## 3 Multi-Parametric Formulations
### Distributionally Robust mp-MILP and Reformulation  
If we only consider the empirical distribution, \(\mathcal{P}^{0}\), the expected renewable energy curtailment, i.e., \(\sum\limits_{s=1}^{S}\rho^{0}\left(\hat{c}^{T}\hat{x}_{s}\right)\), is minimized. As mentioned above, the distributionally robust formulation minimizes the expectation under the worst distribution in the ambiguity set \(\mathcal{M}_{I}^{\varepsilon}(\mathcal{P}^{0})\), giving rise to the following DR-mp-MILP:  
\[ \begin{array}{l}v^{R}(\theta)=\max\limits_{\rho}&\min\limits_{\hat{x}_{s}, \hat{y}_{s},\hat{\gamma}_{s}=1}^{S}\rho_{s}\left(\hat{c}^{T}\hat{x}\right)\\ \text{s.t.}&\hat{A}\hat{x}_{s}+\hat{B}\hat{y}_{s}\leq\hat{b}+\hat{C}w_{s}+\hat {F}\theta,s=1,\cdots S\\ &Q\rho\leq U:\ \lambda\\ &\theta\in\Theta\end{array}  \tag{17}\]  
where the vectors with subscript \(s\) correspond to the representative scenario \(w_{s}\); \(\lambda\) is the vector of dual variables. An important feature in (17) is that, given a \(\theta\), the feasible set of outer maximization problem is disjoint from that of inner minimization problem; thus, dualizing the outer problem results in the single-level parametric problem shown below:  
\[ \begin{array}{l}v^{R}(\theta)=\min\limits_{\lambda,\hat{x}_{s},\hat{y}_{s}, \hat{\gamma}_{s}=1}^{T}U\\ \text{s.t.}\hat{A}\hat{x}_{s}+\hat{B}\hat{y}_{s}\leq\hat{b}+\hat{C}w_{s}+\hat {F}\theta,s=1,\cdots S\\ &Q^{T}\lambda=\left[\hat{c}^{T}\hat{x}_{1},\cdots,\hat{c}^{T}\hat{x}_{S} \right],\lambda\geq 0\\ &\theta\in\Theta\end{array}  \tag{18}\]  
Therefore, to calculate \(v^{R}(\theta)\) entails solving the mp-MILP(18). For brevity, we aggregate all the coefficients, variables, and constraints in (18), to rewrite it as the canonical mp-MILP:  
\[\begin{array}{l}v^{R}(\theta)=\!\!\!\!\!\!\!  
Then, we discuss the properties of \(v^{R}(\theta)\). Binary vector \(y\) denotes the charging and discharging pattern of ESUs, and has a finite number of candidates, written as \(\mathbb{Y}=\{y_{i}\}_{i=1}^{l}\). Given any \(y_{i}\in\mathbb{Y}\), mp-MILP (19) degenerates into an mp-LP as  
\[v_{i}^{R}(\theta)= \underset{x,y}{\min}c^{T}x\] \[\text{s.t.}Ax\leq(b-By_{i})+F\theta:\gamma \tag{20}\] \[\theta\in\Theta\]  
where \(v_{i}^{R}(\theta)\) is the corresponding optimal value function; \(\gamma\) is the vector of dual variables. As a result, \(v^{R}(\theta)\) in mp-MILP (19) can be calculated by comparing \(v_{i}^{R}(\theta),\forall y_{i}\in\mathbb{Y}\) in a pointwise minimum way. Given a \(\theta\), consider the dual problem of the mp-LP (20):  
\[v_{i}^{R}(\theta)=\underset{\gamma\in\Gamma}{\max}\gamma^{T}(b_{i}+F\theta) \theta\in\Theta \tag{21}\]  
where \(\Gamma=\big{\{}\gamma|A^{T}\gamma=c,\gamma\leq 0\big{\}}\), and \(b_{i}=b-By_{i}\). The duality theory of linear programming indicates that the primal problem (20) and the dual problem (21) share the same optimal value \(v_{i}^{R}(\theta)\) for any \(\theta\); as \(\theta\) varies in \(\Theta\), the optimal solution can always be found at one of the extreme points of \(\Gamma\). Thus, \(v_{i}^{R}(\theta)\) can be constructed as follows:  
\[v_{i}^{R}(\theta)=\max\Big{\{}\gamma_{i}^{T}(b_{i}+F\theta)\Big{\}},\forall \gamma_{i}\in\text{vert}(\Gamma),\theta\in\Theta \tag{22}\]  
Therefore, \(v_{i}^{R}(\theta)\) is piecewise linear and convex because the point-wise maximum operation preserves convexity; each piece corresponds to a subset of \(\Theta\), called critical region (CR). Further, \(v^{R}(\theta)\) is also piecewise linear but generally not convex. In Reference [32], a decomposed algorithm is proposed to solve mp-MILP (19), which takes the following four steps:  
* Step 1: Solve MILP by solving MILP (19) by regarding \(\theta\) as decision variables; the optimal solution is \(y_{1}\). * Step 2: With the fixed \(y_{1}\), solve mp-LP (20) by POP(parametric optimization) toolbox [33]. The results include the piecewise linear \(v_{1}^{R}(\theta)\), in detail, the slopes, intercepts, and the corresponding \(M_{1}\) CRs.