## 3 SWIPT-Based Multi-Hop Uneven Clustering Routing Protocol Design for RF EH-CRSNs
### Network Model
\[E_{control}(j)=3L_{1}\times\left(E_{\_}{dec}+E_{\_}{fs}\times R_{\_}{r(j)}^{2} \right)+3L_{1}\times E_{\_}{elec}\times(N_{\_}{r(j)}-1), \tag{3}\]  
\[E_{\_}{instu}(j)=\left(N_{\_}{r(j)}-1\right)\times E_{\_}{dec}\times L_{\_}{2 }+N_{\_}{r(j)}\times E_{\_}{DA}\times L_{\_}{2}+\left(E_{\_}{dec}+E_{\_}{fs} \times d_{\_}{CH(r(j))\rightarrow CH(r(j)-1)}^{2}\right)\times L_{\_}{2}, \tag{4}\]  
\[E_{\_}{format}(j)=\frac{\frac{\sum\limits_{k=r(j)+1}^{2}}{\sum\limits_{\_}{ s(j)}}}{\frac{A_{\_}{r(j)}}{S_{\_}{r(j)}}}\times\left[E_{\_}{dec}\times L_{\_}{2}+ \left(E_{\_}{dec}+E_{\_}{fs}\times d_{\_}{CH(r(j))\rightarrow CH(r(j)-1)}^{2} \right)\times L_{\_}{2}\right], \tag{5}\]  
\[EH_{\_}ESF(j)= \begin{cases}E_{\_}{res}(j)+E_{\_}{EH}(j)-E_{\_}{control}(j)-E_{ \_}{forward}(j)&if\ r(j)=1\\ E_{\_}{res}(j)+E_{\_}{EH}(j)-E_{\_}{control}(j)-E_{\_}{\_}{\text{intra}}(j)-E_{ \_}{forward}(j)&otherwise\end{cases} , \tag{6}\]  
where \(L_{\_}{1}\) and \(L_{\_}{2}\) denote the sizes of the control and data packets, respectively; \(E_{\_}{dec}\) represents the energy consumed by electronic circuitry to send/receive a single bit of data; Under the assumption that the signal transmission adheres to the free-space path loss model, \(E_{\_}{fs}\) is the power amplifier's energy consumption per bit under this model; \(E_{\_}{DA}\) is the energy used to aggregate a bit of data; and \(N_{\_}{r(j)}\) denotes the average node count in a single cluster of ring \(r(j)\). \(A_{\_}{r(j)}\) is the area of ring \(r(j)\), with \(A_{\_}{r(j)}=(2r(j)-1)\pi R_{\_}{r(j)}^{2}\). \(S_{\_}{r(j)}\) indicates the average area of a single cluster in ring \(r(j)\), with \(S_{\_}{i}=\pi R_{\_}{r(j)}^{2}\). Here, \(R_{\_}{r(j)}\) is the cluster radius of ring \(r(j)\), and \(A_{\_}{r(j)}/S_{\_}{r(j)}\) gives the total cluster count in ring \(r(j)\), which corresponds to the total CHs count in that ring. Similarly, \(A_{\_}{k}/S_{\_}{k}\) signifies the total CHs count in ring \(k\). \(\sum\limits_{k=r(j)+1}^{2}A_{\_}{k}/S_{\_}{k}\) represents the number of data packets from outer rings that need to be forwarded by ring \(r(j)\). \(d_{\_}{CH(r(j))\rightarrow CH(r(j)-1)}\) measures the average distance from the CHs of ring \(r(j)\) to the relay CHs located in ring \(r(j)-1\) or to the sink, and the specific calculation is shown in Equation (7). \[d_{\_}{CH(r(j))\rightarrow CH(r(j)-1)}=d_{\_}{CHlosink(r(j))}-d_{\_}{CHlosink( r(j)-1)}, \tag{7}\]  
where \(d_{\_}{CHlosink(r(j))}\) and \(d_{\_}{CHlosink(r(j)-1)}\) represent the average distances from the CHs of ring \(r(j)\) and ring \(r(j)-1\) to the sink, respectively, with \(d_{\_}{CHlosink(r(j))}\) specifically illustrated in Equation (8). \[d_{\_}{CHlosink(r(j))}=\frac{1}{\pi r^{2}(j)R_{\_}{r}^{2}}\int\limits_{\theta= 0}^{2\pi}\int\limits_{r=0}^{r(j)R_{\_}{r}}r^{2}drd\theta=\frac{2}{3}r(j)R_{\_}{t} \tag{8}\]  
Utilizing \(EH_{\_}ESF(j)\), node \(j\) calculates the EH-based CHs selection weight \(EH_{\_}W(j)\), as indicated in Equation (9):  
\[\begin{array}{l}EH_{\_}W(j)=\\ \left\{\begin{array}{cc}[\alpha\times EH_{\_}ESF(j)]^{2}\times\sqrt[3]{C(j)} \times\sqrt{\frac{1}{d_{\_}{\text{initial}}(j)}}\times\sqrt{\frac{1}{\text{ Next}(j)}}\times\sqrt{\frac{1}{\sqrt{\text{energy outside the first ring disseminate their CHs weights within the cluster radius. Nodes receive these weights from neighbors and engage in a comparison of CHs weights. If a node's own weight is less than that of a neighboring node, it broadcasts a message to withdraw from the competition, and surrounding nodes receive this message; if the node has the highest weight among all its neighbors, it becomes a CH and broadcasts a CHs announcement message on CCC, leading to the withdrawal from the competition by the neighboring nodes. This process is repeated until all nodes have either become CHs or withdrawn from the competition. 2.