# Research on Interval Optimal Scheduling Strategy of Virtual Power Plants with Electric Vehicles
## 2 Structure and Principle of Virtual Power Plant with Electric Vehicles
### Principle of a Virtual Power Plant with Electric Vehicles  
(1) Wind power generation model  
Wind power has the characteristics of being random and uncontrollable. Wind speed and wind power have the following relationship [34]:  
\[P_{WPP}=\left\{ \begin{array}{cc}0&0\leq v<v_{in},v>v_{out}\\ \frac{v^{3}-v_{in}^{3}}{v_{rated}^{3}-v_{in}^{3}}P_{rated}&v_{in}\leq v\leq v_{ rated}\\ P_{rated}&v_{rated}\leq v\leq v_{out}\\ \end{array} \right. \tag{1}\]  
where \(P_{WPP}\) is the wind power output, kW, \(P_{rated}\) is the rated wind power, kW/h, and \(v_{int},v_{out}\), and \(v_{rated}\) are the cut-in wind speed, cut-out wind speed, and rated wind speed, m/s, respectively. (2) Photovoltaic power generation model  
The output power model of photovoltaic power generation can be expressed as:  
\[P_{PV}=P_{STC}\frac{G_{INC}}{G_{STC}}(1+k(T_{e}-T_{r})) \tag{2}\]  
where \(G_{INC}\) and \(P_{PV}\) are the actual radiation intensity and actual output power, respectively, \(k\) is the power temperature coefficient, \(T_{r}\) is the standard test condition temperature, and \(T_{e}\) is the actual temperature. (3) Gas turbine model  
The relationship between the electrical power output of a gas turbine, its thermal power, and the natural gas consumption is:  
\[P_{MT}=\frac{H_{MT}\eta_{MT}}{(1-\eta_{MT})\eta_{MT}^{re}K} \tag{3}\]  
\[Q_{MT}=\frac{H_{MT}}{(1-\eta_{MT})\eta_{MT}^{re}K\cdot L_{gas}} \tag{4}\]  
The CO\({}_{2}\) emission volume versus the power generated by the gas turbine is:  
\[Q_{pai}=\frac{c_{co_{2}}\cdot P_{MT}}{\rho_{co_{2}}} \tag{5}\]  
where \(P_{MT}\) and \(H_{MT}\) are the electric and thermal power of the gas turbine, respectively, \(\eta_{MT}\) is the gas turbine conversion efficiency, \(\eta_{MT}^{re}\) is the waste heat recovery efficiency, \(K\) is the heat production coefficient, \(L_{gas}\) is the low heating value of natural gas, \(Q_{MT}\) is the natural gas consumption, \(Q_{pai}\) is the total amount of CO\({}_{2}\) emissions, \(c_{co_{2}}\) is the CO\({}_{2}\) emission factor, and \(\rho_{co_{2}}\) is the density of the CO\({}_{2}\). (4) Electric vehicle charging model  
Assuming that the daily distance driven by electric vehicles satisfies a log-normal distribution, the probability density function is:  
\[f_{L}(d)=\frac{1}{d\sigma_{L}\sqrt{2\pi}}\exp[\frac{-(\ln d-\mu_{L})^{2}}{2{ \sigma_{L}}^{2}}],d>0 \tag{6}\]  
where \(d\) is the daily distance traveled, and \(\mu_{L}\) and \(\sigma_{L}\) are the expected value and variance of the distance traveled, respectively. Suppose the maximum driving distance of the electric vehicle under full charge is \(d_{R}\), then the relationship between the initial SOCst and the daily driving distance of the electric vehicle on the grid is:  
\[\text{SOC}_{\text{st}}=1-\frac{d}{d_{\text{R}}},0<d\leq d_{\text{R}} \tag{7}\]  
Assuming that the EV on-grid time satisfies a normal distribution, the probability density function is:  
\[f_{\text{s}}(t)=\left\{ \begin{array}{l}\frac{1}{\sigma_{s}\sqrt{2\pi}}\exp[ \frac{-(t-\mu_{\text{s}})^{2}}{2\sigma_{s}^{2}}],\mu_{s}-12<t\leq 24\\ \frac{1}{\sigma_{s}\sqrt{2\pi}}\exp[\frac{-(t+24-\mu_{\text{s}})^{2}}{2\sigma _{s}^{2}}],0<t\leq(\mu_{s}-12)\end{array} \right. \tag{8}\]  
where: \(t\) is the time to connect to the grid, and \(\mu_{s}\) and \(\sigma_{s}\) are the expected value, and variance, of \(t\), respectively. The electric vehicle off-grid time is related to the inbound time and the parking duration [35], and, assuming that the parking duration approximately obeys the lognormal distribution, the probability density function is:  
\[f_{\text{p}}(t_{\text{p}})=\frac{1}{t_{\text{p}}\sigma_{\text{p}}\sqrt{2\pi}} \exp[\frac{-(\ln t_{\text{p}}-\mu_{\text{p}})^{2}}{2\sigma_{\text{p}}^{2}}] \tag{9}\]  
where \(t_{\text{p}}\) is the length of the stopping time, \(\mu_{\text{p}}\) and \(\sigma_{\text{p}}\) are the expected value, and variance, of \(t\), respectively, and the electric vehicle off-grid time is: \(t_{\text{q}}=t+t_{\text{p}}\).