## 4 Proposed Energy Neutral Operation
### Energy Neutral Operation  
We explain the proposed ENO based on SWIPT in detail in this section. The proposed ENO is performed on a per-cluster basis. Without loss of generality, we can consider only one cluster assuming that the clustering and CH selection are completed in advance (The clustering and CH selection algorithm is presented in Section 4.4.). Suppose that there are \(N\) sensor nodes in the cluster. Let \(\mathcal{N}\) be the set of sensor nodes in the cluster, i.e., \(\mathcal{N}=\{1,2,\cdots,N\}\). We denote the selected CH node as \(i\), where \(\exists i\in\mathcal{N}\), and denote the other sensor nodes as \(j\), where \(\forall j\in\mathcal{N}\setminus\{i\}\). For simplicity, we assume that there is channel reciprocity between any two nodes. As illustrated in Figure 1, \(h_{j}\) denotes the channel power gain between the HAP and any node \(j\), and \(g_{ij}\) denotes the channel power gain from sensor node \(j\) to CH \(i\). The energy harvested by node \(j\) from HAP is expressed as  
\[E_{j}^{\text{har}}=\zeta_{j}h_{j}PT_{e},\ \forall j\in\mathcal{N} \tag{1}\]  
where \(0<\zeta_{j}<1\) is an EH efficiency of node \(j\), and \(P\) is the constant transmission power of the HAP. The transmission power of sensor node \(j\) for transmitting data to the CH during the SWIPT slot is given by  
\[P_{j}=\frac{\eta_{j}E_{j}^{\text{har}}}{T_{s}}=\frac{\eta_{j}\zeta_{j}h_{j}PT_ {e}}{T_{s}},\ \forall j\in\mathcal{N}\setminus\{i\} \tag{2}\]  
where \(0<\eta_{j}<1\) is the ratio of the energy used for only transmission except reception, processing, and circuit operations to the total energy harvested in node \(j\). By [PERSON]'s theorem, the achievable rate of transmission from sensor node \(j\) to the CH \(i\) is expressed as  
\[R_{ij}=T_{s}\log_{2}\left(1+\frac{g_{ij}P_{j}}{\sigma^{2}}\right),\ \forall j\in\mathcal{N}\setminus\{i\}\ \ \text{(bits/Hz)} \tag{3}\]  
where \(\sigma^{2}\) is the noise power at the receiving side, and the unit of this rate has bits/Hz reflecting the transmission time \(T_{s}\). If SWIPT is not used, as in the conventional method, the CH uses only the energy harvested from the HAP. Thus, the transmission power of CH \(i\) for transmitting data to the HAP during the WIT slot is given by  
\[P_{i}=\frac{\eta_{i}E_{i}^{\text{har}}}{T_{d}}=\frac{\eta_{i}\zeta_{i}h_{i}PT_ {e}}{T_{d}}. \tag{4}\]Then, the achievable rate of transmission from the CH \(i\) to the HAP is given by  
\[R_{i}=T_{d}\log_{2}\left(1+\frac{h_{i}P_{i}}{\sigma^{2}}\right)\ \ \text{(bits/Hz)}. \tag{5}\]  
The maximum rate of sensing data that can be collected in this cluster is limited by the minimum rate of the transmission links [19; 20]. Therefore, when SWIPT is not used, the achievable rate of sensing data in this cluster is determined by  
\[R^{\text{noSwipt}}=\min\left[\min_{j\in\mathcal{N}\setminus\{i\}}\left\{R_{ ij}\right\},R_{i}\right]. \tag{6}\]  
Because the achievable rate of sensing data is determined by the minimum rate of the transmission links, the sensor nodes with higher \(R_{ij}\) can consume only the energy required to satisfy this minimum rate and then transmit the remaining energy to the CH using the SWIPT technique. When PS-based SWIPT is applied, node \(j\) uses power \(\alpha_{j}P_{j}\) for WIT and the remaining power \((1-\alpha_{j})P_{j}\) for WET. Thus, the achievable rate of transmission from sensor node \(j\) to CH \(i\) is expressed as  
\[ \begin{array}{ll}R_{ij}^{\text{PS}}&=T_{s}\log_{2}\left(1+\frac{S_{ij}S_{ij} P_{j}}{\sigma^{2}}\right)\\ &=T_{s}\log_{2}\left(1+\frac{\eta_{j}\zeta_{j}\beta_{ij}\mu_{j}PT_{c}}{\sigma^ {2}T_{s}}\right),\ \forall j\in\mathcal{N}\setminus\{i\}\ \ \text{(bits/Hz)}.\end{array}  \tag{7}\]  
On the other hand, when TS-based SWIPT is applied, node \(j\) uses time \(\alpha_{j}T_{s}\) for WIT and the remaining time \((1-\alpha_{j})T_{s}\) for WET. Hence, the achievable rate of transmission from sensor node \(j\) to CH \(i\) is expressed as  
\[