# RF Energy Harvesting Wireless Communications:
## 4 Power Management in RF-EHWC
### Design Principles of Power Management in RF-EHWC  
#### 4.1.1 Limited Energy Resource  
To facilitate the description of an energy harvesting process, RF waves arrived at an RF-EHWC system are generally modeled as discrete energy packets with varying sizes and random time of arrivals. In conventional offline power management approaches, it is usually assumed that the time and the number of data packets and energy packets that will arrive in the future are known. The accumulative energy harvested by the system is a staircase curve, which is represented by L\({}_{1}\) in Figure 13. Due to the limited capacity of its battery, an additional staircase curve L\({}_{2}\) is obtained by shifting L\({}_{1}\) down by \(e_{max}\), representing the capacity constraint of the battery. The profile of accumulative energy consumption, P, should stay inside the feasible energy tunnel: if it exceeds the upper bound, L\({}_{1}\), the battery will not be able to provide sufficient energy; if the curve P is below L\({}_{2}\), the energy will overflow from the battery. The optimal transmission scheduling is thus formulated as the following problem to find the consumption curve P that maximizes the data throughput given the limited energy resource.  
\[\begin{array}{llll}&\text{\bf P1:}&\arg\max_{r(t)}\int_{0}^{t_{i}}r(t)\, \text{d}t,\\ &\text{\bf s.t.}&\text{\bf C1:}&\int_{0}^{t_{i}}p(t)\,\text{d}t\leq\sum_{j=0}^ {t_{j}\leq t_{i}}e_{j},&i=1,\ldots,N+1,\\ &\text{\bf C2:}&\sum_{j=0}^{t_{j}\leq t_{i}}e_{j}-\int_{0}^{t_{i}}p(t)\,\text{ d}t\,\leq e_{max},&i=1,\ldots,N,\end{array} \tag{1}\]  
where \(e_{i}\) and \(t_{i}\) are the amount of energy and the arrival time of energy packet \(i\), respectively; \(e_{max}\) is the maximum capacity of the battery; and \(p(t)\) is the instant transmission power of the communication module at time \(t\), which is related to the transmission power through a power-rate function, \(p(t)=f(r(t))\). Constrains \(\boldsymbol{C1}\) and \(\boldsymbol{C2}\) represent the energy causality constraint and the maximal battery capacity constraint, respectively.  
The work presented in [53] proves that the optimal transmission power of an RF-EHWC device should remain unchanged between energy harvests in order to maximize the throughput. With such a conclusion, power management is simplified into a piecewise-linear optimization problem. It is pointed out that the optimal transmission power in the feasible energy tunnel has the following properties [54]:  
Figure 13: Power management with limited energy resource.  
**Property 1**.: _The optimal transmission power does not change until the battery is either full or completely depleted._  
**Property 2**.: _The optimal transmission power decreases only at energy arrival instants when the battery is full and increases only at energy arrival instants when the battery is depleted._  
**Property 3**.: _The optimal power management keeps the longest stretches of constant periods in the feasible energy tunnel._  
In addition to the solution from a graphical point of view, a directional water-filling algorithm is proposed in [55], where the energy is considered as \"water\" that has two features: (a) it cannot flow back due to the constraint of energy causality; and (b) it can be stored for future use, but the amount of water flowing to the future each time cannot exceed the maximum capacity of the battery. To maximize the throughput, the directional water-filling algorithm needs to distribute the water as equal as possible along the timeline.