## 4 Theoretical Analyses
### Energy Neutrality Constrains  
We assume that each sensor node transmits its sensing data to the corresponding CH or BS periodically, and sensor nodes in different units may have different data transmission cycles. Let \(T_{i}\) (\(1\leq i\leq m\)) be the data transmission cycle of nodes in unit \(i\), and \(q\) be the length of data packet transmitted to CH or BS by a node during each cycle. As aforementioned, to achieve perpetual network operation, each sensor node should work in an energy neutral state. That is, during each cycle, a sensor node should consume less energy than the total amount of energy which contains the accumulated residual energy before this cycle and the additional energy resource harvested from the ambient environment in the current cycle. Since CHs consume energy much faster than normal nodes, all sensor nodes within a cluster should take turns serving as the CH to distribute the energy consumption. Let \(n_{i}\) (\(1\leq i\leq m\)) be the average number of sensor nodes per cluster in unit \(i\), and then each node in this unit can serve as the CH once every \(n_{i}\) cycles. To maintain every sensor node in unit \(i\) in an energy neutral state, the energy consumption of nodes should satisfy the following constraints:  
\[Ecm_{i}\leq T_{i}P_{e}\left(i=1,2,\cdots,m\right) \tag{4}\]  
\[Ech_{i}\leq n_{i}T_{i}P_{e}-\left(n_{i}-1\right)Ecm_{i}\left(i=1,2,\cdots,m\right) \tag{5}\]  
where \(Ecm_{i}\) is the average amount of energy consumed by a non-cluster head node in unit \(i\) to transmit data to its corresponding CH; \(Ech_{i}\) is the average amount of energy consumed by a CH in unit \(i\) to receive data from its member nodes, aggregate this with its own data and transmit the aggregated data to its routing head or BS; \(P_{e}\) is the energy harvesting rate; \(T_{i}\) is the data transmission cycle of the nodes in unit \(i\); and \(n_{i}\) is the average number of sensor nodes per cluster in unit \(i\). Equation (4) guarantees that a normal node consumes less energy than the amount of energy it can harvest during each cycle. Equation (5) guarantees that a CH consumes less energy than the total amount of energy including the accumulated residual energy before the current cycle and the energy harvested during this cycle. \(Ecm_{i}\) and \(Ech_{i}\) can be computed by the following equations respectively:  
\[Ecm_{i}=q\left(E_{elc}+\varepsilon_{fs}r_{i}^{2}\right) \tag{6}\]\[Ecl_{i}=\frac{\rho\left(m-i\right)\frac{S}{m}aq}{c_{i}}\left(2E_{dec}+\varepsilon_{ fs}d_{i}^{2}\right)+\left(n_{i}-1\right)qE_{elec}+n_{i}q\left(\frac{1}{a}-1\right)E_{DA}+n_{ i}aq\left(E_{elec}+\varepsilon_{fs}d_{i}^{2}\right) \tag{7}\]  
where \(q\) is data packet length of each node; \(r_{i}\) is the average distance between a non-cluster head node and the CH within a cluster in unit \(i\); \(\rho\) is node density; \(m\) is the number of units; \(S\) is the sensor field size; \(a\) is data aggregation ratio; \(c_{i}\) is the number of clusters in unit \(i\); \(n_{i}\) is average number of sensor nodes per cluster in unit \(i\); and \(d_{i}\) is the average distance between a CH in unit \(i\) and its routing head or BS. On the right-hand side of Equation (7), the first term is the average energy consumed by a CH in unit \(i\) to receive and forward the inter-cluster data traffic. In addition, the rest three terms are the average energy consumed by a CH in unit \(i\) to receive, aggregate and forward the intra-cluster data traffic, respectively.