## 3 Proposed Community Solar Operation Strategy for SEC  
As discussed above, the proposed community solar operation strategy is implemented by each member to minimize their electricity bill in a distributed manner. For this purpose, the problem in P0 should be decomposed into a problem for individual members. where \(\
u_{m}\geq 0\) and \(\eta_{m,t}\geq 0\) are the Lagrangian multipliers. The applied Karush-Kuhn-Tucker (KKT) conditions [29] are:  
\[ \begin{array}{l}\
u_{m}\bigg{(}\sum\limits_{t\in\mathcal{T}}e_{m,t}-\gamma_{m }\sum\limits_{t\in\mathcal{T}}g_{t}\bigg{)}=0,\\ \eta_{m,t}(e_{m,t}-\lambda_{m,t})=0,||\forall|t\in\mathcal{T},\\ \frac{\partial\mathcal{L}(e_{m,t}\omega_{m,t}m_{m,t})}{\partial e_{m,t}}= \frac{\partial\mathcal{B}_{m}(e_{m,t})}{\partial e_{m,t}}+\
u_{m}+\eta_{m,t}= 0\end{array}  \tag{13}\]  
The KKT conditions are the conditions that must be satisfied for the solution of a relaxed problem to be an optimal value. In Equation (13), the first and second equations are the complementary slackness conditions, and the third equation expresses the stationarity. In the last condition in Equation (13), the Lagrangian multiplier \(\eta_{m,t}\) acts as a slack variable. Thus, the last condition is expressed as  
\[\
u_{m}\geq-\frac{\partial B_{m}(e_{m,t})}{\partial e_{m,t}}, \tag{14}\]  
and the solution \(e_{m,t}\) is directly calculated as  
\[e_{m,t}= \begin{cases}0,&B_{m}(\cdot)\leq\
u_{m},\\ d_{m,t}-\frac{1}{p_{t}+p_{t}}\
u_{m},&B_{m}(\cdot)\geq\
u_{m}|\text{and Peak time},\\ d_{m,t}-\frac{1}{p_{t}}\
u_{m},&B_{m}(\cdot)\geq\
u_{m}|\text{and Off }-\text{peak time}.\end{cases}  \tag{15}\]  
By substituting Equation (15) into the first condition in Equation (13), \(e_{m,t}\) and \(\
u_{m}\) can be obtained. Moreover, using the second equation, \(e_{m,t}^{*}\) is solved as follows:  
\[e_{m,t}^{*}=\min(e_{m,t},\lambda_{m,t}). \tag{16}\]  
This solution represents one application of the water-filling algorithm [31]. In general, water-filling algorithms increase the objective by adding resources to elements below the water level. However, this study reduces the objective by adding resources to elements above the water level, as shown in Equation (15). This can be expressed as shown in Figure 4 using the negative function. In the figure, the Lagrangian multiplier \(\
u_{m}\) decides water level. The region is flooded to the level which uses a total quantity of water equal to \(\gamma_{m}\sum\limits_{t\in\mathcal{T}}g_{t}\). The height of the water (shown shaded) above each decision time presents the electricity bill saving by the allocation \(e_{m,t}^{*}\) as \(B_{m}(\cdot)-B_{m}\big{(}e_{m,t}^{*}\big{)}\). Considering the basement that is the original electricity bill \(B_{m}(\cdot)\) and the total quantity of water that is the community solar generation, the water level is decided. Based on the water level, the solution is calculated, which is the community solar generation allocated to the member at each decision time. Figure 4: Illustration of the solution as a water-filling algorithm. Using the solution in Equation (16), each member can then decide how to minimize their own electricity bill in a distributed manner; however, future time information is still required. Information on the future time can be obtained based on the demand [32] and solar generation forecasting [33]. However, the forecasted value has an uncertainty, which represents the error between the actual and forecasted values. Therefore, Equation (13) should be updated at each decision time \(t\) as follows:  
\[ \begin{array}{l}\
u_{m}\left(\sum\limits_{t\in\mathcal{T}^{+}}e_{m,t}-\gamma _{m}\sum\limits_{t\in\mathcal{T}^{+}}\hat{g}_{t}+R_{m,t}\right)=0,\\ \eta_{m,t}(e_{m,t}-\lambda_{m,t})=0,||\psi|t\in\mathcal{T},\\ \frac{\partial\mathcal{L}(e_{m,t},\gamma_{m,t},\eta_{m,t})}{\partial e_{m,t}}= \frac{\partial B_{m}^{\pm}(e_{m,t})}{\partial e_{m,t}}+\
u_{m}+\eta_{m,t}=0\end{array}