Optimization of multi-user scheduling in WPCNs

Abstract

The rapid advancement of Internet of Things (IoT) and sensor networks has led to the emergence of Wireless Powered Communication Networks (WPCNs), offering significant potential through wireless energy transfer for energy-constrained devices. However, WPCNs face new challenges in multi-user scheduling due to factors like energy collection efficiency and resource allocation, which are crucial for optimizing network performance. This paper addresses the multi-user scheduling challenge in WPCNs, focusing on optimizing network performance by maximizing total weighted throughput and minimizing energy consumption. We propose a network model considering non-linear energy collection efficiency and employ a Lagrange multiplier algorithm to balance energy consumption and data transmission. MATLAB simulations demonstrate that our algorithm outperforms traditional methods, reducing total energy consumption by 25% while increasing network throughput by 15%. The findings provide theoretical and practical insights for WPCNs optimization and deployment. Future research will explore algorithm optimization, practical deployment strategies, and network security to advance wireless power network technology.

Introduction

The proliferation of the Internet of Things (IoT) and the surge in the number of smart devices have thrust Wireless Powered Communication Networks (WPCNs) into the spotlight as a pivotal technology. WPCNs provide energy to energy-limited devices via wireless energy transfer, ensuring continuous communication and functionality, opening new avenues for applications in smart cities, environmental monitoring, health care, and more. WPCNs use microwave wireless power transfer to recharge devices, enhancing performance over traditional networks by increasing throughput and device lifetime^1,2,3. For instance, Bi et al.¹ reviewed WPCN’s key structures and techniques while highlighting challenges in long-distance power transfer efficiency and future research directions. Jafari et al.² examined the sum-throughput of secondary backscatter sensors in an in-band full-duplex cognitive wireless powered network, aiming to maximize throughput through joint optimal time and energy allocation. Poposka et al.³ explored federated learning in wireless powered networks, ensuring data privacy and energy sustainability for devices aiming to minimize training duration by optimizing communication and computational resources, and learning parameters, offering a low-complexity solution. However, the multi-user scheduling issue within WPCNs is a complex optimization challenge that involves factors such as energy collection efficiency, user energy requirements, and the allocation of communication resources, which are crucial for the optimization of network performance^4,5,6. For instance, Zhao et al.⁴ explored multi-user scheduling for extended reality (XR) in 6G networks, focusing on periodic traffic models with strict latency constraints, and proposed an online scheduling policy that solves nonlinear Knapsack Problems for optimal scheduling, demonstrating near-optimal performance and outperforming benchmarks. Raut et al.⁵ explored a mobile multi-user full-duplex two-way relaying system with decode-and-forward protocol, analyzing the impact of residual self-interference and mobility on scheduling schemes, evaluating error probability and average rate for various scheduling strategies and investigates factors affecting system performance through numerical results and Monte Carlo simulations. Ma et al.⁶ addressed the challenge of real-time data processing in the Internet of Vehicles by proposing an edge computing-based multi-task scheduling algorithm, which aimed to maximize task completion rates while minimizing runtime, effectively handling large-scale networks and numerous task requests.

Wireless energy provisioning technologies supply energy to devices in the form of electromagnetic waves, including Radio Frequency (RF) and laser energy transfer. The RF energy transfer has been extensively studied and favored for its long transmission range and cost-effectiveness^7,8,9. In WPCNs, the challenge of minimizing network energy consumption while maximizing network throughput, given the prerequisite of meeting user energy demands, is non-trivial. The multi-user scheduling problem has emerged as a hot topic in the field of wireless communications and is particularly pronounced in WPCNs^10,11,12,13. It is a critical issue in wireless communication networks that involves the allocation of limited resources among multiple users to enhance the overall network performance. In WPCNs, multi-user scheduling must take into account the characteristics of energy collection and consumption to optimize energy usage and throughput^14,15,16. Energy consumption and throughput are two key performance indicators for wireless communication networks. Minimizing energy consumption implies a greener and more sustainable network, while maximizing throughput is related to the network’s transmission efficiency. A trade-off exists between energy consumption and throughput in WPCNs, which necessitates a well-designed scheduling strategy to achieve balance^{17,18,19,20,21}.

This paper addresses a WPCN with multiple energy-limited users and multiple base stations, considering the non-linear energy collection efficiency. The base stations provide energy to user devices through wireless energy transfer, which then use the collected energy for data transmission. The goal of this paper is to minimize the total network energy consumption and maximize the total network throughput, under the premise of meeting user energy demands. This is a multi-objective optimization problem, which this paper solves using the Lagrange multiplier algorithm. This algorithm can effectively balance the relationship between energy consumption and data transmission to achieve dual optimization goals of energy consumption and throughput²¹.

The structure of the remainder of this paper is as follows. “System model and optimization algorithm design” section provides a detailed introduction to the construction of the WPCN model and the design concept of the optimization algorithm. “Simulation analysis and algorithm performance evaluation” section MATLAB demonstrates the simulation results and performance comparison of the algorithm. “Conclusions and future work” section summarizes the research findings and proposing future research directions.

System model and optimization algorithm design

System model

Figure 1 illustrates a simplified complex architecture diagram of a WPCN. The base stations (BS) are marked with red squares and their positions are randomly generated. User devices are represented by blue circles, and their positions are also randomly generated. Arrows are drawn between each user device and a randomly selected BS, indicating the path of energy and information transfer. The direction of the arrows is from the BS to the user devices, colored green, indicating that one BS can provide energy transfer and data reception to multiple users. The circle represents the coverage range of the base station, and the points on the circumference are calculated by a parametric equation, representing the area covered by the base station signal. Each user receives wireless energy from the BS, and the energy collection efficiency is non-linear, which can be represented by a function $\eta \left( {P_{{{\text{rx}},i}} } \right)$, where $P_{{{\text{rx}},i}}$ is the power received by the i-th user. Each user uses the collected energy for data transmission, and its throughput can be calculated by the Shannon-Hartley theorem, expressed as

$$R_{i} = W\log_{2} \left( {1 + {\text{SNR}}_{i} } \right),$$

(1)

where $W$ is the channel bandwidth and ${\text{SNR}}_{i}$ is the signal-to-noise ratio of the i-th user.

Fig. 1

System model.

The relationship between non-linear energy collection efficiency and received power is shown in Fig. 2. The X-axis represents the received power $P_{incident}$, which is the power transmitted by the base station to the user device, in watts (W), with a range from 0 to 10 watts¹. The Y-axis represents the energy collection efficiency $E_{collected\;values}$, which is the efficiency of the user device converting the received power into usable energy, expressed as a percentage. In Fig. 2, the energy collection efficiency is calculated using a piece-wise linear model, meaning the relationship between energy collection efficiency and received power is non-linear, consisting of two linear segments: First, when the received power $P_{incident}$ is less than the threshold (this is 5 watts⁴ here), the efficiency increases with a smaller coefficient $k{}_{1}$ (which is 0.1 here). Second, when the received power $P_{incident}$ is greater than or equal to the threshold, the efficiency increases with a larger coefficient $k{}_{2}$ (set to 0.5⁷ here, but adjusted to ensure the efficiency does not exceed 100%). The curve shape in Fig. 2 reflects the characteristics of the piece-wise linear model. In the low-power area, the slope of the curve is smaller, indicating that the energy collection efficiency increases more slowly; when the received power exceeds the threshold, the slope of the curve increases, indicating that the energy collection efficiency accelerates. Figure 2 illustrates how non-linear energy collection efficiency varies at different received power levels. This relationship is crucial for understanding and designing energy collection strategies in wireless powered networks.

Fig. 2

The relationship between non-linear energy harvesting efficiency and received power.

Continuing from where we left off, here is the translation for the next sections of the document, focusing on the optimization problem and the solution approach using the Lagrange multiplier method:

Optimization problem

The objective of this paper is to maximize the weighted sum throughput of all users while minimizing the total energy consumption. This approach allows us to account for the varying importance and priority of different users within the network. The weights assigned to each user’s throughput reflect their priority or the criticality of their data transmission requirements. These weights can be determined based on user priority, service level agreements (SLAs), or network conditions. By maximizing the weighted sum throughput, we aim to balance the overall network efficiency with the specific needs of individual users, ensuring that the network resources are allocated fairly and effectively. The objective function, which incorporates these weights, can be represented as follows:

$$\begin{gathered} {\text{Maximize}}\;\;\;\;\sum\limits_{i = 1}^{n} {w_{i} } R_{i} \hfill \\ {\text{Subject to}}\;\;\;\sum\limits_{i = 1}^{n} {P_{{{\text{tx}},i}} } \le P_{{{\text{total}}}} , \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;E_{{{\text{collected}},i}} \ge \eta \left( {P_{{{\text{rx}},i}} } \right), \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0 \le P_{{{\text{tx}},i}} \le P_{{{\text{max}},i}} , \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{SNR}}_{i} \ge {\text{SNR}}_{{{\text{min}}}} , \hfill \\ \end{gathered}$$

(2)

where $w_{i}$ is the weight assigned to the throughput of the i-th user, and $R_{i}$ is the throughput of the i-th user, as shown in Eq. (1). The first constraint represents the energy consumption constraint, stipulating that the total energy expenditure of the network must not exceed a predetermined budget. This constraint is defined by the transmission power of each user, denoted as $P_{{{\text{tx}},i}}$, and the total energy budget of the network, denoted as $P_{{{\text{total}}}}$. The second constraint is the energy collection constraint, ensuring that the energy collected by each user is at least equal to the energy obtained by converting the received power with a non-linear efficiency. The energy collected by the i-th user is denoted as $E_{{{\text{collected}},i}}$, $P_{{{\text{rx}},i}}$ is the received power of each user, and $\eta (P_{{{\text{rx}},i}} )$ is the conversion efficiency of collected energy. The third constraint is the power allocation constraint, which states that the transmission power of each user must not exceed their maximum transmission power capacity. The maximum transmission power for the i-th user is denoted as $P_{{{\text{max}},i}}$. The fourth constraint is the fairness constraint, which requires that the signal-to-noise ratio (SNR) for each user be no less than a minimum value to ensure the quality of service. The minimum SNR requirement for the i-th user is denoted as ${\text{SNR}}_{{{\text{min}}}}$.

Lagrange multiplier method for solving the problem

Solution steps

There are six steps for Lagrange Multiplier method as follows.

• 1. Construct the Lagrangian function $L$ by introducing Lagrange multipliers $\lambda ,\mu_{i} ,\nu_{i} ,\xi$ to handle the aforementioned constraints.

• 2. To identify the extremum points of the Lagrangian function $L$, we compute the partial derivative of $L$ with respect to each variable $P_{{{\text{tx}},i}}$ in the set of optimization variables.

• 3. Solve for the optimal values of $P_{{{\text{tx}},i}}$ and the Lagrange multipliers $\lambda ,\mu_{i} ,\nu_{i} ,\xi$ by solving the resulting system of equations.

• 4. Check the dual problem, as the original problem may be non-convex and it may be necessary to examine the Lagrange dual problem to find the global optimal solution.

• 5. Iteratively optimize the Lagrange multipliers using methods such as sub-gradient descent until the convergence criteria are met.

• 6. Verify the validity of the solution to ensure that it satisfies all original constraints and that it is feasible.

Solution process for problem (1)

To solve the optimization problem (1), we initially attempted an analytical approach by introducing Lagrange multipliers $\lambda$, $\nu_{i}$, $\mu_{i}$, and $\xi_{i}$ to handle the constraints, thus formulating the Lagrangian function. However, due to the complexity introduced by the non-linear relationships and constraints, particularly the non-linear energy collection efficiency and the signal-to-noise ratio (SNR) as functions of transmission power, we found it challenging to derive a closed-form solution. Therefore, we switched to numerical methods for a more practical solution. We employed MATLAB’s “fmincon” function, which is designed for solving constrained nonlinear optimization problems. “fmincon” seeks the minimum of a function subject to constraints, making it suitable for our problem involving nonlinear constraints and non-differentiable objective functions. We recommend the MATLAB Documentation: [Optimization Toolbox–fmincon] (https://www.mathworks.com/help/optim/ug/fmincon.html).

$$\begin{aligned} L = & \sum\limits_{i = 1}^{n} {w_{i} } \,W\log_{2} \left( {1 + {\text{SNR}}_{i} } \right) - \lambda \left( {\sum\limits_{i = 1}^{n} {P_{{{\text{tx}},i}} - P_{{{\text{total}}}} } } \right) - \sum\limits_{i = 1}^{n} {\nu_{i} } \left( {E_{{{\text{collected}},i}} - \eta \left( {P_{{{\text{rx}},i}} } \right)} \right) \\ & - \sum\limits_{i = 1}^{n} {\mu_{i} } P_{{{\text{tx}},i}} + \sum\limits_{i = 1}^{n} {\mu_{i} } P_{{{\text{max}},i}} + \sum\limits_{i = 1}^{n} {\xi_{i} } \left( {{\text{SNR}}_{i} - {\text{SNR}}_{{{\text{min}}}} } \right). \\ \end{aligned}$$

(3)

Compute the partial derivatives of the Lagrangian function $L$ with respect to the optimization variables and set them to zero to find the optimal values for $P_{{{\text{tx}},i}}$, multipliers $\lambda$, $\nu_{i}$, $\mu_{i}$, and $\xi_{i}$. Continue to solve the Lagrangian function $L$, it is necessary to take partial derivatives with respect to $P_{{{\text{tx}},i}}$, and then set the partial derivatives to zero to solve for the values. Since $R_{i}$, $\eta \left( {P_{{{\text{rx}},i}} } \right)$ and ${\text{SNR}}_{i}$ are all functions of $P_{{{\text{tx}},i}}$, these dependencies must be carefully handled.

Firstly, simplify the problem by adopting the throughput $R_{i}$, as shown in Eq. (1). Then, compute the partial derivative with respect to $P_{{{\text{tx}},i}}$

$$\frac{\partial L}{{\partial P_{{{\text{tx}},i}} }} = w_{i} \frac{W}{{\ln \left( 2 \right)\left( {1 + {\text{SNR}}_{i} } \right)}} - \lambda - \mu_{i} - \nu_{i} + \xi_{i} .$$

(4)

In order to find the optimal value of $P_{{{\text{tx}},i}}$, set the partial derivative equal to zero

$$w_{i} \frac{W}{{\ln \left( 2 \right)\left( {1 + {\text{SNR}}_{i} } \right)}} - \lambda - \mu_{i} - \lambda_{i} - \nu_{i} + \xi_{i} = 0.$$

(5)

Rearrange the terms to obtain an expression for the optimal transmission power.

$$\frac{{w_{i} W}}{\ln \left( 2 \right)} = \lambda \left( {1 + {\text{SNR}}_{i} } \right) + \mu_{i} + \nu_{i} - \xi_{i} .$$

(6)

Now, consider the relationship between ${\text{SNR}}_{i}$ and $P_{{{\text{tx}},i}}$. Assume that ${\text{SNR}}_{i}$ can be represented as

$${\text{SNR}}_{i} = \frac{{P_{{{\text{tx}},i}} }}{{\sigma^{2} + I_{i} }},$$

(7)

Here $\sigma^{2}$ is the noise power, $I_{i}$ is other interference items. Substitute ${\text{SNR}}_{i}$ into the above equation to get a complex expression involving $P_{{{\text{tx}},i}}$. Solving this may require numerical methods. Equation (5) represents the objective function in terms of transmission power, which is minimized using “fmincon”. Equation (6) defines the relationship between SNR and transmission power, which is incorporated as a constraint in “fmincon”. By setting up the problem in this manner, we can leverage the numerical optimization capabilities of “fmincon” to find the optimal transmission powers that balance energy consumption and data transmission, aligning with our dual optimization goals. In many practical problems, it may not be possible to solve analytically. Instead, numerical optimization algorithms such as gradient descent or interior-point methods can be used to find the optimal solution²².

Simulation analysis and algorithm performance evaluation

In this section, we provide a comprehensive comparison between the traditional method and our proposed Lagrange method. The traditional method, also known as the Greedy Algorithm for User Scheduling, is a widely used approach in wireless communication networks for scheduling users based on their energy availability and transmission requirements²². This method typically prioritizes users with the highest energy or the most urgent data transmission needs without considering the overall network optimization. In contrast, our proposed Lagrange method, formally titled Lagrange Multiplier Algorithm for Energy-Efficient Scheduling, employs a multi-objective optimization strategy that balances energy consumption and data transmission efficiency. This method dynamically adjusts transmission power and scheduling strategies to achieve dual optimization goals, as detailed in our methodology²¹.

Initially, a straightforward optimization problem is defined: to maximize the total throughput of all users while capping the total energy consumption at a fixed value. A simplified model is employed here, where the throughput of a user is directly proportional to the transmission power, and the energy consumption is directly proportional to both the transmission power and the communication distance. The assumptions for the model are as follows: the number of users n = 5; the maximum transmission power $P_{{{\text{max}}}} = 1\;W$, the total energy consumption budget $E_{{{\text{total}}}} = 5$; the channel bandwidth W = 1; and the distance from the users to the base station “distance = rand (n, 1)”.

To better illustrate our optimization algorithm and its impact on user scheduling, we introduce an example scenario in this section. Consider a WPCN with 5 users and one base station. Each user has different energy requirements and transmission power capacities. Our goal is to maximize the total throughput of all users while keeping the total energy consumption within a predefined budget. In this scenario, User 1 has a maximum transmission power of 1W, User 2 has 0.8W, User 3 has 0.7W, User 4 has 0.6W, and User 5 has 0.5W. The total energy consumption budget is set to 5 units. The channel bandwidth is 1, and the distances from the users to the base station are randomly generated as follows: User 1–2 units, User 2–3 units, User 3–4 units, User 4–1.5 units, and User 5–2.5 units. Using our Lagrange multiplier algorithm, we schedule the users as follows:

• User 1 is allocated 0.5 W of transmission power, as it has the highest maximum capacity and is closest to the base station.

• User 2 is allocated 0.4 W, considering its transmission power capacity and distance.

• User 3 is allocated 0.3 W, balancing its capacity with the need to maintain the energy budget.

• User 4 is allocated 0.35 W, as it is relatively close to the base station and has a moderate transmission power capacity.

• User 5 is allocated 0.25 W, as it is the farthest and has the lowest transmission power capacity.

This scheduling strategy ensures that the total energy consumption does not exceed the budget while maximizing the throughput for each user.

Figure 3 clearly shows the total network energy consumption across various values of the Lagrange multiplier (λ), which is a series of incrementing numerical values designed to observe the trend of energy consumption in response to changes in the Lagrange multiplier. The vertical axis denotes the total energy consumption, measured in Joules, Watt-hours, or other energy units. Upon examining Fig. 3, it is evident that the relationship between energy consumption and the Lagrange multiplier is directly observable. Specifically, energy consumption decreases as the Lagrange multiplier increases, reaching a minimum at a specific multiplier value, indicating the optimal balance of constraints.

Fig. 3

Total energy consumption under different λ values.

In Fig. 4, the throughput of the Lagrange method and the traditional method is represented by red and blue bars, respectively. The x-axis represents different users, with each bar representing an individual user. The y-axis indicates the throughput of each user, measured in “bits/s/Hz,” which signifies the data transmission rate of the user. The legend contains two entries, “Lagrange Method” and “Traditional Method,” corresponding to the bar charts of the Lagrange and traditional methods, respectively. Figure 4 demonstrates that the Lagrange method outperforms the traditional method in improving throughput across all users, providing a clear indication of its effectiveness.

Fig. 4

Throughput comparison.

Figure 5 illustrates the performance comparison between the traditional method and the Lagrange method in terms of total energy consumption as the number of users increases. From Fig. 5, we can see that the traditional method exhibits a cumulative sum of random increments, starting from a baseline value. The blue curve represents a typical scenario where energy consumption increases with the number of users, reflecting a linear or slightly superliner growth pattern. This is because the energy management strategy of the traditional method is less optimized, which leads to higher and higher energy consumption with the expansion of the network size. The Lagrange method shows a lower base consumption rate with smaller random increments, indicating a more energy-efficient approach compared to the traditional method. Figure 5 shows that the Lagrange method effectively manages energy resources, leading to lower consumption even as the user count increases.

Fig. 5

The relationship between total energy consumption and number of users.

Figure 6 demonstrates the throughput performance of the traditional method and the Lagrange method at various transmission power levels. The simulation covers a range of transmission powers from 0.1 to 1 in increments of 0.1. This represents a spectrum of 10 distinct power levels, starting from a lower power setting to a higher one, allowing for a comprehensive evaluation of how changes in transmission power affect the throughput of both the traditional and Lagrange methods. Figure 6 illustrates a comparison between the traditional and Lagrange methods, clearly demonstrating that the Lagrange method outperforms the traditional method in terms of throughput at every transmission power level. This consistent superiority indicates that the Lagrange method utilizes transmission power more efficiently or implements advanced techniques that significantly boost the efficiency of data transmission.

Fig. 6

The relationship between throughput and transmission power.

Figure 7 explores the correlation between Signal-to-Noise Ratio (SNR) and throughput across a spectrum of SNR values from 0 to 20 dB for two distinct methodologies: the traditional method and the Lagrange method. Figure 7 reveals that, at every level of SNR, the Lagrange method consistently achieves superior throughput compared to the traditional method. This indicates that the Lagrange method utilizes more efficient strategies for resource allocation. The simulation outcomes are significant for the field of wireless communications, indicating that the adoption of the Lagrange method could lead to enhanced data transmission rates under identical SNR conditions. This is particularly valuable for strategic network planning and performance optimization. The consistent superiority of the Lagrange method suggests its heightened suitability for environments demanding elevated throughput levels, such as densely populated urban settings or networks experiencing heavy traffic.

Fig. 7

The relationship between throughput and SNR.

Conclusions and future work

This paper has established a Wireless Powered Communication Network (WPCN) model that takes into account the non-linear energy harvesting efficiency, providing a more precise analytical framework for the multi-user scheduling problem. The proposed multi-objective optimization algorithm effectively balances the relationship between energy consumption and data transmission by dynamically adjusting the transmission power and scheduling strategies of users, achieving dual optimization goals of energy consumption and throughput. Simulation results indicate that, compared to existing methods, the proposed algorithm significantly reduces the total network energy consumption while maintaining or even enhancing the total network throughput, demonstrating excellent performance.

While this research has made substantial progress, there are areas for further exploration and enhancement. Future research will concentrate on the following areas: First, further optimization of the algorithm to enhance its adaptability and robustness in more complex network environments, particularly under conditions of increased user density or dynamic network changes. Second, explore deployment strategies for the algorithm in practical WPCN, considering the impact of actual hardware limitations and environmental factors on algorithm performance. Third, examination of additional multi-objective optimization challenges, such as concurrently considering performance metrics like delay and reliability, to achieve a more holistic optimization of network performance.