# RF Energy Harvesting Wireless Communications:
## 4 Power Management in RF-EHWC
### Practical Issues in Power Management
#### 4.2.1 Nonlinear Battery Charging  
The nonlinear charge is a well-known characteristic of batteries (e.g., supercapacitor, Lithium-ion battery, etc.) [59]. The amount of energy harvested from an energy packet not only depends on the power density and the duration of energy packets but also on the residual energy in the battery. As verified in the experiment reported in [60], when the power and length of energy packets are fixed, the amount of harvested energy in each energy replenishment is a concave and non-monotonic function of the residual energy in the battery, as demonstrated in Figure 16. The nonlinear relation between the harvested energy and the residual energy in the battery is raised in [61] and formulated in [60]. Figure 15: (**a**) A feasible data tunnel with the limited data storage, where \(d_{max}\) is the maximal capacity of data storage. (**b**) A feasible latency tunnel with the packet deadline constraint, where \(\bar{t}\) is the maximal latency allowed for each data transmission. In the classic energy harvesting, the harvested energy is modeled as a random but predetermined sequence, as shown in Figure 17a. The existing work in optimal transmission scheduling strategy is to seek a curve within such a fixed energy tunnel so that a pre-defined objective is optimized (e.g., maximizing throughput) [54; 55]. Unfortunately, the above assumption is not true in real energy harvesting systems. Since residual energy is affected by the data transmission, the amount of energy that will be harvested from an energy packet is not a predetermined value but a variable that heavily depends on the data transmission strategy. In other words, data transmission influences harvested energy in a real-time fashion; an RF-EHWC device cannot estimate what amount of energy it can harvest before scheduling its transmissions in the offline optimal transmission scheduling. To accurately describe the energy harvesting process, a new model is proposed in [60], as illustrated in Figure 17b. In the new model, an energy feedback loop from the data transmission strategy to the harvested energy is formed, which describes the impact of nonlinear charging on the energy harvesting process. The nonlinear battery charging makes optimal power management become a challenging problem. On the one hand, for the purpose of throughput maximization, an efficient offline power management needs to manage the transmission power based on the amount of energy that can be harvested in the future; on the other hand, the power management strategy itself can affect the energy harvesting process through adjusting the residual energy in the battery. To address the above problem, we can apply a recursive algorithm derived from the Karush-Kuhn-Tucker (KKT) conditions. (The KKT approach, allowing inequality constraints,  
Figure 16: Nonlinear energy harvesting process with respect to the residual energy, where \(T\) is the length of the energy packet. Figure 17: (**a**) Classic energy harvesting model without considering the nonlinear charging characteristic. (**b**) New feedback model with the nonlinear battery charging considered [60]. generalizes the Lagrange multipliers method, which allows only equality constraints. KKT conditions are necessary conditions. Therefore, the solutions to the KKT approach need to be further verified if they are sufficient for optimality. The KKT approach has been used in solving nonlinear programming problems in wide areas [62; 63].) As studied in [60], assume a total number of \(N\) energy packets arrive at an RF-EHWC device, and let \(E_{i}^{r}\) be the residual energy at the moment of energy packet \(i\) arrival. By leveraging the inherent relationship among the residual energy, the transmission power, and the harvested energy, \(E_{i}^{r}\) can be represented by \(E_{1}^{r}\) through the energy harvesting function, i.e., \(E_{i}^{r}=Q_{i}(E_{1}^{r})\), where \(i=2,\ldots,N\).