## Appendix A Modeling Expectation of Load Shedding  
Load shedding occurs when the power transmitted to the load is less than the amount of the load. The forecasted amount of load shedding is the difference between the load amount and the power transmitted to the load, and it can be defined as in Equation (A1).  
\[LS_{t}^{FC}=max(Load_{t}-P_{t}^{LL},0)\] (A1)  
Since the forecast values for load and renewable energy are used at the operational planning stage, the actual load shedding value considering the forecast error is as Equation (A2). \(\epsilon_{t}\) represents the sum of the load forecast error and renewable energy forecast error.  
\[LS_{t}^{actual}=max((Load_{t}-P_{t}^{LL})-\epsilon_{t},0)=max(LS_{t}^{FC}- \epsilon_{t},0)\] (A2)  
\(P_{t}^{ELS}\) represents the conditional expected load shedding that exists when \(LS_{t}^{FC}\) is greater than \(\epsilon_{t}\), and \(P_{t}^{ELS}\) is calculated as in Equation (A3). We assumed that \(\epsilon_{t}\) follows a Gaussian distribution. In Equation (A3), \(\phi_{\epsilon_{t}}\) represents the PDF of \(\epsilon_{t}\) and \(\Phi_{\epsilon_{t}}\) represents the CDF of \(\epsilon_{t}\).  
\[\begin{split} P_{t}^{ELS}(LS_{t}^{FC})&=\frac{1}{ \mathbb{P}(LS_{t}^{FC}-\epsilon_{t}>=0)}\int_{-\inf}^{LS_{t}^{FC}}(LS_{t}^{FC} -\epsilon_{t})\cdot\phi_{\epsilon_{t}}d_{\epsilon_{t}}\\ &=\frac{1}{\Phi_{\epsilon_{t}}(LS_{t}^{FC})}\Big{\{}\int_{-\inf }^{LS_{t}^{FC}}LS_{t}^{FC}\cdot\phi_{\epsilon_{t}}d_{\epsilon_{t}}-\int_{-\inf }^{LS_{t}^{FC}}\epsilon_{t}\cdot\phi_{\epsilon_{t}}d_{\epsilon_{t}}\Big{\}}\\ &=LS_{t}^{FC}+\frac{\sigma^{2}\cdot\phi_{\epsilon_{t}}(LS_{t}^{ FC})}{\Phi_{\epsilon_{t}}(LS_{t}^{FC})}\end{split}\] (A3)  
\[\left\{\begin{array}{l}\text{where }\int_{-\inf}^{LS_{t}^{FC}}LS_{t}^{FC} \cdot\phi_{\epsilon_{t}}d_{\epsilon_{t}}=LS_{t}^{FC}\cdot\Phi_{\epsilon_{t}}( LS_{t}^{FC})-LS_{t}^{FC}\cdot\Phi_{\epsilon_{t}}(-\inf)\\ \text{where }\int_{-\inf}^{LS_{t}^{FC}}\epsilon_{t}\cdot\phi_{ \epsilon_{t}}d_{\epsilon_{t}}=-\sigma^{2}\cdot(\phi_{\epsilon_{t}}(LS_{t}^{ FC})-\phi_{\epsilon_{t}}(-\inf))\end{array}\right.\] (A4)  
Here, \(P^{ELS}\) represents the nonlinear nonconvex function. However, the ECLS included in the objective function of optimization problem is a nonlinear and convex function. The ECLS is defined as multiplying \(P^{ELS}\) by \(C^{LS}\) and \(\Phi_{\epsilon_{t}}\), which means the probability that \(LS_{t}^{FC}\) is greater than \(\epsilon_{t}\), \(P_{t}^{ELS}\). The ECLS is expressed as Equation (A4).  
\[\begin{split} C_{t}^{ELS}&=C^{LS}\cdot\Phi_{\epsilon_{t} }\cdot P^{ELS}(LS_{t}^{FC})\\ &=C^{LS}\Big{\{}LS_{t}^{FC}\cdot\Phi_{\epsilon_{t}}(LS_{t}^{FC})+ \sigma^{2}\cdot\phi_{\epsilon_{t}}(LS_{t}^{FC})\Big{\}}\end{split}\] (A5)  
Equation (A5) represents first derivative for Equation (A4). The second derivative for Equation (A4) is expressed as Equation (A6). The \(C^{LS}\) is a positive parameter and \(\phi_{\epsilon_{t}}(LS_{t}^{FC})\) is always positive. Therefore, the modeled \(C_{t}^{ELS}\) is convex for all \(LS_{t}^{FC}\).  
\[\left\{\begin{array}{l}\frac{dC_{t}^{ELS}}{dLS_{t}^{FC}}=C^{LS}\Big{\{} \Phi_{\epsilon_{t}}(LS_{t}^{FC})+LS_{t}^{FC}\cdot\phi_{\epsilon_{t}}(LS_{t}^{ FC})-LS_{t}^{FC}\cdot\phi_{\epsilon_{t}}(LS_{t}^{FC})\Big{\}}\\ \text{where }\frac{d\phi_{\epsilon_{t}}(LS_{t}^{FC})}{dLS_{t}^{FC}}=\frac{ -LS_{t}^{FC}\cdot\phi_{\epsilon_{t}}(LS_{t}^{FC})}{\sigma^{2}}\end{array}\right.\] (A6)\[\frac{dC_{t}^{ELS}}{d(LS_{t}^{FC})^{2}}=C^{LS}\cdot\phi_{\epsilon_{t}}(LS_{t}^{FC}) \tag{10}\]