Non-Commutative Composite Water-Fillings 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 power gridis often included to serve as a supplementary source to regulate the overall energy supply of the system. Further, the powers from the power grid are often subject to the constraints of peak powers and the energy budget. These constraints lead to more difficulties in solving the optimal power allocation problems. In this paper, we extend our recently proposed geometric water-filling (GWF) and recursive geometric water-filling (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 orders and followed by a power adjustment step of the powers from the grid. The allocation order is not commutative for optimal solution due to specific structure of the target problems. The proposed algorithms can computethe 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 the composite operation of the two water-fillings is non-commutative.