# Multi-User AF Relay Networks with Power Allocation and Transfer: A Joint Approach

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. System Model

#### 2.2. Energy Harvesting and Power Transfer Model

#### 2.3. Problem Formulation and Transformation

**(P.1)**as follows:

**(P.2)**) are convex, while $(C.3)$ and $(C.4)$ become non-convex. Let ${\Omega}_{k,m}={\sum}_{i=1}^{m}\left({e}^{{\overline{P}}_{k,i}}+{P}_{k,i}^{c}\right)$ and ${\Omega}_{r,m}={\sum}_{i=1}^{m}\left({e}^{{\overline{W}}_{r,i}}+{P}_{r,i}^{c}\right)$, the problem

**(P.2)**is equivalently written as

**(P.3)**is still non-concave, and thus, an SCA method is utilized here in order to transform the non-tractable objective function of $(\mathit{P}.\mathit{3})$ into a tractable one as follows:

## 3. Proposed Power Allocation and Transfer Algorithm

#### 3.1. Subproblem Solution

#### 3.2. Master Problem

Algorithm 1: Proposed algorithm for given values of ${\Omega}_{s}$ and ${\Omega}_{r}$. |

1: Set ${I}_{max}$ as the maximum number of iterations with step sizes ${\u03f5}_{1},{\u03f5}_{2},\cdots ,{\u03f5}_{6}$ 2: Initialize $t=0$, ${\rho}_{j,i}^{\left(t\right)}=1$ and ${\beta}_{j,i}^{\left(t\right)}=0$; 3: Initialize ${\overline{\mathbf{P}}}_{s}^{\left(t\right)}$, ${\overline{\mathbf{W}}}_{r}^{\left(t\right)}$, ${\mathit{\lambda}}^{\left(t\right)}$, ${\mathit{\mu}}^{\left(t\right)}$, ${\mathit{\eta}}^{\left(t\right)}$, ${\mathit{\nu}}^{\left(t\right)}$, ${\mathit{\vartheta}}^{\left(t\right)}$, and ${\mathit{\kappa}}^{\left(t\right)}$. 4: repeat 5: repeat(Solving (P4))6: Compute ${\overline{\mathbf{P}}}_{s}$ and ${\overline{\mathbf{W}}}_{r}$ and $\mathit{\delta}$; 7: Compute $\mathit{\lambda}$, $\mathit{\mu}$, $\mathit{\eta}$, $\mathit{\nu}$, $\mathit{\vartheta}$, and $\mathit{\kappa}$ using (30)–(35). 8: until convergence;9: Compute ${\rho}_{j,i}^{(t+1)}$ and ${\beta}_{j,i}^{(t+1)}$ using (26) and (27) 10: Set ${\overline{\mathbf{P}}}_{s}^{(t+1)}\leftarrow {\overline{\mathbf{P}}}_{s}^{\u2605\left(t\right)}$, ${\overline{\mathbf{W}}}_{r}^{(t+1)}\leftarrow {\overline{\mathbf{W}}}_{r}^{\u2605\left(t\right)}$, and ${\mathit{\delta}}^{(t+1)}\leftarrow {\mathit{\delta}}^{\u2605\left(t\right)}$ and $t\leftarrow t+1$. 11: until convergence or $t>{I}_{max}$. |

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**P.4**) by assuming ${\Omega}_{k,m}^{\left(0\right)}=\infty $ and ${\Omega}_{r,m}^{\left(0\right)}=\infty $, for $m=1,\dots ,{N}_{T}-1$, and $k=1,\dots ,N$, and find the optimal $\left({\overline{\mathbf{P}}}_{s}^{\u2605\left(0\right)},{\overline{\mathbf{W}}}_{r}^{\u2605\left(0\right)},{\mathit{\delta}}^{\u2605\left(0\right)}\right)$ using the Algorithm 1. In the next step, we recompute the optimal solution by updating ${\Omega}_{k,m}^{\left(1\right)}={\sum}_{i=1}^{m}\left({e}^{{\overline{P}}_{k,i}^{\u2605\left(0\right)}}+{P}_{k,i}^{c}\right)$ and ${\Omega}_{r,m}^{\left(1\right)}={\sum}_{i=1}^{m}\left({e}^{{\overline{W}}_{r,i}^{\u2605\left(0\right)}}+{P}_{r,i}^{c}\right)$, for $m=1,\dots ,{N}_{T}-1,\phantom{\rule{0.277778em}{0ex}}\forall k$. This enables us to rectify the solutions by following the track of ${\Omega}_{k,m}^{\left(1\right)}$ and ${\Omega}_{r,m}^{\left(1\right)}$. □

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Convergence behavior of the proposed algorithm ($N=4$, ${d}_{sr}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$ and ${d}_{rd}=50\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$).

**Figure 3.**Scenario 1 (${E}_{s}={E}_{r}$): Average sum rate performance vs. time index ($N=4$, ${d}_{sr}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$ and ${d}_{rd}=50\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$).

**Figure 4.**Scenario 2 (${E}_{s}\ge {E}_{r}$): Average sum rate performance vs. time index ($N=4$, ${d}_{sr}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$ and ${d}_{rd}=50\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$).

**Figure 5.**Average sum rate performance vs. time index when ${E}_{s}<{E}_{r}$ ($N=4$, ${d}_{sr}=1\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$ and ${d}_{rd}=50\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$).

**Figure 6.**Impact of the relay position on the average sum rate performance for different values of energy harvesting (EH) profiles with ${d}_{SR}+{d}_{RD}=2\phantom{\rule{0.277778em}{0ex}}\mathrm{m}$.

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**MDPI and ACS Style**

Yadav, R.; Singh, K.; Biswas, S.; Kumar, A.
Multi-User AF Relay Networks with Power Allocation and Transfer: A Joint Approach. *Energies* **2019**, *12*, 3157.
https://doi.org/10.3390/en12163157

**AMA Style**

Yadav R, Singh K, Biswas S, Kumar A.
Multi-User AF Relay Networks with Power Allocation and Transfer: A Joint Approach. *Energies*. 2019; 12(16):3157.
https://doi.org/10.3390/en12163157

**Chicago/Turabian Style**

Yadav, Ramnaresh, Keshav Singh, Sudip Biswas, and Ashwani Kumar.
2019. "Multi-User AF Relay Networks with Power Allocation and Transfer: A Joint Approach" *Energies* 12, no. 16: 3157.
https://doi.org/10.3390/en12163157