## 5 Results
### Results of Deterministic mp-Milp  
Given different scenarios of wind power, (10) turns into a canonical mp-MILP like (19), and it can be solved by the algorithm in Section 4. Taking Scenario 3 as an example, the optimal value function is drawn in Figure 4.  
Figure 3: Ten representative scenarios of wind power.  
Figure 2: Load curve.  
It can be seen that, without an ESU, the total renewable energy curtailment under Scenario 3 reaches 45.5 MWh. By employing an ESU whose power/energy capacity is higher than 22.75 MW/81.83 MWh, respectively, the curtailment reduces to zero.  
The parameter set is divided into three critical regions, named CR1, CR2, and CR3. The polyhedral representations of them are as follows:  
\[CR1 =\left\{\begin{array}{c|c}\theta=[P^{s},E^{s}]^{T}&\left|\begin{array}{ c}0\leq P^{s}\leq 100\\ 0\leq E^{s}\leq 81.83\\ E^{s}\leq 3.60P^{s}\end{array}\right.\end{array}\right\}\] \[CR2 =\left\{\begin{array}{c|c}\theta=[P^{s},E^{s}]^{T}&\left|\begin{array} []{c}0\leq P^{s}\leq 22.75\\ 0\leq E^{s}\leq 150\\ 3.60P^{s}\leq E^{s}\end{array}\right.\end{array}\right\} \tag{25}\] \[CR3 =\left\{\theta=[P^{s},E^{s}]^{T}&\left|\begin{array}{c}22.75\leq P ^{s}\leq 100\\ 81.83\leq E^{s}\leq 150\end{array}\right.\right\}\]  
Then, the corresponding optimal value functions are the following:  
\[v^{R}(\theta)=\left\{\begin{array}{c}45.5-[0,0.56]\theta,\;\theta\in CR1\\ 45.5-[2.0,0]\theta,\;\;\theta\in CR2\\ 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; 0\in CR3\end{array}\right. \tag{26}\]  
Figure 4: Results in Scenario 3 of wind power (**a**) optimal value function and (**b**) critical regions.  
From (26), we see that the sensitivity of renewable energy curtailment to the capacities of ESU is different as \(\theta\) changes. When \(\theta\) falls into CR1, 1 MWh increment of energy capacity will lead to a 0.56 MWh decline of curtailment, while the increase of power capacity makes no sense, implying that the impact of power capacity has saturated; similarly, when \(\theta\) falls into CR2, a 1 MWh increment of power capacity will cause a 2 MWh decline of curtailment, while raising the energy capacity does not take effect. Moreover, when \(\theta\) falls into CR3, the curtailment is zero.  
Furthermore, the boundary between CR1 and CR2, the red line in Figure 4b, gives valuable information. The points on this boundary are efficient in fully using the capacities of ESUs; from any point on this boundary, increasing power or energy capacity alone will not help reduce the curtailment. Specifically, on this boundary, the ratio of energy capacity to power capacity is 3.60; such a ratio is useful in sizing ESUs in practice.  
From an economic perspective, consider the investment cost of ESU as \(\Pi=\kappa^{P}P^{s}+\kappa^{E}E^{s}\), where \(\kappa^{P}/\kappa^{E}\) is the unit cost of power/energy capacity. Regardless of the value of \(\kappa^{P}\) and \(\kappa^{E}\), the isocost curves are a cluster of parallel straight lines, as seen from the dashed lines in Figure 5. Combining with Figure 4a, we can find that, given a budget, the parameter on the red line enjoys the minimum renewable energy curtailment, as compared with other points that cost the same.