## Abstract

In this paper, we generalize conventional time division multiple access (TDMA) wireless networks to a new type of wireless networks coined generalized wireless powered communication networks (g-WPCNs). Our prime objective is to optimize the design of g-WPCNs where nodes are equipped with radio frequency (RF) energy harvesting circuitries along with constant energy supplies. This constitutes an important step towards a generalized optimization framework for more realistic systems, beyond prior studies where nodes are solely powered by the inherently limited RF energy harvesting. Towards this objective, we formulate two optimization problems with different objective functions, namely, maximizing the sum throughput and maximizing the minimum throughput (maxmin) to address fairness. First, we study the sum throughput maximization problem, investigate its complexity and solve it efficiently using an algorithm based on alternating optimization approach. Afterwards, we shift our attention to the maxmin optimization problem to improve the fairness limitations associated with the sum throughput maximization problem. The proposed problem is generalized, compared to prior work, as it seemlessly lends itself to prior formulations in the literature as special cases representing extreme scenarios, namely, conventional TDMA wireless networks (no RF energy harvesting) and standard WPCNs, with only RF energy harvesting nodes. In addition, the generalized formulation encompasses a scenario of practical interest we introduce, namely, WPCNs with two types of nodes (with and without RF energy harvesting capability) where legacy nodes without RF energy harvesting can be utilized to enhance the system sum throughput, even beyond WPCNs with all RF energy harvesting nodes studied earlier in the literature. We establish the convexity of all formulated problems which opens room for efficient solution using standard techniques. Our numerical results show that conventional TDMA wireless networks and WPCNs with only RF energy harvesting nodes are considered as lower bounds on the performance of the generalized problem setting in terms of the maximum sum throughput and maxmin throughput. Moreover, the results reveal valuable insights and throughput-fairness trade-offs unique to our new problem setting.

This is a preview of subscription content, log in to check access.

## Notes

- 1.
Note that slot time allocations are assumed to take continuous values. This, in turn, requires accurate synchronization methods to implement such scheme in realistic systems.

- 2.
The assumption that CSI is perfectly pre-estimated at the BS in the beginning of each slot is an idealization of actual practical systems. This calls for the necessity of using estimators with high accuracy to sufficiently reduce the potential estimation errors.

- 3.

## References

- 1.
Ho, C. K., & Zhang, R. (2012). Optimal energy allocation for wireless communications with energy harvesting constraints.

*IEEE Transactions on Signal Processing*,*60*(9), 4808–4818. - 2.
Ozel, O., Tutuncuoglu, K., Yang, J., Ulukus, S., & Yener, A. (2011). Transmission with energy harvesting nodes in fading wireless channels: Optimal policies.

*IEEE Journal on Selected Areas in Communications*,*29*(8), 1732–1743. - 3.
Paradiso, J., Starner, T., et al. (2005). Energy scavenging for mobile and wireless electronics.

*IEEE Pervasive Computing*,*4*(1), 18–27. - 4.
Rabaey, J. M., Ammer, M. J., da Silva, J. L., Patel, D., & Roundy, S. (2000). Picoradio supports ad hoc ultra-low power wireless networking.

*Computer*,*33*(7), 42–48. - 5.
Raghunathan, V., Schurgers, C., Park, S., & Srivastava, M. B. (2002). Energy-aware wireless microsensor networks.

*IEEE Signal Processing Magazine*,*19*(2), 40–50. - 6.
Roundy, S. J. (2003).

*Energy scavenging for wireless sensor nodes with a focus on vibration to electricity conversion*, Ph.D. dissertation, University of California, Berkeley. - 7.
Zhao, N., Yu, F. R., & Leung, V. C. (2015). Opportunistic communications in interference alignment networks with wireless power transfer.

*IEEE Wireless Communications*,*22*, 88–95. - 8.
Zhao, N., Yu, F. R., & Leung, V. C. (2015). Wireless energy harvesting in interference alignment networks.

*IEEE Communications Magazine*,*53*, 72–78. - 9.
Guo, J., Zhao, N., Yu, F. R., Liu, X., & Leung, V. C. (2017). Exploiting adversarial jamming signals for energy harvesting in interference networks.

*IEEE Transactions on Wireless Communications*,*16*, 1267–1280. - 10.
Zhao, N., Zhang, S., Yu, R., Chen, Y., Nallanathan, A., & Leung, V. (2017). Exploiting interference for energy harvesting: A survey, research issues and challenges.

*IEEE Access*,*5*, 10403–10421. - 11.
Varshney, L. R. (2008). Transporting information and energy simultaneously. In

*IEEE international symposium on information theory*(pp. 1612–1616). - 12.
Grover, P., & Sahai, A. (2010). Shannon meets Tesla: Wireless information and power transfer. In

*IEEE international symposium on information theory*(pp. 2363–2367). - 13.
Huang, K., & Larsson, E. (2013). Simultaneous information and power transfer for broadband wireless systems.

*IEEE Transactions on Signal Processing*,*61*(23), 5972–5986. - 14.
Zhang, R., & Ho, C. K. (2013). Mimo broadcasting for simultaneous wireless information and power transfer.

*IEEE Transactions on Wireless Communications*,*12*(5), 1989–2001. - 15.
Huang, K., & Larsson, E. (2013). Simultaneous information-and-power transfer for broadband downlink systems. In

*IEEE acoustics speech and signal processing conference*(pp. 4444–4448). - 16.
Zhou, X., Zhang, R., & Ho, C. K. (2013). Wireless information and power transfer: Architecture design and rate-energy tradeoff.

*IEEE Transactions on Communications*,*61*(11), 4754–4767. - 17.
Park, S., Heo, J., Kim, B., Chung, W., Wang, H., & Hong, D. (2012). Optimal mode selection for cognitive radio sensor networks with RF energy harvesting. In

*IEEE international symposium on personal indoor and mobile radio communications (PIMRC)*. IEEE (pp. 2155–2159). - 18.
Park, S., Kim, H., & Hong, D. (2013). Cognitive radio networks with energy harvesting.

*IEEE Transactions on Wireless Communications*,*12*(3), 1386–1397. - 19.
Lee, S., Zhang, R., & Huang, K. (2013). Opportunistic wireless energy harvesting in cognitive radio networks.

*IEEE Transactions on Wireless Communications*,*12*(9), 4788–4799. - 20.
Huang, K., & Lau, V. K. (2014). Enabling wireless power transfer in cellular networks: Architecture, modeling and deployment.

*IEEE Transactions on Wireless Communications*,*13*(2), 902–912. - 21.
Xie, L., Shi, Y., Hou, Y. T., Lou, W., Sherali, H. D., & Midkiff, S. F. (2012). On renewable sensor networks with wireless energy transfer: The multi-node case. In

*IEEE communications society conference on sensor, mesh and ad hoc communications and networks*. IEEE (pp. 10–18). - 22.
Ko, S.-W., Yu, S. M., & Kim, S.-L. (2013). The capacity of energy-constrained mobile networks with wireless power transfer.

*IEEE Communications Letters*,*17*(3), 529–532. - 23.
Doost, R., Chowdhury, K. R., & Felice, M. D. (2010). Routing and link layer protocol design for sensor networks with wireless energy transfer. In

*IEEE global telecommunications conference (GLOBECOM)*. IEEE (pp. 1–5). - 24.
Nintanavongsa, P., Naderi, M. Y., & Chowdhury, K. R. (2013). Medium access control protocol design for sensors powered by wireless energy transfer. In

*IEEE INFOCOM*. IEEE (pp. 150–154). - 25.
Ju, H., & Zhang, R. (2014). Throughput maximization in wireless powered communication networks.

*IEEE Transactions on Wireless Communications*,*13*(1), 418–428. - 26.
Ju, H., & Zhang, R. (2014). User cooperation in wireless powered communication networks. In

*IEEE global telecommunications conference (GLOBECOM)*(pp. 1430–1435). - 27.
Ju, H., & Zhang, R. (2014). Optimal resource allocation in full-duplex wireless-powered communication network.

*IEEE Transactions on Communications*,*62*(10), 3528–3540. - 28.
Kang, X., Ho, C. K., & Sun, S. (2015). Full-duplex wireless-powered communication network with energy causality.

*IEEE Transactions on Wireless Communications*,*14*(10), 5539–5551. - 29.
Lee, S., & Zhang, R. (2015). Cognitive wireless powered network: Spectrum sharing models and throughput maximization.

*IEEE Transactions on Cognitive Communications and Networking*,*PP*(99), 1–11. - 30.
Bi, S., & Zhang, R. (2015). Node placement optimization in wireless powered communication networks. ArXiv preprint arXiv:1505.06530.

- 31.
Abd-Elmagid, M. A., ElBatt, T., & Seddik, K. G. (2015). Optimization of wireless powered communication networks with heterogeneous nodes. In

*IEEE global communications conference (GLOBECOM)*. - 32.
Knopp, R., & Humblet, P. A. (1995). Information capacity and power control in single-cell multiuser communications. In

*IEEE international conference on communications*(Vol. 1, pp. 331–335). - 33.
Li, L., & Goldsmith, A. J. (2001). Capacity and optimal resource allocation for fading broadcast channels. I. Ergodic capacity.

*IEEE Transactions on Information Theory*,*47*, 1083–1102. - 34.
Tse, D. N., & Hanly, S. V. (1998). Multiaccess fading channels. I. Polymatroid structure, optimal resource allocation and throughput capacities.

*IEEE Transactions on Information Theory*,*44*(7), 2796–2815. - 35.
Nintanavongsa, P., Muncuk, U., Lewis, D. R., & Chowdhury, K. R. (2012). Design optimization and implementation for rf energy harvesting circuits.

*IEEE Journal on Emerging and Selected Topics in Circuits and Systems*,*2*(1), 24–33. - 36.
Roberg, M., Reveyrand, T., Ramos, I., Falkenstein, E. A., & Popovic, Z. (2012). High-efficiency harmonically terminated diode and transistor rectifiers.

*IEEE Transactions on Microwave Theory and Techniques*,*60*(12), 4043–4052. - 37.
Boyd, S., & Vandenberghe, L. (2004).

*Convex optimization*. Cambridge: Cambridge University Press. - 38.
Boyd, S. (2011). Alternating direction method of multipliers. In

*Talk at NIPS workshop on optimization and machine learning*. - 39.
Jain, R., Chiu, D.-M., & Hawe, W. (1984). A quantitative measure of fairness and discrimination for resource allocation in shared computer systems.

*DEC Research Report, Technical Report TR-301*. - 40.
Julian, D., Chiang, M., O’Neill, D., & Boyd, S. (2002). QoS and fairness constrained convex optimization of resource allocation for wireless cellular and ad hoc networks. In

*IEEE INFOCOM*(pp. 477–486).

## Author information

### Affiliations

### Corresponding author

## Appendices

### Appendix A

Thanks to the fact that the perspective function of a concave function is also a concave function [37]. \(\tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right)\) is the perspective function of the concave function \(\log _{2} \left( 1 + \alpha _{i} E_{i}\right)\) which preserves the concavity of \(R_{i}\) with respect to \((E_{i},\tau _{i})\). Since the non-negative weighted sum of concave functions is also concave [37], then the objective function of \(\mathbf P1\), which is the non-negative weighted summation of concave functions, i.e., \(R_{i}\) for \(i=1, \ldots ,K\), is a concave function in \((\mathbf {E},\pmb {\tau })\). In addition, all constraints of \(\mathbf P1\) are affine in \((\mathbf {E},\pmb {\tau })\). This establishes the proof.

### Appendix B

For a given \(\mathbf {E}\) that satisfies (5) and \(0 \le E_{i} < E^b_{i} + \eta _{i} P_{B} h_{i}, i=1, \ldots ,K\), **P1** reduces as follows.

It can be easily shown that \(R_{i} = \tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right)\) is a monotonically increasing function in \((E_{i}, \tau _{i})\) [25, Lemma 3.2], \(i=1,\ldots ,K\). Therefore, the constraint in (28) should hold with equality at the optimality (otherwise, the objective function can be further increased by increasing some \(\tau _{i}\)’s). Hence, from (30), the optimal harvesting time duration is given by

Hence, \(\mathbf P1 ^{\prime }\) reduces to

Recall that \(\pmb {\tau ^{\prime }}=[\tau _{1}, \ldots ,\tau _{K}]\). Based on Theorem 1, \(\mathbf P1 ^{\prime \prime }\) is a convex optimization problem and its Lagrangian is given by

where \(R_{sum}\left( \pmb {\tau ^{\prime }}\right) = \sum _{i=1}^{K}{\tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right) }\) and \(\mu\) is the Lagrangian dual variable associated with the total slot duration constraint (32). It can be easily shown that there exists a \(\pmb {\tau ^{\prime }}\) that strictly satisfies all constraints of \(\mathbf P1 ^{\prime \prime }\). Hence, according to Slater’s condition [37], strong duality holds for this problem; therefore, the KKT conditions are necessary and sufficient for the global optimality of \(\mathbf P1 ^{\prime \prime }\), which are given by

where \(\pmb {\tau ^{\prime *}}\) and \(\mu ^{*}\) denote, respectively, the optimal primal and dual solutions of \(\mathbf P1 ^{\prime \prime }\). Therefore, from (35) and (36), we have

Thus from (37), the optimal time allocations are given by

This establishes the proof.

### Appendix C

For a given \(\pmb {\tau }\) that satisfies (6)–(8), **P1** reduces as follows.

Recall that \(R_{i} = \tau _{i} \log _{2} \left( 1 + \alpha _{i} \dfrac{E_{i}}{\tau _{i}}\right)\) is a monotonically increasing function in \((E_{i}, \tau _{i})\), \(i=1,\ldots ,K\). Therefore, when \(E_{max} \ge \sum _{j = 1}^{K}{\left( E^b_{j} + \eta _{j} P_{B} h_{j} \tau _{0}\right) }\), \(\mathbf P1 ^{\dagger }\) has a trivial solution that \(E_{i}^{*} = E^b_{i} + \eta _{i} P_{B} h_{i} \tau _{0}\), \(i=1, \ldots ,K\). On the other hand, when \(E_{max} < \sum _{j = 1}^{K}{\left( E^b_{j} + \eta _{j} P_{B} h_{j} \tau _{0}\right) }\), the optimal solution of \(\mathbf P1 ^{\dagger }\) can be characterized as follows. First, the constraint in (39) should hold with equality at the optimality (otherwise, the objective function can be further increased by increasing some \(E_{i}\)’s). Based on Theorem 1, \(\mathbf P1 ^{\dagger }\) is a convex optimization problem and its Lagrangian is given by

where \(\lambda\) is the Lagrangian dual variable associated with the total allowable consumed energy per slot constraint (39). The strong duality holds for \(\mathbf P1 ^{\dagger }\); therefore, the KKT conditions are necessary and sufficient for the global optimality of \(\mathbf P1 ^{\dagger }\), which are given by

where \(\mathbf {E}^{*}\) and \(\lambda ^{*}\) denote, respectively, the optimal primal and dual solutions of \(\mathbf P1 ^{\dagger }\). Therefore, from (42), we have

Taking into account the constraints in (40), the optimal energy allocations are given by

where \(\lambda ^{*}\) satisfies the equality constraint \(\sum _{i=1}^{K}{E_{i}^{*}} = E_{max}\). This establishes the proof.

### Appendix D

**P4** is a convex optimization problem and its Lagrangian is given by

where \(\mu\) and \(\lambda\) are the Lagrangian dual variables associated with the slot duration and the total allowable consumed energy per slot constraints, respectively, and \(R_{sum}\left( \pmb {\tau ^{\prime \prime }},\bar{E}\right) = \sum _{i=1}^{M}{R_{1,i}\left( \tau _{0},\tau _{1,i}\right) } + \sum _{j=1}^{N}{R_{2,j}\left( \bar{E},\tau _{2,j}\right) }\). Hence, the dual function can be expressed as

where \(\mathcal {S}\) is the feasible set specified by \(\pmb {\tau ^{\prime \prime }} \succeq \mathbf {0}\) and \(\bar{E} \ge 0\). It can be easily shown that there exists a \((\pmb {\tau ^{\prime \prime }},\bar{E})\) that strictly satisfies all constraints of **P4**. Hence, according to Slater’s condition [37], strong duality holds for this problem; therefore, the KKT conditions are necessary and sufficient for the global optimality of **P4**, which are given by

where \(\left( \pmb {\tau ^{\prime \prime *}},\bar{E}^{*}\right)\) and \(\left( \lambda ^{*},\mu ^{*}\right)\) denote, respectively, the optimal primal and dual solutions of **P4**. Since \(R_{sum}\left( \pmb {\tau ^{\prime \prime }},\bar{E}\right)\) is a monotonic increasing function in \(\left( \pmb {\tau ^{\prime \prime }},\bar{E}\right)\), therefore \(\tau _{0}^{*}+\sum _{i=1}^{M}{\tau _{1,i}^{*}}+\sum _{j=1}^{N}{\tau _{2,j}^{*}} = 1\) and \(a\tau _{0}^{*} + N \bar{E}^{*} = E_{max}\) must hold. From (52)–(55), we have

Therefore, from (57) and (58), we have

From \(\tau _{0}^{*}+\sum _{i=1}^{M}{\tau _{1,i}^{*}}+\sum _{j=1}^{N}{\tau _{2,j}^{*}} = 1\) and (60), \(\tau _{1,i}^{*}\) and \(\tau _{2,j}^{*}\) can be expressed, respectively, by

where \(A_{1} = \sum _{i=1}^{M}{\gamma _{i}}\) and \(A_{2} = \sum _{j=1}^{N}{\theta _{j}}\). From (56) and (59), it follows that

By substituting with \(\mu ^{*}\) into (57), we have

From (61) and (62), it is clear that \(x_{1} > 1\) if \(A_{1} > 0\), \(A_{2} > 0\) and \(0< \tau _{0}^{*} < 1\). According to [25, Lemma 3.2], there exists a unique solution \(x_{1}^{*} > 1\) for (65) if \(A_{1} \ge \dfrac{a}{N} A_{2}\), otherwise the total slot time and the total allowable consumed energy per slot will be assigned to the Type II nodes for uplink information transmissions. Thus from (60)–(65), the optimal time and energy allocations are given by

From (66)–(69), for \([\tau _{0}^{*}, \tau _{1,1}^{*}, \ldots ,\tau _{1,M}^{*}, \tau _{2,1}^{*}, \ldots ,\tau _{2,N}^{*}, \bar{E}^{*}] \succeq \mathbf {0}\), we must have \(\dfrac{a(x_{1}^{*} - 1)}{A_{1} + x_{1}^{*} - 1} \le E_{max} \le \dfrac{N}{A_{2}}(x_{1}^{*} - 1)\). If \(E_{max} > \dfrac{N}{A_{2}}(x_{1}^{*} - 1)\), then we have \([\tau _{0}^{*}, \tau _{1,1}^{*}, \ldots ,\tau _{1,M}^{*}] \prec \mathbf {0}\). Hence, the total slot time and the total allowable consumed energy per slot will be assigned to the Type II nodes for uplink information transmissions. Therefore, from (50) and (58), the optimal time and energy allocations are given by (21)–(24).

On the other hand, if \(E_{max} < \dfrac{a(x_{1}^{*} - 1)}{A_{1} + x_{1}^{*} - 1}\), then we have \([\tau _{2,1}^{*}, \ldots ,\tau _{2,N}^{*}, \bar{E}^{*}] \prec \mathbf {0}\). Hence, the total slot time and the total allowable consumed energy per slot will be assigned to the Type I nodes for uplink information transmissions. Therefore, from (50), (56) and (57), the optimal time and energy allocations are given by (21)–(24). This establishes the proof.

## Rights and permissions

## About this article

### Cite this article

Abd-Elmagid, M.A., ElBatt, T. & Seddik, K.G. Optimization of energy-constrained wireless powered communication networks with heterogeneous nodes.
*Wireless Netw* **25, **713–730 (2019). https://doi.org/10.1007/s11276-017-1587-x

Published:

Issue Date:

### Keywords

- Cellular networks
- Green communications
- RF energy harvesting
- Convex optimization
- Numerical results