## 3 The Construction Model of Virtual Power Plant
### Constraint Condition  
1 Virtual Power Plant Scale Constraint  
The VPP can aggregate the DG, ES and controllable load into a whole, so that it can participate in the operation of the electricity market and the auxiliary service market, and the number of resourcesand the size of the power consumption scale should not be too small for the construction of the VPP, which satisfy:  
\[\left\{\begin{array}{l}\sum\limits_{i=1}^{m}\left(X_{i}^{DG}+X_{i}^{ES}+X_{i}^{ DR}\right)\geq M_{\text{min}}\\ \sum\limits_{i=1}^{m}\sum\limits_{k=1}^{n}\left(P_{ki}^{ES}X_{i}^{ES}+P_{00} \right)\geq Q_{\text{kmin}}\end{array}\right. \tag{18}\]  
where \(M_{\text{min}}\) is the minimum number of resources; and \(Q_{\text{min}}\) is the minimum power consumption in time period \(k\).  
2. Fusion Space Distance Constraint  
There is a fusion space distance between any two areas, and the fusion space distance should be controlled within a certain range, not too large, which satisfies:  
\[\text{max}d(i,j)\leq d_{\text{max}} \tag{19}\]  
where \(d_{\text{max}}\) is the maximum fusion space distance.  
3. Virtual Power Plant Output Constraint  
The VPP output constraint can be written as:  
\[P_{\text{kmin}}\leq\sum\limits_{i=1}^{m}\left(P_{ki}^{DG}X_{i}^{DG}-P_{ki}^{ES} X_{i}^{ES}-P_{ki}^{DR}X_{i}^{DR}\right)\leq P_{\text{kmax}} \tag{20}\]  
where \(P_{k\text{min}}\) and \(P_{k\text{max}}\) are the minimum and maximum values of VPP output at time period \(k\), respectively.  
4. Curve Volatility Constraint  
In order to make the actual output curve closer to the planned output curve and improve the operation stability of the VPP, the volatility of the curve \(L\) should not be too large, which is described as:  
\[\frac{\sigma}{\mu}\leq\theta_{\text{max}} \tag{21}\]  
where \(\theta_{\text{max}}\) is the upper limit of the volatility.  
5. Resource Complementarity Constraint  
In order to minimize the influence of the uncertain factors on the VPP and improve the stability of the system, the complementarity among various resources should be fully considered in the construction of VPP. The measurement of resource complementarity is the measurement of the difference between different load curves in the same time period. The dissimilarity and similarity are corresponding to each other. The higher the similarity between the curves is, the lower the complementarity is, and vice versa [28]. The correlation among all kinds of resources is as follows:  
\[\beta_{xy}=\left[\sum\limits_{k=1}^{n}\frac{\Lambda(k)}{1+\left|\Delta x(k) /(\sigma_{k}\Delta k)-\Delta y(k)/(\sigma_{k}\Delta k)\right|}\right]/\left[ \sum\limits_{k=1}^{n}\Lambda(k)\right] \tag{22}\]  
where \(\Lambda(k)\) is the weighting function of active power at different time periods; \(\sigma_{x}\) is the standard deviation of the resource sequence \(x(k)\); and \(\sigma_{y}\) is the standard deviation of the resource sequence \(y(k)\).  
The resource complementarity coefficient between resources \(x\) and \(y\) is:  
\[\gamma_{xy}=1-\beta_{xy} \tag{23}\]In this paper, the DG, ES and controllable load are involved in the construction of VPP. Therefore, the sum of the complementarity coefficients among different resources is taken as the complementary coefficient of the whole VPP, which is recorded as \(\gamma_{VPP}\).  
In order to guarantee the complementarity among various resources, the condition should be restrained and is shown as following:  
\[\gamma_{VPP}\geq\gamma_{\min} \tag{24}\]  
where \(\gamma_{\min}\) is the lower limit of complementarity.