{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:59:51Z","timestamp":1760234391674,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,17]],"date-time":"2021-05-17T00:00:00Z","timestamp":1621209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, to maximize the energy efficiency (EE) in the two-hop multi-relay cooperative decoding and forwarding (DF) system for simultaneous wireless information and power transmission (SWIPT), an optimal power allocation algorithm is proposed, in which the relay energy harvesting (EH) adopts a nonlinear model. Under the constraints, including energy causality, the minimum transmission quality of information and the total transmission power at the relays, an optimization problem is constructed to jointly optimize the transmit power and power-splitting (PS) ratios of multiple relays. Although this problem is a nonlinear fractional programming problem, an iterative algorithm is developed to obtain the optimal power allocation. In particular, the joint power allocation at multiple relays is first decoupled into a single relay power allocation, and then single-relay power allocation is performed by the Dinkelbach iteration algorithm, which can be proven that it is a convex programming problem. Its closed form solutions for different polylines of EH models are obtained by using mathematical methods, such as monotonicity, Lagrange multipliers, the KKT condition and the Cardan formula. The simulation results show the superiority of the power allocation algorithm proposed in this paper in terms of EE.<\/jats:p>","DOI":"10.3390\/a14050155","type":"journal-article","created":{"date-parts":[[2021,5,17]],"date-time":"2021-05-17T05:45:13Z","timestamp":1621230313000},"page":"155","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Energy-Efficient Power Allocation in Non-Linear Energy Harvesting Multiple Relay Systems"],"prefix":"10.3390","volume":"14","author":[{"given":"Huifang","family":"Pan","sequence":"first","affiliation":[{"name":"Jiangsu Key Laboratory of Wireless Communications, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}]},{"given":"Qi","family":"Zhu","sequence":"additional","affiliation":[{"name":"Engineering Research Center of Health Service System Based on Ubiquitous Wireless Networks, Nanjing University of Posts and Telecommunications, Nanjing 210003, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,17]]},"reference":[{"key":"ref_1","unstructured":"Fettweis, G., and Zimmermann, E. (2008, January 27\u201329). ICT energy consumption-trends and challenges. Proceedings of the 11th International Symposium on Wireless Personal Multimedia Communications, Lisbon, Portugal."},{"key":"ref_2","first-page":"1","article-title":"A Survey of Green Networking Research","volume":"99","author":"Bianzino","year":"2010","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/MPRV.2005.12","article-title":"Energy Harvesting and Conservation","volume":"4","author":"Want","year":"2005","journal-title":"IEEE Pervasive Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4971","DOI":"10.1109\/TWC.2018.2834528","article-title":"Toward Optimal Power Control and Transfer for Energy Harvesting Amplify-and-Forward Relay Networks","volume":"17","author":"Singh","year":"2018","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ozel, O., Tutuncuoglu, K., Yang, J., Ulukus, S., and Yener, A. (2011, January 10\u201315). Resource management for fading wireless channels with energy harvesting nodes. Proceedings of the IEEE Infocom, Shanghai, China.","DOI":"10.1109\/INFCOM.2011.5935203"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Gurrala, K.K., Navya, K., Sravya, M., and Ramakrishna, E. (2019, January 22\u201324). Maximized Energy Efficiency Based Power Allocation Strategy in Wireless Cooperative Network. Proceedings of the 2019 TEQIP III Sponsored International Conference on Microwave Integrated Circuits, Photonics and Wireless Networks (IMICPW), Tiruchirappalli, Tamil Nadu, India.","DOI":"10.1109\/IMICPW.2019.8933221"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3640","DOI":"10.1109\/ACCESS.2017.2670641","article-title":"Energy-Efficient Power Allocation in Energy Harvesting Two-Way AF Relay Systems","volume":"5","author":"Zhang","year":"2017","journal-title":"IEEE Access"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Orhan, O., and Erkip, E. (2012, January 21\u201323). Optimal Transmission Policies for Energy Harvesting Two-hop Networks. Proceedings of the Information Sciences & Systems, Princeton, NJ, USA.","DOI":"10.1109\/CISS.2012.6310831"},{"key":"ref_9","first-page":"874","article-title":"AF Relaying with Energy Harvesting Source and Relay","volume":"66","author":"Chen","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"6352","DOI":"10.1109\/TWC.2013.103113.130470","article-title":"Wireless information and power transfer: Energy efficiency optimization in OFDMA systems","volume":"12","author":"Ng","year":"2013","journal-title":"IEEE Trans. Wireless Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2082","DOI":"10.1109\/LCOMM.2015.2478460","article-title":"Practical Non-Linear Energy Harvesting Model and Resource Allocation for SWIPT Systems","volume":"19","author":"Boshkovska","year":"2015","journal-title":"IEEE Commun. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1109\/LSP.2019.2906463","article-title":"Energy Efficiency Maximization for SWIPT Enabled Two-Way DF Relaying","volume":"26","author":"Shi","year":"2019","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"26617","DOI":"10.1109\/ACCESS.2018.2834225","article-title":"Maximum Throughput of TS\/PS Scheme in an AF Relaying Network with Non-Linear Energy Harvester","volume":"6","author":"Lu","year":"2018","journal-title":"IEEE Access"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"972","DOI":"10.1109\/TVT.2015.2402274","article-title":"Outage Probability of Energy Harvesting Relay-Aided Cooperative Networks Over Rayleigh Fading Channel","volume":"65","author":"Li","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Shi, L., Zhao, L., Liang, K., Chu, X., Wu, G., and Chen, H. (2017, January 21\u201325). Profit maximization in wireless powered communications with improved non-linear energy conversion and storage efficiencies. Proceedings of the 2017 IEEE International Conference on Communications (ICC), Paris, France.","DOI":"10.1109\/ICC.2017.7997317"},{"key":"ref_16","unstructured":"Guo, J., and Zhu, X. (2012, January 17\u201322). An improved analytical model for RF-DC conversion efficiency in microwave rectifiers. Proceedings of the 2012 IEEE\/MTT-S International Microwave Symposium Digest, Montreal, QC, Canada."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Grover, P., and Sahai, A. (2010, January 13\u201318). Shannon meets Tesla: Wireless information and power transfer. Proceedings of the IEEE International Symposium on Information Theory Proceedings, Austin, TX, USA.","DOI":"10.1109\/ISIT.2010.5513714"},{"key":"ref_18","first-page":"1","article-title":"Adaptive Relay Selection and Power Allocation for OFDM Cooperative Underwater Acoustic Systems","volume":"99","author":"Ebrahimzadeh","year":"2017","journal-title":"IEEE Trans. Mobile Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1287\/mnsc.13.7.492","article-title":"On nonlinear fractional programming","volume":"13","author":"Dinkelbach","year":"1967","journal-title":"Manag. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1109\/TGCN.2019.2906616","article-title":"Time-Switching EH-Based Joint Relay Selection and Resource Allocation Algorithms for Multi-User Multi-Carrier AF Relay Networks","volume":"3","author":"Gupta","year":"2019","journal-title":"IEEE Trans. Green Commun. Netw."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Li, C., Wu, P., and Xia, M. (May, January 28). Energy Efficiency Maximization of AF Relaying SWIPT Systems with Energy Recycling. Proceedings of the 2019 IEEE 89th Vehicular Technology Conference (VTC2019-Spring), Kuala Lumpur, Malaysia.","DOI":"10.1109\/VTCSpring.2019.8746645"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/5\/155\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:02:31Z","timestamp":1760162551000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/14\/5\/155"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,17]]},"references-count":21,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["a14050155"],"URL":"https:\/\/doi.org\/10.3390\/a14050155","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2021,5,17]]}}}