SCIENCE CHINA Information Sciences, Volume 63, Issue 2: 122302(2020) https://doi.org/10.1007/s11432-019-9936-y

## Joint utility optimization for wireless sensor networks with energy harvesting and cooperation

• AcceptedJul 17, 2019
• PublishedJan 16, 2020
Share
Rating

### Abstract

In wireless sensor networks (WSNs), the limited battery capacity restricts the lifetime of the sensor nodes and thus degrades the system performance. The energy harvesting and cooperation techniques are promising solutions to prolonging the battery life, by collecting energy from ambient environment and exchanging energy among sensor nodes. This paper studies the joint utility maximization problem for WSNs in consideration of energy harvesting and cooperation. We first derive an upper bound on the Lyapunov drift for the network stability, and then formulate the optimization as a stochastic optimization problem. Furthermore, we propose an energy harvesting and energy transfer, data transmission, power control, routing and scheduling (EDPR) online algorithm by combining Lyapunov optimization technique with drift-plus-penalty method and perturbation technique. It contributes to optimal utility in a distributed manner along with a balanced trade-off between network utility and queue backlog, with no need for any statistical information about dynamic systems and no concern of curse of dimensionality under large queue backlog. Simulation results also show the practicality of the proposed algorithm in real implementation since data transmission has a linear relationship with battery life.

### Acknowledgment

This work was supported in part by National Science and Technology Major Project of China (Grant No. 2018ZX03001008-002), in part by Natural Science Foundation of Jiangsu Province (Grant No. BK20180011), and in part by National Natural Science Foundation of China (Grant Nos. 61571120, 61871122).

### Supplement

Appendix

Proof of Theorem th1

The Lyapunov function 15 consists of two parts, the data stream queue and the energy queue. We separately handle these two parts in order to obtain the upper bound of the Lyapunov drift 16.

For one thing, we discuss the energy queue part. By introducing $\theta_n$, squaring both sides of 12 and rearranging it, we then have \begin{align} &\frac{1}{2}{{\left[ {{e}_{n}}(t+1)-{{\theta }_{n}} \right]}^{2}} -\frac{1}{2}{{\left[ {{e}_{n}}(t)-{{\theta }_{n}} \right]}^{2}} \\ & \le {{\left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{\left( {{P}_{nm}}( t )+{{\varepsilon}_{nm}}( t ) \right)} \right]}^{2}} +{{\left[ \sum\limits_{k\in \mathcal{N}_{n}^{(\rm in)}}{{{\varepsilon}_{kn}}( t )}+{{h}_{n}}( t ) \right]}^{2}} \\ & - \left[ {{e_n}(t) - {\theta _n}} \right]\left[ {\sum\limits_{m \in \mathcal{{\cal N}}_n^{(\rm out)}} {{P_{nm}}( t )} - {h_n}( t )} \right] -\left[ {{e}_{n}}(t)-{{\theta }_{n}} \right] \left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{{{\varepsilon}_{nm}}( t )-\sum\limits_{k\in \mathcal{N}_{n}^{(\rm in)}}{{{\widehat{\varepsilon}}_{kn}}( t )}} \right]. \tag{33} \end{align} Since ${{P}_{nm}}(~t~)$, ${{\varepsilon}_{nm}}(~t~)$, ${{h}_{n}}(~t~)$ have upper bounds ${P}_{\rm~max}$, ${\varepsilon}_{\rm~max}$, ${h}_{\rm~max}$ respectively, we can obtain that $${{\left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{\left( {{P}_{nm}}( t )+{{\varepsilon}_{nm}}( t ) \right)} \right]}^{2}}+{{\left[ \sum\limits_{k\in \mathcal{N}_{n}^{(\rm in)}}{{{\widehat{\varepsilon}}_{kn}}( t )}+{{h}_{n}}( t ) \right]}^{2}} \le {{\left[ {{D}_{\max }}\left( {{P}_{\max }}+{{\varepsilon}_{\max }} \right) \right]}^{2}}+{{\left[ {{D}_{\max }}{{\varepsilon}_{\max }}+{{h}_{\max}} \right]}^{2}}. \tag{34}$$

Let $B_1$ denote ${{[~{{D}_{\max~}}(~{{P}_{\max~}}+{{\varepsilon}_{\max~}}~)~]}^{2}}+{{[~{{D}_{\max~}}{{\varepsilon}_{\max~}}+{{h}_{\max}}~]}^{2}}$. Hence, we sum both sides of 34 over $n$ and it can be recast as \begin{align} \frac{1}{2}\sum\limits_{n=1}^{N}{{{\left[ {{e}_{n}}(t+1)-{{\theta }_{n}} \right]}^{2}}}-\frac{1}{2}\sum\limits_{n=1}^{N}{{{\left[ {{e}_{n}}(t)-{{\theta }_{n}} \right]}^{2}}}\le & N{{B}_{1}} -\sum\limits_{n=1}^{N}{\left[ {{e}_{n}}(t)-{{\theta }_{n}} \right]}\left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{{{P}_{nm}}( t )-{{h}_{n}}( t )} \right] \\ & -\sum\limits_{n=1}^{N}{\sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{{{\varepsilon}_{nm}}( t )\left[ {{\widehat{\varepsilon}}_{n}}( t )-{{\eta }_{nm}}{{\widehat{\varepsilon}}_{m}}(t) \right]}}. \tag{35} \end{align} We define ${{\widehat{E}}_{n}}(t)={{e}_{n}}(~t~)-{{\theta~}_{n}}$. And due to the fact that $0\le~{{\eta~}_{kn}}\le~1$, we have $(~1-{{\eta~}_{nm}}~){{\widehat{E}}_{n}}(~t~)\ge~-(~1-{{\eta~}_{nm}}~){{\theta~}_{n}}$. Let $B_2$ denote ${{B}_{2}}=N{{B}_{1}}+\sum\nolimits_{n=1}^{N}{\sum\nolimits_{m=1}^{N}{(~1-{{\eta~}_{nm}}~)}}{{\varepsilon}_{\max~}}$, and thus 35 can be rewritten as \begin{align} \frac{1}{2}\sum\limits_{n=1}^{N}{{{\left[ {{e}_{n}}(t+1)-{{\theta }_{n}} \right]}^{2}}}-\frac{1}{2}\sum\limits_{n=1}^{N}{{{\left[ {{e}_{n}}(t)-{{\theta }_{n}} \right]}^{2}}}\le & {{B}_{2}} -\sum\limits_{n=1}^{N}{{{\widehat{E}}_{n}}( t )}\left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{{{P}_{nm}}( t )-{{h}_{n}}( t )} \right] \\ &-\sum\limits_{n=1}^{N}{\sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{{{\eta }_{nm}}{{\varepsilon}_{nm}}( t )}}\big[ {{\widehat{E}}_{n}}( t )-{{\widehat{E}}_{m}}(t) \big]. \tag{36} \end{align} For another, the data queue part is studied and proved. Since the inequality $(~{{[~f(x)~]}^{+}}~)\le~{{f}^{2}(x)}$ holds for any $x\in~\mathbb{R}$, Eq. 9 can be converted into \begin{align} {{\big[ q_{n}^{(d)}(t+1) \big]}^{2}} -{{\big[ q_{n}^{(d)}(t) \big]}^{2}}\le & -2q_{n}^{(d)}(t)\left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{\lambda _{nm}^{( d )}(t)}-\sum\limits_{k\in \mathcal{N}_{n}^{(\rm in)}}{\lambda _{kn}^{( d )}(t)}-R_{n}^{(d)} \right] \\ &+{{\left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{\lambda _{nm}^{( d )}(t)} \right]}^{2}}+{{\left[ \sum\limits_{k\in \mathcal{N}_{n}^{(\rm in)}}{\lambda _{kn}^{( d )}(t)}+R_{n}^{(d)} \right]}^{2}}. \tag{37} \end{align} Rearrange 37 similarly to how we deal with that in the energy queue part, and it can be written as $$\frac{1}{2}{{\big[ q_{n}^{(d)}(t+1) \big]}^{2}}-\frac{1}{2}{{\big[ q_{n}^{(d)}(t) \big]}^{2}}\le {{B}_{3}} -q_{n}^{(d)}(t)\left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{\lambda _{nm}^{( d )}(t)}-\sum\limits_{k\in \mathcal{N}_{n}^{(\rm in)}}{\lambda _{kn}^{( d )}(t)}-R_{n}^{(d)} \right], \tag{38}$$ where ${{B}_{3}}=\frac{3}{2}D_{\max~}^{2}C~_{\max~}^{2}+R_{\max~}^{2}$.

Hence, we sum 38 over all $(n,d)$, and then obtain the following bound combined with 35. \begin{align} L({\boldsymbol Q}(t+1))-L({\boldsymbol Q}(t)) \le & B -\sum\limits_{n,d}{q_{n}^{(d)}}(t)\left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{\lambda _{nm}^{( d )}(t)}-\sum\limits_{k\in \mathcal{N}_{n}^{(\rm in)}}{\lambda _{kn}^{( d )}(t)}-R_{n}^{(d)} \right] \\ & -\sum\limits_{n=1}^{N}{{{{\hat{E}}}_{n}}( t )}\left[ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{{{P}_{nm}}( t )-{{h}_{n}}( t )} \right] -\sum\limits_{n=1}^{N}{\sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{{{\eta }_{nm}}{{\varepsilon}_{nm}}( t )}}[ {{{\hat{E}}}_{n}}( t )-{{{\hat{E}}}_{m}}(t) ], \tag{39} \end{align} where \begin{align} B & ={{N}^{2}}{{B}_{3}}+{{B}_{2}} \\ &={{N}^{2}}\left( \frac{3}{2}D_{\max }^{2}C _{\max }^{2}+R_{\max }^{2} \right) + N \big\{ {{[ {{D}_{\max }}( {{P}_{\max }}+{{\varepsilon}_{\max }} )]}^{2}}+{{[ {{D}_{\max }}{{\varepsilon}_{\max }}+{{h}_{\max}} ]}^{2}} \big\} +\sum\limits_{n=1}^{N}{\sum\limits_{m=1}^{N}{( 1-{{\eta }_{nm}} )}}{{\varepsilon}_{\max }}. \tag{40} \end{align} According to 16, we take expectations over the network state ${\boldsymbol~Q}(t)$ on both sides of 39, and then obtain 18.

Theorem th1 is thus proved.

Proof of Theorem th2

First, we use mathematical induction (MI) to prove the boundary constraint of $q_{n}^{(d)}(t)$ in 29 and $e_{n}(t)$ in 30.

As for $q_{n}^{(d)}(t)$, it is clear to see that Eq. 29 holds for $t=0$, because $q_{n}^{(d)}(0)=0$ for all $n,d~\in~\mathcal{N}$. Then, we assume that at time slot $t$, $q_{n}^{(d)}(t)$ obeys $q_{n}^{(d)}(t)\le~\beta~V+{{R}_{\max~}}$, $\forall~n,d~\in~\mathcal{N}$. Hence, two cases are considered in the next time slot $t+1$ classified by whether to receive data stream.

If node $n$ does not receive any data stream from other nodes through links at $t$, then it comes to $~q_{n}^{(d)}(t+1)\le~\beta~V+{{R}_{\max~}}$ referring to 9.

If node $n$ receives data stream from node $k$, with $~q_{k}^{(d)}(t)\le~\beta~V+{{R}_{\max~}}$, then it comes to $$q_{n}^{(d)}(t) \le q_{k}^{(d)}(t)-\tau \le (\beta V+{{R}_{\max }})-({{D}_{\max }}{{C}_{\max }}+{{R}_{\max }}) \le \beta V. \tag{41}$$ Since the transmitted data cannot exceed ${{R}_{\max~}}$, we have $q_{n}^{(d)}(t+1)\le~\beta~V+{{R}_{\max~}}$.

If node $n$ receives data packets from the outside of the network, we have $q_{n}^{(d)}(t)\le~Vf'(~0~)=\beta~V$ and get the same result as the above case.

Hence, $~q_{n}^{(d)}(t+1)\le~\beta~V+{{R}_{\max~}}$ holds for any $n,d~\in~\mathcal{N}$, and thus 29 is proved.

As for $e_{n}(t)$, since we have made the assumption that there is no energy in the initial state, we can get $e_{n}(t)=0$ for any $n~\in~\mathcal{N}$. For node $n$, we assume $e_{n}^{{}}(t)\le~{{\theta~}_{n~}}+{{h}_{\max~}}+{{D}_{\max~}}{{\varepsilon}_{\max~}}{{\eta~}_{\max~}}$ at time slot $t$.

If $e_{n}^{{}}(t)~<~{{\theta~}_{n}}$, $e_{n}(t+1)$ holds due to the limit of the input energy, no more than ${{h}_{\max~}}+{{D}_{\max~}}{{\varepsilon}_{\max~}}{{\eta~}_{\max~}}$.

If $e_{n}^{{}}(t)\ge~{{\theta~}_{n}}$, node $n$ does not harvest energy but may receive energy from other nodes. As an example, $e_{m}(t)~\le~{{\theta~}_{m}+h_{\max}+{{D}_{\max~}}{{\varepsilon}_{\max~}}{{\eta~}_{\max~}}}$ and energy is transferred from node $m$ to node $n$, which implies that \begin{align} e_{n}^{{}}(t) & < e_{m}^{{}}(t)-{{\theta }_{m}}-\frac{\zeta }{{{\eta }_{nm}}}+{{\theta }_{n}} \\ & \le e_{m}^{{}}(t)+{{\theta }_{n}}-\zeta \\ & \le ({{\theta }_{m}+h_{\max}+{{D}_{\max }}{{\varepsilon}_{\max }}{{\eta }_{\max }}})+{{\theta }_{n}} -({{\theta }_{\max}} +{{D}_{\max }}{{\varepsilon}_{\max }}{{\eta }_{\max }}) \\ & \le {{\theta }_{n}}+ h_{\max}. \tag{42} \end{align} Since the transferred energy cannot surpass ${{D}_{\max~}}{{\varepsilon}_{\max~}}{{\eta~}_{\max~}}$, we then have $e_{n}(t+1)~<~{{\theta~}_{n}+h_{\max}+{{D}_{\max~}}{{\varepsilon}_{\max~}}{{\eta~}_{\max~}}}$.

Hence, $e_{n}(t+1)~<~{{\theta~}_{n}+h_{\max}+{{D}_{\max~}}{{\varepsilon}_{\max~}}{{\eta~}_{\max~}}}$ holds for any $n~\in~\mathcal{N}$, and thus 30 is proved.

Therefore, the boundary constraint of $q_{n}^{(d)}(t)$ and $e_{n}(t)$ are provided in 29 and 30.

Next we prove 31 by contradiction when the data transmission or energy transfer action is in progress, that is $P_{nm}>0$ or $e_{\max}>0$.

In the power control step of the EDPR algorithm, we strive to choose the optimal $\varepsilon_{nm}^{*}(~t~)$ and $P_{nm}^{*}(~t~)$ to solve the energy-availability problem 28 as \begin{align} G({{\varepsilon}_{nm}}(t),{{P}_{nm}}(t))=& \sum\limits_{n}{\left\{ \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{{{\varepsilon}_{nm}}(t)}\big[ {{\eta }_{nm}}\big( {{{\hat{E}}}_{n}}(t)-{{{\hat{E}}}_{m}}(t) \big)-\zeta \big] \right.} \\ &+\sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{C_{nm}^{{}}(t)}\left[ q_{n}^{(d)}(t)-q_{m}^{(d)}(t)-\tau \right] \left. +{{{\hat{E}}}_{n}}(t)\sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{{{P}_{nm}}(t)} \right\}. \tag{43} \end{align} If there exist the optimal power allocation ${{\boldsymbol\varepsilon}^{*}}$ and the transferred energy ${{{\boldsymbol~P}}^{*}}$ that maximize 43, it is sure that they satisfy $P_{nm}^{*}(t)>0$ and ${\varepsilon}_{nm}^{*}(t)>0$ on link $(n,m)$. We now create a new solution $({{\boldsymbol\varepsilon}^{*}},{{{\boldsymbol~P}}^{*}})$ to the problem, by setting ${{P}_{nm}}^{*}(t)=0$, $\forall~k\ne~n$, $\forall~m$ and ${{\varepsilon}_{nm}}(~t~)$, $\forall{n,m}$ remains the same. Then, we have $$G({{{\boldsymbol P}}^{*}})-G({\boldsymbol P})=\sum\limits_{n}\left\{ {{{\hat{E}}}_{n}}( t ){{{\boldsymbol P}}^{*}} + \sum\limits_{m\in \mathcal{N}_{n}^{(\rm out)}}{\left[ {{C}_{nm}}\left( {{{\boldsymbol P}}^{*}} \right)-{{C}_{nm}}\left( {\boldsymbol P} \right) \right]}{{\widetilde{W}}_{( n,m )}}(t) \right\}. \tag{44}$$

Contrary to Theorem th2 we assume $e_{n}^{{}}(t)<{{P}_{\max~}}+{{\varepsilon}_{\max~}}$ in such case that node $n$ cannot transfer energy to other nodes with energy transferred weight ${{W}_{(~n,m~)}}<0$. Combined with 22, then ${{\widehat{E}}_{n}}(~t~)<\delta~(~\beta~V+{{R}_{\max~}}~)$. In addition, we observe it from 29 that the backlog of all data queue is no more than $\beta~V+{{R}_{\max~}}$, so the backlog weight of data stream over link $(~n,m~)$ satisfies $\widetilde{W}_{(n,m)}^{{}}(t)\le~\beta~V-{{D}_{\max~}}{{C}_{\max~}}$ at any time slot. Hence, the above problem can be simplified into \begin{align} G(P_{nm}^{*}(t))-G({{P}_{nm}}(t))&={{C}_{nm}}P_{nm}^{*}(t)\text{ }\widetilde{W}_{( n,m )}^{{}}(t)+{{\hat{E}}_{n}}( t )P_{nm}^{*}( t ) \\ &\le\left( \beta V-{{D}_{\max }}{{C}_{\max }} \right)\delta P_{nm}^{*}(t)-\delta \left( \beta V+{{R}_{\max }} \right)P_{nm}^{*}(t) \\ &=\delta P_{nm}^{*}(t)\left({{D}_{\max }}{{C}_{\max }}+{{R}_{\max }}\right) \\ &<0. \tag{45} \end{align} This implies that ${{{\boldsymbol~P}}^{*}}$ is not the optimal solution of the problem 44 since it conflicts with our assumption. The proof of the optimal transferred energy $\varepsilon_{nm}^{*}(t)$ can be performed similarly. Hence, $e_{n}^{{}}(t)\ge~{{P}_{\max~}}+{{\varepsilon}_{\max~}}$, namely 31, holds whether node $n$ transfers any non-negative energy or allocates any non-zero power over its subsequent nodes. What's more, it turns out that all the power allocation and energy transfer actions are feasible, and the “energy-availability" constraint is indeed redundant.

Therefore, part (a) of Theorem th2 is proved. And the proof of part (b) is similar to the proof of Theorem 5.1 in [26], so we omit it here for brevity.

### References

[1] Almalkawi I T, Guerrero Zapata M, Al-Karaki J N. Wireless Multimedia Sensor Networks: current trends and future directions.. Sensors, 2010, 10: 6662-6717 CrossRef PubMed Google Scholar

[2] Zhang Y, He Q, Xiang Y. Low-Cost and Confidentiality-Preserving Data Acquisition for Internet of Multimedia Things. IEEE Internet Things J, 2018, 5: 3442-3451 CrossRef Google Scholar

[3] Zhu B, Xie L, Han D. A survey on recent progress in control of swarm systems. Sci China Inf Sci, 2017, 60: 070201 CrossRef Google Scholar

[4] Zhang Y, Xiang Y, Zhang L Y. Secure Wireless Communications Based on Compressive Sensing: A Survey. IEEE Commun Surv Tutorials, 2019, 21: 1093-1111 CrossRef Google Scholar

[5] Yousaf R, Ahmad R, Ahmed W. A unified approach of energy and data cooperation in energy harvesting WSNs. Sci China Inf Sci, 2018, 61: 082303 CrossRef Google Scholar

[6] Alippi C, Galperti C. An Adaptive System for Optimal Solar Energy Harvesting in Wireless Sensor Network Nodes. IEEE Trans Circuits Syst I, 2008, 55: 1742-1750 CrossRef Google Scholar

[7] Yang H H, Lee J, Quek T Q S. Heterogeneous Cellular Network With Energy Harvesting-Based D2D Communication. IEEE Trans Wireless Commun, 2016, 15: 1406-1419 CrossRef Google Scholar

[8] Dhillon H, Li Y, Nuggehalli P. Fundamentals of heterogeneous cellular networks with energy harvesting. IEEE Trans Wireless Commun, 2014, 13: 2782-2797 CrossRef Google Scholar

[9] Vaze R. Transmission capacity of wireless ad hoc networks with energy harvesting nodes. In: Proceedings of Global Conference on Signal and Information Processing (GlobalSIP), Austin, 2013. 353--358. Google Scholar

[10] Michelusi N, Stamatiou K, Zorzi M. Transmission Policies for Energy Harvesting Sensors with Time-Correlated Energy Supply. IEEE Trans Commun, 2013, 61: 2988-3001 CrossRef Google Scholar

[11] Prabuchandran K J, Meena S K, Bhatnagar S. Q-Learning Based Energy Management Policies for a Single Sensor Node with Finite Buffer. IEEE Wireless Commun Lett, 2013, 2: 82-85 CrossRef Google Scholar

[12] Hsu R C, Liu C T, Wang H L. A Reinforcement Learning-Based ToD Provisioning Dynamic Power Management for Sustainable Operation of Energy Harvesting Wireless Sensor Node. IEEE Trans Emerg Top Comput, 2014, 2: 181-191 CrossRef Google Scholar

[13] Yu H, Neely M J. Learning aided optimization for energy harvesting devices with outdated state information. In: Proceedings of IEEE Conference on Computer Communications, 2018. 1853--1861. Google Scholar

[14] Neely M J. Super-fast delay tradeoffs for utility optimal fair scheduling in wireless networks. IEEE J Sel Areas Commun, 2006, 24: 1489-1501 CrossRef Google Scholar

[15] Huang L, Moeller S, Neely M J. LIFO-Backpressure Achieves Near-Optimal Utility-Delay Tradeoff. IEEE/ACM Trans Networking, 2013, 21: 831-844 CrossRef Google Scholar

[16] Huang L. Optimal Sleep-Wake Scheduling for Energy Harvesting Smart Mobile Devices. IEEE Trans Mobile Comput, 2017, 16: 1394-1407 CrossRef Google Scholar

[17] Neely M J. Stochastic Network Optimization with Application to Communication and Queueing Systems. Synthesis Lectures Communication Networks, 2010, 3: 1-211 CrossRef Google Scholar

[18] Eryilmaz A, Srikant R. Joint congestion control, routing, and MAC for stability and fairness in wireless networks. IEEE J Sel Areas Commun, 2006, 24: 1514-1524 CrossRef Google Scholar

[19] Liu J, Shroff N B, Xia C H. Joint Congestion Control and Routing Optimization: An Efficient Second-Order Distributed Approach. IEEE/ACM Trans Networking, 2016, 24: 1404-1420 CrossRef Google Scholar

[20] Le L B, Modiano E, Shroff N B. Optimal Control of Wireless Networks With Finite Buffers. IEEE/ACM Trans Networking, 2012, 20: 1316-1329 CrossRef Google Scholar

[21] Xu W Q, Zhang Y S, Shi Q J, et al. Dynamic optimization for heterogeneous powered wireless multimedia sensor networks with correlated sources and network coding. 2014,. arXiv Google Scholar

[22] You C, Huang K, Chae H. Energy Efficient Mobile Cloud Computing Powered by Wireless Energy Transfer. IEEE J Sel Areas Commun, 2016, 34: 1757-1771 CrossRef Google Scholar

[23] Bi S, Ho C K, Zhang R. Wireless powered communication: opportunities and challenges. IEEE Commun Mag, 2015, 53: 117-125 CrossRef Google Scholar

[24] Huang L, Neely M J. Utility Optimal Scheduling in Energy-Harvesting Networks. IEEE/ACM Trans Networking, 2013, 21: 1117-1130 CrossRef Google Scholar

[25] Gutiérrez M A, Steen K. Stochastic finite element methods. In: Encyclopedia of Computational Mechanics. Hoboken: Wiley, 2018. Google Scholar

[26] Tapparello C, Simeone O, Rossi M. Dynamic Compression-Transmission for Energy-Harvesting Multihop Networks With Correlated Sources. IEEE/ACM Trans Networking, 2014, 22: 1729-1741 CrossRef Google Scholar

• Figure 1

(Color online) (a) The energy harvesting and data receiving process of a source node; (b) the topology of a WSN.

•

Algorithm 1 EDPR algorithm with optimal utility at time slot $t$

Require:Initialize $V$ and obtain the corresponding ${\theta~}_n$ (${n~\in~\mathcal{N}}$) for nodes according to 22;

Output:Calculate the optimal amount of data transmission ${R_n^{(~d~)}(t)}$, the energy transfer ${{\varepsilon}_{nm}}(~t~)$, and the power allocation ${{P}_{nm}}(t)$.

Determine ${{h}_{n}}(t)$ for node $n$ through energy management;

Determine $R_{n}^{(~d~)}(t)$ according to 23 through data transmission;

if the link from node $n$ to node $m$ exists then

Calculate the weight ${{W}_{n}}(t)$ and $~{{\widetilde{W}}_{(~n,m~)}}(t)$ according to 26 and 27 through power control and routing scheduling;

end if

Obtain ${{\varepsilon}_{nm}(t)}$, $P_{nm}(t)$, $\lambda~_{nm}^{{d}}(~t~)$ based on the value of $e_{n}(t)$, $\theta_{n}$, $W_{n}(t)$;

Update $q_{n}^{(~d~)}(t)$ and $e_n(t)$ according to 9 and 12.

• #### 0

Citations

• Altmetric

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有