ [V^{ex}_{gas,t}]\cdot c^{selt}_{gas,t}+[P^{ex}_{grid,t}]\cdot c^{buy}_{grid,t}&V^{ ex}_{gas,t}\leq 0,P^{ex}_{grid,t}>0\\ [V^{ex}_{gas,t}]\cdot c^{buy}_{gas,t}+[P^{ex}_{grid,t}]\cdot c^{selt}_{grid,t}&V^{ ex}_{gas,t}>0,P^{ex}_{grid,t}\leq 0\\ [V^{ex}_{gas,t}]\cdot c^{buy}_{gas,t}+[P^{ex}_{grid,t}]\cdot c^{buy}_{grid,t}&V^{ ex}_{gas,t}>0,P^{ex}_{grid,t}>0\end{cases} \tag{25}\]  
\[\left[C^{DGS}_{op}\right]=\sum_{t=1}^{24}\left([P_{WPP,t}]\cdot c_{WPP}+[P_{PV,t }]\cdot c_{PV}+P_{PMT,t}\cdot c_{PMT}+P_{vs,t}\cdot c_{es}+P_{P2G,t}\cdot c_{ r2G}+V_{gs,t}\cdot c_{gs}\right) \tag{26}\]  
where \(P_{ev,t}\) denotes the electric vehicle charging power at time \(t\), \(P_{load,t}\) denotes the residential load power at time \(t\), kW, \([P^{ex}_{grid,t}]\) and \([V^{ex}_{gas,t}]\) denote the energy interaction power between the virtual power plant and the grid and the natural gas network at time \(t\), respectively, and \(c^{buy}_{gas,t},\,^{selt}_{gas,t},\,^{buy}_{grid,t}\) and \(c^{selt}_{grid,t}\) denote the purchase/sale price of gas from the virtual power plant to the natural gas network and the grid at time \(t\), respectively.  
Environmental objectives  
Assuming that the virtual power plant is equivalent to coal-fired power generation in its interaction with the grid, the calculation of CO\({}_{2}\) emissions mainly considers the emissions corresponding to the power interaction with the grid (39), the emissions from gas turbine power generation, and the absorption of power-to-gas (40). The environmental objective function is then:  
\[\min\left[f_{2}\right]=\sum_{t=1}^{24}\left([P^{ex}_{grid,t}]\cdot\lambda_{grid }+[V^{ex}_{gas,t}]\cdot L_{gas}\cdot\lambda_{gas}-Q_{pa,t}\cdot\lambda_{P2G}\right) \tag{27}\]  
Grid load smoothing target  
By reducing the fluctuation of the load curve of the virtual power plant interacting with the grid power (41), the objective function of grid load smoothing becomes:  
\[\min[f_{3}]=\sum_{t=1}^{24}\left|P_{P2G,t}+P_{es,t}+P_{ev,t}+[P_{load,t}]-[P_{ WPP,t}]-[P_{PV,t}]-P_{PMT,t}\right| \tag{28}\]