Robust Power Allocation for NOMA Heterogeneous Networks with EH under Imperfect CSI

: A robust power allocation is proposed for downlink non-orthogonal multiple access (NOMA) heterogeneous networks with EH (Energy harvesting) under imperfect channel state information (CSI). In order to achieve green communication, an EH-aided scheme by leveraging energy from macro base station (MBS) signal and interference signal transmitted from other SBSs is proposed, which reduces the power burden and energy consumption of the SBS. In order to conform to the actual communication scenario, we construct an energy efficiency optimization function under imperfect CSI with considering the constraint of the outage probability interference power in macro cell user (MCU). However, the formulated optimization problem is non-convex due to the fractional form of the objective function and the probabilistic constraints of the outage probability limit. To cope with this problem, we propose a robust power allocation scheme. Firstly, the probabilistic problem is converted into a robust non-probabilistic problem by the minimax probability machine (MPM) and robust optimization theory. Then, the robust non-probabilistic problem can be transformed into the convex optimization problem via Dinkelbach method and sequential convex programming. Finally, the optimal transmission powers of the small cell users (SCUs) are obtained by Lagrange dual approach. The simulation results show that the robust power allocation scheme for NOMA heterogeneous networks with EH under imperfect CSI can significantly improve energy efficiency compared with traditional power allocation algorithms.


Introduction
Nowadays, with the explosive growth of traffic demand, such as the popularity of mobile terminal devices, there are more and more demands for high data rate and energy efficiency in the heterogeneous networks. [1]- [2]. And in order to further improve the spectrum efficiency, NOMA technology is integrated into heterogeneous networks.
The resource allocation algorithms can improve system preform in NOMA heterogeneous networks and have been widespread concerned. Then the energy-efficient resource allocation algorithm in the coordinated downlink multicell network system is studied with perfect CSI [3], the user scheduling, data rate adaptation, and power allocation are jointly designed to maximize the system energy efficiency under the maximum transmitted power constraint and power allocation is proposed in a downlink NOMA single-cell network with imperfect CSI [9], in which the non-convex problem is solved by the iterative algorithm. In [10], with the imperfect CSI, a power allocation algorithm about small cell and SCUs is proposed for downlink NOMA heterogeneous networks. To restrain the impact of parameter uncertainties for multiuser underlay cognitive radio networks, a robust adaptive power allocation algorithm is investigated in [11], and then an effective algorithm is developed that aims at maximizing the overall throughput of secondary users under the robust interference temperature and signal to interference and noise ratio constraints. In [12], a power adjustment algorithm is proposed for heterogeneous networks which considers varying QoS requirements of users. In order to maximize energy efficiency [13], a robust power allocation scheme for a downlink NOMA heterogeneous network is designed with considering imperfect CSI.
A robust resource allocation scheme for maximizing the interference efficiency of users in heterogeneous networks is proposed [14], and the closed-form solution is obtained by using Dinkelbach's method, the logarithmic transformation method, and the successive convex approximation method.
But the finiteness of energy resources and excessive energy consumption will cause a lot of pollution to the environment currently [15]- [17].
Traditional EH techniques allow mobile terminals or BS to harvest energy from wireless transmission environment, which is a very effective technique, and is extensive concerned [18]- [20]. In [21], the energy beamforming scheme for an EH multiple input single output (MISO) HetNets is proposed, which maximizes the sum harvested energy. In [22], an EHrate maximization-based resource allocation (RA) scheme for heterogeneous macrocell-smallcell networks with simultaneous wireless information and power transfer (SWIPT) is addressed to obtain the optimal power splitting (PS) and time-switching variables where the minimum throughput constraint of the macrocell user is considered. A robust power allocation and PS scheme is proposed for downlink SWIPT-enabled HetNets with EH [23]. In [24], a robust secure transmission scheme is proposed for wireless information and power transfer in HeNets under the deterministic and stochastic CSI errors, respectively.
Based on above analysis, we can find that there are few schemes for jointly considering imperfect CSI and EH, so in this paper, we propose a robust power allocation for NOMA heterogeneous networks with EH under imperfect CSI. Different from [21]- [23], which installing an EH device at the mobile terminal to reduce the energy consumption of the system, but in this paper, the EH device is installed at the BS to collect energy. And in contrast to [11], [13], [15], we add the interference constraints about macro users to protect macro users. Then a robust power allocation algorithm is designed to maximize energy efficiency.
First, we formulate the optimization problem with outage probabilistic and then convert probabilistic constraints to non-probabilistic constraints by MPM, at last, the original optimization problem is converted into convex problem by using the Dinkelbach method and the quadratic transformation approach and, the optimal solution is obtained by Lagrange algorithm.
The rest of the paper is organized as follows: Section 2 builds the system model of NOMA heterogeneous networks and formulates energy efficiency maximization optimization. Section 3 transforms the optimization problem into a robust optimization problem. Section 4 shows the power allocation algorithm. Section 5 demonstrates the numerical results. Finally, the paper is concluded in Section 6.

System model and problem formulation
As shown in Figure 1, we consider a two-layer heterogeneous networks.  (4) According to Shannon's formula, the data rate of the nth SCU in the kth SBS can be written as represent channel gain from the MBS and the lth SBS to the kth SBS, respectively. Therefore, the total power consumption can be described as According the equation (5) and (7), the optimization problem is formulated as where k P is the maximum transmit power of the kth SBS, , nm h denotes channel gain between nth SCU and mth MCU, and thr I is the total interference threshold from all SBS to mth MCU. 1 C is the transmission power constraint for the BS, 2 C ensures that the power of SCU is non-negative, 3 C ensures the QoS of SCU, and 4 C limits the interference power received by the MCU.

Robust power allocation formulation
In order to provide more protection to the Due to the introduction of probability constraint 4 C , the optimization problem (9) is an NP-hard problem, which is difficult to obtain the optimal value in a polynomial time. To solve this problem simply, we transform the probabilistic problem into a nonprobabilistic problem.
Due to the channel uncertainty, , nm h in 4 C of (9) can be formulated as Besides, due to the influence of channel fading, the perfect CSI and accurate statistical models of these random parameters cannot be obtained directly. In order to solve the problem, the outage probability constraint of macro user interference is transformed into deterministic constraint by MPM method without knowledge of exact models of uncertainties [25].
We consider a probability-constraint problem as follows: (10) where inf means the lower bound operation, a represents the optimal object, y is the true value of the uncertain parameter, y is the estimated value of the uncertain parameter, b is a constant value, E denotes the variance of y , and [0,1]

 
represents the outage probability threshold. From inequation (10), we note that the maximum outage , and thus problem (10) is converted into a sup form: where sup means the upper bound operation.
According to the principle of MPM, we transform the problem (10) into (12): According to (11) and (12), we obtain According to (13) and (14), we obtain the following inequation a y (15) where ( )  (20) According to (19) and (20) The robust optimization objective function (29) with respect to , nk p is no-convex and it is difficult to obtain the optimal solution. In order to solve the optimization problem (29), we propose a reasonable sub-optimal power allocation algorithm to maximize the energy efficiency.

Power Allocation Algorithm
In this section, we can obtain an optimal allocated power of SCU by solving the optimization problem According to the Cauchy-Schwarz inequality In particularly, we can obtain 0 T  z Hz , which means that the Hessian matrix H is positive semi-definite.
Therefore, () f Q is a convex function with respect to Q . As a result, the objective function (38) is concave with respect to Q . Furthermore, the objective function is concave and the constraints in this problem (33) are concave. Therefore, the optimization problem (33) is a concave optimization problem.
Proof end.
To solve the optimization problem (33), we can obtain the optimal allocated power of each SCU by the Lagrangian dual method. According to (33), we get the Lagrangian function as following , 2     . From Figure 5, we note that the total of energy efficiency of the proposed algorithm improves with the increasing of energy conversion efficiency  . That is because we use the energy conversion efficiency  , the collected energy is increase when  is increasing, then the overall power consumption is decrease, and the system energy efficiency is increase. Meanwhile the proposed algorithm can improve the energy efficiency performance as the k P increases.  . It is shown that the total energy efficiency of our scheme decreases with the increasing of m P . This is because the cross-layer interference increases with the increase of m P , then the SINR decreases, at last the system energy efficiency also decreases. We note that the output performance with NOMA is better as compared with OFDMA. For example, when the number of small cell is six, the energy efficiency with NOMA is 1.83Mbit/J larger than the energy efficiency with OFDMA. It is because NOMA technology can not only allow multiple users to reuse the same channel, but also allow the strong user to remove the interference from the weak users.