## 3 Design of MCSS
### Length Determination and Cell Allocation for WPT Slofframe
\begin{array}{c}n_{overcell,j}n_{init\text{PC},i}\\ \sum\limits_{i=1}^{n}(n_{minPC,i}+n_{minDC,i})\end{array} \right)\ (0<i\leq n_{ node,j},\ 0<j\leq M), \tag{7}\]  
\[n_{overDC,i}=round\left( \begin{array}{c}n_{overcell,j}n_{init\text{DC},i}\\ \sum\limits_{i=1}^{n}(n_{minPC,i}+n_{minDC,i})\end{array} \right)\ (0<i\leq n_{ node,j},\ 0<j\leq M), \tag{8}\]  
where \(n_{overcell,j}\) represents the total number of overprovisioned cells in the \(j\)-th HAP cluster. Further, \(n_{overcell,j}\) is determined based on the least common multiple (LCM) of multiple concurrent slotframes of the \(j\)-th HAP cluster (\(l_{LCM,j}\)), calculated as follows:  
\[n_{overcell,j}=\left[ \begin{array}{c}l_{WPT,j}n_{totalcell,j}\\ l_{LCM,j}n_{node,j}\end{array} \right]\ (0<j\leq M), \tag{9}\]  
where \(l_{WPT,j}\) represents the initial WPT slotframe length for the \(j\)-th HAP cluster. In addition, \(n_{totalcell,j}\) denotes the total number of cells expected to be allocated to the MCSS schedule for the \(j\)-th HAP cluster during \(l_{LCM,j}\) timeslots, calculated as follows:  
\[n_{totalcell,j}=n_{CMcell,j}+n_{HAPcell,j}-n_{dupcell,CMeHAP,j}\ (0<j\leq M) \tag{10}\]  
where \(n_{CMcell,j}\) and \(n_{HAPcell,j}\) are the respective numbers of cells expected to be independently allocated to the CM and HAP slotframes of the \(j\)-th HAP cluster during \(l_{LCM,j}\) timeslots. Additionally, \(n_{dupcell,CMeHAP,j}\) is the number of cells expected to be redundantly allocated between the CM and HAP slotframes in the \(j\)-th HAP cluster during \(l_{LCM,j}\) timeslots. Moreover, \(n_{CMcell,j}\) and \(n_{HAPcell,j}\) in Equation (10) are calculated as follows:  
\[n_{CMcell,j}=\frac{l_{LCM,j}}{L_{CM}}\ (0<j\leq M), \tag{11}\]  
\[n_{HAPcell,j}=\frac{l_{LCM,j}n_{HAP,j}}{L_{HAP}}\ (0<j\leq M), \tag{12}\]  
where \(L_{CM}\) and \(L_{HAP}\) denote the length of CM and HAP slotframes, respectively. The term \(n_{HAP,j}\) refers to the total number of HAPs in the \(j\)-th HAP cluster. In addition, \(n_{dupcell,CMeHAP,j}\) in Equation (10) is determined based on the relationship between repeated cycles of multiple concurrent slotframes for the same period, calculated as follows:  
\[n_{dupcell,CMeHAP,j}=\frac{l_{LCM,j}n_{HAP,j}}{LCM(L_{CM},L_{HAP})}\ (0<j\leq M) \tag{13}\]  
where \(LCM(L_{CM},L_{HAP})\) represents the LCM of the lengths of the CM and HAP slotframes. Furthermore, \(LCM(x_{1},x_{2},\cdots,x_{w})\) refers to the LCM of \(x_{1}\), \(x_{2}\),..., \(x_{w}\), where \(w\) is the number of multiple concurrent slotframes. Consequently, the numbers of required power and data cells for the \(i\)-th cluster member in the HAP cluster (\(n_{reqPC,i}\) and \(n_{reqDC,i}\)) are calculated as follows:  
\[n_{reqPC,i}=n_{minPC,i}+n_{overPC,i}\ (0<i\leq n_{node,j},\ 0<j\leq M), \tag{14}\]  
\[n_{reqDC,i}=n_{minDC,i}+n_{overDC,i}\ (0<i\leq n_{node,j},\ 0<j\leq M). \tag{15}\]For the WPT slotframe length determination, each HAP maintains a HAP cluster table (HCT) containing information about the cluster members. Specifically, the table entry of the HCT for the \(j\)-th HAP (i.e., the \(j\)-th HCT) includes the ID list (\(\textbf{ID}_{(j)}\)) and numbers of required power and data cell lists (\(\textbf{N}_{reqPC(j)}\) and \(\textbf{N}_{reqDC(j)}\)) of its cluster members of the \(j\)-th HAP. Finally, \(\textbf{ID}_{(j)}\), \(\textbf{N}_{reqPC(j)}\), and \(\textbf{N}_{reqDC(j)}\) are represented as follows:  
\[\textbf{ID}_{(j)}=\Big{[}id_{(1)},id_{(2)},\cdots,id_{(i)}\Big{]},\ \ (0<i\leq n_{node,j},\ 0<j\leq M), \tag{16}\]  
\[\textbf{N}_{reqPC(j)}=\big{[}n_{reqPC,1},n_{reqPC,2},\cdots,n_{reqPC,i}\big{]},\ \ (0<i\leq n_{node,j},\ 0<j\leq M),\ \text{and} \tag{17}\]  
\[\textbf{N}_{reqDC(j)}=\big{[}n_{reqDC,1},n_{reqDC,2},\cdots,n_{reqDC,i}\big{]},\ \ (0<i\leq n_{node,j},\ 0<j\leq M), \tag{18}\]  
where \(id_{(i)}\) indicates ID of the \(i\)-th cluster member in the \(j\)-th HAP cluster. Algorithm 1 presents the procedure of the WPT slotframe length determination for each HAP.