## 4 Robust Programming Model for DG Planning
### The Objective Function  
The objective function of this paper is to minimize the annual comprehensive cost. It includes the fixed annual investment cost of \(DG\)\(C_{I}\), the annual operation and maintenance cost of \(DG\)\(C_{OM}\), the network loss cost \(C_{loss}\), and the power purchase cost from the superior grid \(C_{up}\). In particular, the restructuring costs \(C_{C}\) is considered. At the same time, according to the characteristics of ADN, the operation process of ADN is considered in the planning stage, so the active management cost of \(DG\)\(C_{AM}\) should also be included in the objective function. The objective function expression is  
\[\text{min}F=C_{I}+C_{OM}+C_{loss}+C_{up}+C_{AM} \tag{3}\]  
The specific expression of each part of the cost in Equation (3) is as follows. (1) \(DG\) investment cost \(C_{I}\):  
\[C_{I}=R_{WTG}\sum_{i=1}^{N_{bias}}\left(c_{WTG,i}^{l}\times p_{WTG,i}^{r})+R_ {PVG}\sum_{i=1}^{N_{bias}}\left(c_{PVG,i}^{l}\times p_{PVG,i}^{r}\right) \tag{4}\]  
\[R_{WTG}=\frac{d(1+d)^{y_{WTG}}}{(1+d)^{y_{WTG}}-1} \tag{5}\]  
\[R_{PVG}=\frac{d(1+d)^{y_{PVG}}}{(1+d)^{y_{PVG}}-1} \tag{6}\]  
where \(R_{WTG}\) and \(R_{PVG}\) are denoted as the present value to equivalent annual coefficients of \(WTG\) and \(PVG\), respectively. \(d\) is the discount rate. \(y_{WTG}\) and \(y_{PVG}\) are denoted as the economic service life of \(WTG\) and \(PVG\), respectively. \(N_{bus}\) is denoted as the number of distribution network nodes. \(c_{INTG,i}^{l}\) and \(c_{PVG,i}^{l}\) are denoted as the fixed investment costs per unit capacity of \(WTG\) and \(PVG\) installed at node \(i\), respectively. \(p_{WTG,i}^{r}\) and \(p_{PVG,i}^{r}\) are denoted as the rated capacities of the \(WTG\)s and \(PVG\)s installed at node \(i\), respectively. (2) Operation and maintenance cost of \(DG\)\(C_{OM}\):  
\[C_{OM}=\sum_{s=1}^{N_{\rm s}}\sum_{i=1}^{N_{\rm hs}}\left(c_{WTG,i}^{OM}p_{WTG,i,s}+c_{PVG,i}^{OM}p_{PVG,i,s}\right)t_{s} \tag{7}\]  
where \(N_{\rm S}\) is denoted as the number of ADN operation scenarios. \(t_{s}=8760\times p_{s}\) is denoted as the duration of scenario \(s\) in a year, and 8760 is the number of hours in a year; \(p_{s}\) is the probability of occurrence of scenario \(s\); \(c_{WTG,i}^{OM}\) and \(c_{PVG,i}^{OM}\) are denoted as the operation and maintenance costs of the unit electricity generated by the \(WTG\) and \(PVG\) installed at node \(i\), respectively. \(p_{WTG,i,s}\) and \(p_{PVG,i,s}\) are denoted as the active outputs of \(WTG\)s and \(PVG\)s installed at node \(i\) under scenario \(s\), respectively; \(N_{s}\) is denoted as the number of scenarios. (3) Network loss cost \(C_{loss}\):  
\[C_{loss}=\sum_{s=1}^{N_{\rm s}}\sum_{h=1}^{H}c_{t,h}^{loss}p_{t,h,s}^{loss} \cdot t_{s} \tag{8}\]  
where \(h\) is denoted as the number of daily operation period; \(H\) is denoted as the total number of daily periods; \(c_{t,h}^{loss}\) is denoted as the network loss price in \(t\) period; \(p_{t,h,s}^{loss}\) is denoted as the network loss in period \(t\). (4) Power purchase cost from the power system \(C_{up}\):  
\[C_{up}=\sum_{s=1}^{N_{\rm s}}\rho_{s}p_{subs}t_{s} \tag{9}\]  
where \(p_{sub,s}\) is denoted as the active power of purchasing electricity from the superior power grid in scenario \(s\). \(\rho_{s}\) is denoted as power purchase cost for ADN to purchase power from the power grid in scenario \(s\). (5) Reconstruction cost \(C_{\rm C}\):  
\[C_{\rm C}=\sum_{j=1}^{N_{\rm s}}\sum_{i=1}^{N}E_{B_{j}}\big{|}s_{j,i}-s_{j,i-1} \big{|} \tag{10}\]  
where \(E_{B_{j}}\) refers to the cost caused by changing the primary opening and closing state of the switch numbered \(j\); \(S_{j,i}\) refers to the on-off state of the switch numbered \(j\) in the \(i\) th period. \(S_{j,i}=1\) indicates that the switch is closed, and \(S_{j,i}=0\) indicates that the switch is open.