## 4 Robust Programming Model for DG Planning
### Constraints  
The constraints in this paper are system power flow constraints, node voltage constraints, branch power constraints, \(DG\) operation constraints, switching times constraints, and network topology constraints. In particular, according to [27], one of the characteristics of ADN different from traditional distribution networks is that it has a certain proportion of distributed controllable resources. Therefore, in the planning stage of ADN, the proportion of \(DG\) input should be guaranteed to meet the needs of ADN, so as to ensure more stable operation of the power grid. Therefore, \(DG\) permeability constraints should be considered. The specific formulas for each constraint are shown as following. 1. Power flow constraint: A linearized AC power flow model proposed is utilized here. The linearization method has been proved applicable in [28]. The linearized AC power flow model can be expressed as \[k_{ij\_1}=\frac{x_{ij}}{r_{ij}^{2}+x_{ij}^{2}},\ k_{ij\_2}=\frac{r_{ij}}{r_{ij}^{ 2}+x_{ij}^{2}}\] (12) \[P_{ij}^{s,t}=k_{ij\_1}\cdot\left(\delta_{i}^{s,t}-\delta_{j}^{s,t}\right)+k_{ ij\_2}\cdot\left(V_{i}^{s,t}-V_{j}^{s,t}\right)\] (13) \[Q_{ij}^{s,t}=-k_{ij\_2}\cdot\left(\delta_{i}^{s,t}-\delta_{j}^{s,t}\right)+ k_{ij\_1}\cdot\left(V_{i}^{s,t}-V_{j}^{s,t}\right)\] (14) \[P_{i}^{s,t}=\sum\limits_{i=1,i,j\
eq j}^{M}k_{ij\_1}\cdot\left(\delta_{i}^{ s,t}-\delta_{j}^{s,t}\right)+k_{ij\_2}\cdot\left(V_{i}^{s,t}-V_{j}^{s,t}\right)\] (15) \[Q_{i}^{s,t}=\sum\limits_{i=1,i\
eq j}^{M}-k_{ij\_2}\cdot\left(\delta_{i}^{ s,t}-\delta_{j}^{s,t}\right)+k_{ij\_1}\cdot\left(V_{i}^{s,t}-V_{j}^{s,t}\right)\] (16) \[\left|P_{ij}^{s,t}\right|\leq P_{ij}^{\max,t}\] (17) \[\left|Q_{ij}^{s,t}\right|\leq Q_{ij}^{\max,t}\] (18)  
where \(x_{ij}\) and \(r_{ij}\) are denoted as reactance and resistance of the branch _ij_. \(\delta_{i}^{s,t}\) and \(V_{i}^{s,t}\) are voltage angle and voltage magnitudes at bus m. \(P_{i}^{\max,t}\) and \(Q_{i}^{\max,t}\) are denoted as maximum active power and maximum reactive power between node \(i\) and \(j\). (2) _WTG_ operation constraints:  
\[\left\{ \begin{array}{l}(1-W_{\text{WTG},i}^{t})P_{\text{WTG},i,s}^{\max}\leq P _{\text{WTG},i,s}^{t}\leq P_{\text{WTG},i,s}^{\max}\\ 0\leq W_{\text{WTG},i}^{t}\leq W_{\text{WTC},i}^{\max}\end{array} \right. \tag{19}\]  
where \(W_{\text{WTG},i}^{t}\) is the removal ratio of the active output of the _WTG_ installed at node \(i\). (3) _PVG_ operation constraints:  
\[\left\{ \begin{array}{l}(1-W_{\text{PVG},i}^{t})P_{\text{PVG},i,s}^{\max}\leq P _{\text{PVG},i,s}^{t}\leq P_{\text{PVG},i,s}^{\max}\\ 0\leq W_{\text{PVG},i}^{t}\leq W_{\text{PVG},i}^{\max}\end{array} \right. \tag{20}\]  
where \(W_{\text{PVG},i}^{t}\) is denoted as the removal ratio of the active output of the _PVG_ installed at node \(i\). (4) Power flow balance on each node:  
\[P_{\text{WTG},i,s}^{t}+P_{\text{PVG},i,s}^{t}+P_{i}^{s,t}-L_{i,load}^{t}=0 \tag{21}\]  
where \(L_{i,load}^{t}\) is denoted as power load at node \(i\) and time \(t\). (5) Switching times constraint:  
\[\left\{ \begin{array}{l}\sum\limits_{n=1}^{N}\left|\omega_{n+1,j}-\omega_{n,l }\right|\leq T_{s}\\ \sum\limits_{l=1}^{n_{\text{P}}}\sum\limits_{n=1}^{N}\left|\omega_{n+1,l}- \omega_{n,l}\right|\leq T_{t}\end{array} \right. \tag{22}\]  
where \(T_{s}\) is denoted as the upper limit value of switching action times of the first branch; \(T_{t}\) is the upper limit value of all switch operation times in the whole period of time. (6) \(DG\) permeability constraint:  
\(DG\) permeability limit at the planning stage can enable \(DG\) to be absorbed locally. Therefore, the penetration rate of clean energy increases and network loss reduces.