## 4 Proposed Energy Neutral Operation
### Energy Neutral Operation
\begin{array}{ll}R_{ij}^{\text{TS}}&=\alpha_{j}T_{s}\log_{2}\left(1+\frac{S_ {ij}P_{j}}{\sigma^{2}}\right)\\ &=\alpha_{j}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{8}\]  
In both the PS and TS methods, the amount of energy that CH \(i\) additionally obtains is given by  
\[E_{i}^{\text{add}}=\sum_{j\in\mathcal{N}\setminus\{i\}}\zeta_{i}S_{ij}(1- \alpha_{j})P_{j}T_{s}. \tag{9}\]  
Then, when SWIPT is used, the transmission power of CH \(i\) is updated as  
\[P_{i}^{\text{swipt}}=\frac{\eta_{i}(E_{i}^{\text{har}}+E_{i}^{\text{add}})}{ T_{d}}. \tag{10}\]  
Moreover, the achievable rate of transmission from CH \(i\) to the HAP is represented as  
\[ \begin{array}{ll}R_{i}^{\text{swipt}}&=T_{d}\log_{2}\left(1+\frac{h_{i}P_{ i}^{\text{swipt}}}{\sigma^{2}}\right)\\ &=T_{d}\log_{2}\left\{1+\frac{\eta_{j}h_{i}(E_{i}^{\text{har}}+E_{i}^{\text{add }})}{\sigma^{2}T_{d}}\right\}\\ &=T_{d}\log_{2}\left\{1+\frac{\eta_{j}\zeta_{j}h_{i}PT_{d}\left(h_{i}P_{j} \sum_{i\in\mathcal{N}\setminus\{i\}}\eta_{j}\zeta_{j}(1-\alpha_{j})g_{ij}\mu_{j }\right)}{\sigma^{2}T_{d}}\right\}\ \ \text{(bits/Hz)}.\end{array}  \tag{11}\]  
Finally, the achievable rates of sensing data when SWIPT is used according to PS and TS are respectively expressed as  
\[R^{\text{swipt}}=\left\{ \begin{array}{ll}\min\left[\min_{j\in\mathcal{N} \setminus\{i\}}\left\{R_{ij}^{\text{PS}}\right\},R_{i}^{\text{swipt}}\right],\ \ \text{if PS is used}\\ \min\left[\min_{j\in\mathcal{N}\setminus\{i\}}\left\{R_{ij}^{\text{TS}}\right\},R_{i}^{\text{swipt}}\right],\ \ \text{if TS is used}.\end{array} \right. \tag{12}\]In WSNs, energy dissipation can occur because the sensor node does not need to consume more energy than the energy required to attain the achievable rate of sensing data. This metric is important for measuring the energy efficiency in WSNs [43]. The amount of energy dissipated in the sensor node is defined as the difference between the energy harvested and the energy required to satisfy the achievable rate. In the case without using SWIPT, all of the nodes only need to satisfy \(R^{\text{noSwipt}}\) given as Equation (6). Thus, the amounts of energy dissipated in sensor node \(j\) and CH \(i\) are respectively calculated as  
\[\mathcal{E}_{j}^{\text{noSwipt}} =\left\{P_{j}-\frac{\sigma^{2}}{g_{ij}}\left(2^{\frac{\text{n} \text{n}\text{o}\text{swipt}}{I_{i}}}-1\right)\right\}T_{s},\ \forall j\in\mathcal{N} \setminus\{i\} \tag{13}\] \[\mathcal{E}_{i}^{\text{noSwipt}} =\left\{P_{i}-\frac{\sigma^{2}}{h_{i}}\left(2^{\frac{\text{n} \text{n}\text{o}\text{swipt}}{I_{d}}}-1\right)\right\}T_{d}. \tag{14}\]  
Therefore, the total amount of energy dissipated in the cluster is expressed as  
\[\mathcal{E}^{\text{noSwipt}}=\mathcal{E}_{i}^{\text{noSwipt}}+\sum_{j\in \mathcal{N}\setminus\{i\}}\mathcal{E}_{j}^{\text{noSwipt}}. \tag{15}\]  
On the other hand, when SWIPT is used, all of the nodes only need to satisfy \(R^{\text{PS}}\) or \(R^{\text{TS}}\) depending on the SWIPT method used. However, SWIPT is not necessary if \(R_{i}\geq\min_{j}\left\{R_{ij}\right\}\) because the rate of the CH is sufficiently high that the CH does not need to receive energy from its member nodes. In this case, the energy dissipated in the cluster is the same as Equation (15). However, in the opposite case, i.e., if \(R_{i}<\min_{j}\left\{R_{ij}\right\}\), the energy dissipated in sensor node \(j\) is zero because it transfers all of the remaining energy to the CH. Thus, the energy dissipated in sensor node \(j\) is given by  
\[\mathcal{E}_{j}^{\text{swipt}}=\left\{ \begin{array}{ll}0,&\text{if }R_{i}<\min_{j}\left\{R_{ij}\right\}\\ \mathcal{E}_{j}^{\text{noSwipt}},&\text{otherwise}.\end{array} \right.