## 3 Problem Formulation
### Proposed Problem Formulation  
Robust optimization is an optimization framework in which a certain measure of robustness is sought against uncertainty, represented as variability in the variables [29; 30; 31; 32]. However, robust optimization has a significant drawback in that its solutions are highly conservative. This is because it seeks the solution that is feasible whatever the uncertainty is, or equivalently, the solution that is feasible in the worst-case scenario. On the other hand, two-stage stochastic programming is well known as a typical stochastic programming and treats the impact of uncertainty as an additional cost [33; 34; 35]. In two-stage stochastic programming, the minimum cost that is additionally necessary to deal with the constraint violations, which is the recourse function, is calculated and then aggregated into the original objective function in the first stage. Afterward, the additional cost is minimized in the second stage. Here we regard the PV outputs as the stochastic variables and assume their likely ranges as  
\[\widetilde{PV_{j}}\in\left[0,\xi PV_{j}^{\max}\right],\,j\in S, \tag{10}\]  
where \(\widetilde{PV_{j}}\) is the stochastic PV output and an element of the vector of \(\widetilde{PV}\) and \(\xi\) is the confidence coefficient (\(\in[0,1]\)). In theory, \(\widetilde{PV_{j}}\) can take any positive value up to \(PV_{j}^{\max}\). We, however, assume that the distribution network operators have confidence that the value is included in the smaller range of (10). Variation sets of the section loads are represented as  
\[U_{j}=\left\{I_{j}\middle|I_{j}^{\text{low}}\leq I_{j}\leq I_{j}^{\text{high}} \right\},\,j\in S, \tag{11}\]  
\[\left\{ \begin{array}{c}I_{j}^{\text{low}}=I_{j}^{\prime}-\xi PV_{j}^{\max} \\ I_{j}^{\text{high}}=I_{j}^{\prime}+\xi PV_{j}^{\max}\end{array} \right.. \tag{12}\]  
The constraint violations caused by the uncertainty are, when they occur, eliminated by additional switch operations, which leads to extra power outage that is the cost or recourse function we have to pay to deal with uncertainty. Now, the objective function is reformulated as  
\[f(x)=f_{0}(x)+E_{\widetilde{PV}}\Big{[}Q\Big{(}x,\widetilde{PV}\Big{)}\Big{]}, \tag{13}\]  
\[f_{0}(x)=\sum_{j\in S^{\prime}(x)}\left(\max_{\widetilde{PV_{j}}\in[0,\xi PV_{ j}^{\max}]}\middle|I_{j}^{\dagger}\right|\Bigg{)}+\alpha\sum_{h\in W}\lvert x_{h}-x_{h }^{*}\rvert, \tag{14}\]  
\[E_{\widetilde{PV}}\Big{[}Q\Big{(}x,\widetilde{PV}\Big{)}\Big{]}=\int_{\xi PV_ {k}^{\max}}^{PV_{k}^{\max}}\left\{\sum_{k\in S^{\prime}(x,PV_{k}^{\dagger})} \Big{(}\Big{|}I_{k}^{\dagger}+PV_{k}^{\dagger}\Big{|}\cdot c_{k}\Big{(}PV_{k} ^{\dagger}\Big{)}\Big{)}\right\}\!\mathrm{d}PV_{k}^{\dagger}, \tag{15}\]where \(I_{j}^{\dagger}\) is the realization value of the section load of section \(j\), \(PV_{j}^{\dagger}\) is the realization value of the PV output of section \(j\), \(c_{j}\Big{(}PV_{j}^{\dagger}\Big{)}\) is the probability distribution of the PV output of section \(j\) and \(S\Big{(}x,PV_{j}^{\dagger}\Big{)}\) is the set of extra unrestored loads of section \(j\). Robust optimization deals with the uncertain PV output \(PV_{j}^{\prime*}\) in the range of \(\Big{[}0,\xi PV_{j}^{\max}\Big{]}\), and the first term of (14) expresses its worst-case load outage. Beyond this range of uncertain PV output, i.e., in the range of \(\Big{(}\xi PV_{j}^{\max},\ PV_{j}^{\max}\Big{]}\), the two-stage stochastic programming takes over as shown in (15). The operational constraints (5) and (6) are redefined as  
\[\left\{ \begin{array}{ll}\sum_{j^{\prime}\in L_{ij}}\Big{(}-I_{j^{\prime}}^{ \text{low}}y_{i,j^{\prime}}(x)\Big{)}&\leq C_{j}\\ \sum_{j^{\prime}\in L_{ij}}\Big{(}I_{j^{\prime}}^{\text{high}}y_{i,j^{\prime} }(x)\Big{)}&\leq C_{j}\end{array} \right.,\text{ for}\;i\in F,\;j\in S\backslash S ^{\prime}(x), \tag{16}\]  
\[\left\{