## 2 Method
### Problem Formulation  
The main purpose of participating in community solar services is to reduce electricity bills. This study adopts a Time-of-Use (ToU) tariff for billing. Therefore, taking \(d_{m,t}\) as the demand for member \(m\) at a time \(t\), the electricity bill of member \(m\) is measured as:  
\[B_{m}(\cdot) =p_{d}\underset{t\in\mathcal{T}}{\max}d_{m,t}+\sum\limits_{t\in \mathcal{T}}p_{e,t}d_{m,t}, \tag{1}\]  
where \(\mathcal{T}=\{1,\dots,t,\dots,T\}\) is the billing period such as one month, \(p_{d}\) is the demand price for the billing period, and \(p_{e,t}\) is the energy price at time \(t\). This study assumes that all members use the same electricity tariff. The demand price is the price charged for peak demand that occurs during the billing period and is the factor that lowers peak demand. The energy price is a charge imposed on energy usage during the billing period, and it has the effect of demand shifting by setting the price for each time differently. Moreover, when \(e_{m,t}\) is the solar generation allocated to member \(m\) at time \(t\), the electricity bill is modified as  
\[B_{m}(e_{m,t})=p_{d}\underset{t\in\mathcal{T}}{\max}(d_{m,t}-e_{m,t})+\sum \limits_{t\in\mathcal{T}}p_{e,t}(d_{m,t}-e_{m,t}). \tag{2}\]  
Subsequently, the SESP allocates the solar generation to minimize the overall electricity bill. Letting \(\mathcal{M}=\{1,\dots,m,\dots,M\}\) be a community member set, the objective function is expressed as:  
\[\mathcal{O}(e_{m,t})=\sum\limits_{m\in\mathcal{M}}B_{m}(e_{m,t}). \tag{3}\]  
The objective, the electricity bill, is reduced by the allocation of the solar generation. However, the utility of solar generation that is the electricity bill saving varies with time. This is because the energy price at each time is difference. Moreover, if the time at which peak demand occurs is different for each member, the utility of solar generation for each member will also be different. This is because the electricity bill is determined by peak demand as well as energy consumption. Therefore, even if the same amount of solar generation is used, the electricity bill saving of each member depends on when it is allocated. As a result, the community solar operation to minimize the electricity bill presented as the objective function in Equation (3) is the solar generation resource allocation problem that can be formulated as a mixed-integer linear programming problem [27]. The allocation for each member should be less than the solar generation. Therefore, the generation constraint is considered as:  
\[\sum\limits_{m\in\mathcal{M}}e_{m,t}\leq g_{t},||\forall|t\in\mathcal{T}, \tag{4}\]  
Figure 2: Constitution of a community solar system for a smart energy community. where \(g_{t}\) is the solar generation at time \(t\). Moreover, resource fairness was considered in this study. Resource fairness is defined as the resource allocation of a member proportional to some pre-existing claims [28]. The pre-existing claim can be set to various values such as total demand and investment cost. In this study, it is assumed that the resource, i.e., the solar generation, is allocated in proportion to the total demand of each member. Letting \(\gamma_{m}\) be the resource indicator for member \(m\) that is the fraction of total solar generation distributed to each member during the billing period, and \(\sum\limits_{m\in\mathcal{M}}\gamma_{m}=1\), the resource fairness constraint is expressed as:  
\[\sum\limits_{t\in\mathcal{T}}e_{m,t}=\gamma_{m}\sum\limits_{t\in\mathcal{T}}g _{t},||\forall|m\in\mathcal{M}. \tag{5}\]  
Using Equations (3)-(5), the community solar operation problem is:  
\[ \begin{array}{l}\text{P0:}||||||||||||\min\limits_{e_{m,t}}\quad\sum \limits_{m\in\mathcal{M}}B_{m}(e_{m,t})\\ \text{subject to}|\quad\sum\limits_{\begin{subarray}{c}m\in\mathcal{M} \end{subarray}}e_{m,t}\leq g_{t},||\forall|t\in\mathcal{T},\\ \sum\limits_{t\in\mathcal{T}}e_{m,t}=\gamma_{m}\sum\limits_{t\in\mathcal{T}}g _{t},||\forall|m\in\mathcal{M}.\end{array}