Noncommutative Composite Water-Filling for Energy Harvesting and Smart Power Grid Hybrid System With Peak Power Constraints

Energy harvesting makes use of energy from the environment. However, since harvesting energy depends on natural conditions, it is not a stable energy source. As a result, the energy from the power grid is often included to serve as a supplementary source to regulate the overall energy supply of the system. Further, the power from the power grid is often subject to the constraints of peak power and the energy budget. These constraints lead to more difficulties in solving optimal power allocation problems. In this paper, we extend our recently proposed geometric water-filling (GWF) and recursive GWF (RGWF) algorithms to solve the throughput maximization problem and transmission completion time minimization problems for this kind of hybrid energy source system. Our investigation shows that the optimal power allocation for throughput maximization is the result of a sequence of water-filling algorithms for smart power grid and harvested energy, in that order, followed by a power adjustment step of the power from the grid. The allocation order is not commutative for an optimal solution due to the specific structure of the target problems. The proposed algorithms can compute the exact (optimal) solutions to the problems via finite computation with low computational complexity. Numerical examples are presented to illustrate the detailed procedures to efficiently obtain the optimal power allocation solutions using the proposed algorithms. The results also illustrate that the composite operation of the two water-fillings is noncommutative.