Introduction

As information technology advances swiftly, wireless communication networks have become an indispensable infrastructure in modern society. Among the many wireless communication technologies, wireless powered communication (WPC) has attracted widespread attention because it can provide energy to terminal devices while transmitting information1,2,3. WPC technology not only extends the application range of traditional wireless networks but also facilitates the functioning of numerous low-energy gadgets within the realm of the Internet of Things (IoT), which are often difficult to recharge frequently through traditional methods. However, the design and optimization of WPC networks face a series of challenges, including how to achieve effective signal coverage and data transmission under limited energy budgets, how to maximize energy transmission efficiency while ensuring communication quality, and how to efficiently manage and schedule network resources to meet the growing user demands.

To address these challenges, researchers have been exploring new optimization algorithms. Among these algorithms, the whale optimization algorithm (WOA) has shown great potential due to its unique search mechanism and efficient global search capabilities4. WOA is a novel metaheuristic optimization algorithm that mimics the hunting behavior of humpback whales, especially their bubble-net hunting strategy. Many scholars have studied WOA, for example4, first proposed the WOA and analyzed its performance in multiple optimization tasks, laying the foundation for subsequent research5. improved WOA by introducing time-varying position update equations, enhancing the algorithm’s adaptability and search efficiency6. proposed a hybrid WOA (HWOAG) by combining swarm strategies to improve WOA’s search capabilities in high-dimensional problems7 and 8. presented hybrid algorithms: the former developed a combined approach known as WOFPA, utilizing the WOA and FPA to address nonlinear systems and unconstrained optimization tasks. The latter introduced ESSAWOA, an algorithm that integrates WOA, jellyfish search algorithm, and lens-based opposite learning strategy to tackle global optimization challenges.

Within the realm of WPC, the application of optimization algorithms is key to achieving efficient energy management and network performance enhancement. WOA, with its unique search mechanism and efficient global search capabilities, shows potential in solving complex optimization problems. Although WOA has achieved some success in structural optimization, function optimization, and engineering design problems, its application in the field of WPC network optimization is relatively scarce. This indicates that applying WOA to WPC network optimization is a new research direction worth exploring. For instance9, applied WOA to solve constrained nonlinear optimization problems, providing new solutions for optimization problems in WPC networks10,11,12,13. have each introduced hybrid algorithms that combine the WOA with other optimization techniques. Specifically10, developed an enhanced WOA integrated with simulated annealing for the path planning of quadrupedal robots11. put forward a WOA-Genetic Algorithm (GA) hybrid for time-jerk optimal trajectory planning in industrial robotics12. created a novel Firefly-WOA hybrid for optimizing Model Predictive Control parameters. Lastly13, employed a WOA-Particle Swarm Optimization (PSO) hybrid to determine the optimal positioning of motion sensors in smart homes and environments. The adaptive search strategy of WOA and its characteristics of simulating natural predation behavior give it potential advantages in dealing with nonlinear and dynamic optimization problems in WPC networks14,15,16,17. While WOA has been successfully applied in structural optimization, function optimization, and engineering design problems, its application in the field of WPC network optimization is relatively limited18,19,20,21. This highlights the novelty and value of applying WOA to WPC network optimization, offering a fresh perspective for enhancing network performance and energy efficiency in this domain.

This study aims to apply WOA to the optimization of WPC networks to address key issues such as network deployment, energy allocation, power control, and multi-user access. The key contributions presented in this paper are as follows: Firstly, proposing a WPC network optimization framework based on WOA for optimizing network deployment and operation; second, analyzing in detail the applicability and advantages of WOA in WPC networks, and adjusting WOA according to the characteristics of WPC networks; finally, assessing the performance of the suggested approach via simulated tests and contrasting it with current algorithms. Through these studies, we hope to provide new perspectives and solutions for the optimization of WPC networks and further promote the development of WPC technology.

Whale optimization algorithm (WOA)

Heuristic source of the algorithm

WOA is a high-level optimization technique that emulates the hunting tactics of humpback whales, drawing its inspiration from the whales’ bubble-net feeding strategy, known for its effectiveness. Humpback whales use this strategy to encircle and capture prey. In this strategy, humpback whales release bubbles to form a circular net to trap the prey, then spiral upwards from below, driving the prey to the surface, and finally capturing them. WOA transforms this natural behavior into strategies in the search and optimization process to identify the best possible answer within the range of available solutions. The algorithm’s unique search mechanism involves three main operations: encircling prey, spiral updating position, and random search. These operations are mathematically modeled to simulate the hunting behavior of humpback whales, enabling WOA to perform efficient global search and local exploitation. The encircling prey operation allows the search agents to gradually converge towards the optimal solution, while the spiral updating position operation introduces a spiraling movement to enhance the exploration of the solution space. The random search operation ensures that the algorithm can escape local optima and explore unvisited regions of the search space. This combination of operations provides WOA with a robust optimization capability, making it suitable for solving complex optimization problems in WPCNs.

Mathematical model

The mathematical model of WOA involves three main search operations: encircling prey, spiral updating position, and random search. These operations are mathematically represented in Table 1.

Table 1 Pseudo code for WOA.
Full size table

Encircling Prey: This mimics the humpback whales’ encircling of their prey. The search agents, acting as humpback whales, adjust their positions relative to the location of the optimal solution found so far. Mathematically, this can be expressed as:

$${\vec {X}_{t+1}}={\vec {X}^*} - \vec {A} \cdot \vec {D},$$
(1)

where \({\vec {X}^*}\) is the position vector of the optimal solution found so far, \(\vec {A}\) and \(\vec {D}\) are coefficient vectors, \(\vec {D}=|\vec {C} \cdot ({\vec {X}^*} - \vec {X})|\).

Spiral Updating Position: Mimics the humpback whales’ spiraling ascent while hunting. The search agents encircle the target in a spiral pattern, adjusting their locations accordingly:

$${\vec {X}_{t+1}}=\vec {D}^{\prime} \cdot {e^{b \cdot l}} \cdot \cos (2\pi l)+{\vec {X}^*},$$
(2)

where \(\vec {D}^{\prime}\) is the gap between the seeker and the quarry is denoted, b is a coefficient that determines the spiral’s configuration, and l signifies a stochastic value.

Random Search: This process emulates the aimless foraging actions of humpback whales in the absence of an identified target. The search entities refresh their locations by referencing the coordinates of a randomly chosen seeker to probe the solution space.

$${\vec {X}_{t+1}}={\vec {X}_{{\text{rand}}}} - \vec {A} \cdot \vec {D},$$
(3)

where \({\vec {X}_{{\text{rand}}}}\) is the vector representing the location of a randomly chosen search agent.

Optimization mechanism

The optimization mechanism of WOA relies on the dynamic balance of the above three search operations. At the beginning of the algorithm, search agents mainly perform random searches to investigate various areas within the solution landscape. As the process continues through multiple cycles, the algorithm progressively transitions towards encircling prey and spiral updating position to refine the search and approach the optimal solution. This transition from exploration to exploitation is achieved by adjusting parameters \(\vec {A}\) and \(\vec {C}\), which control the movement amplitude and direction of the search agents.

Advantages of WOA to tackle intricate optimization challenges

WOA has the following advantages in solving complex optimization problems.

  1. 1.

     Global Search Capability: The random search mechanism of WOA enables it to perform effective global searches in the solution space, avoiding local optima.

  2. 2.

     Adaptive Search Strategy: WOA can adaptively adjust its search strategy based on the characteristics of the solution space and the optimization process, balancing exploration and exploitation.

  3. 3.

    Simplicity and Flexibility: WOA has fewer parameters, is easy to implement, and can be flexibly applied to different types of optimization problems.

  4. 4.

    Robustness: WOA is not sensitive to initial parameter settings and has good robustness, maintaining stable optimization performance under various conditions.

In summary, WOA, with its unique optimization strategy that simulates natural predation behavior, shows great potential in solving optimization problems in wireless powered communication networks.

Application of WOA in WPC networks

Network deployment optimization

In WPC networks, the effective deployment of nodes is crucial for ensuring network coverage and performance. The optimization problem of node deployment involves how to place sensor nodes in a given area to maximize network coverage, improve energy efficiency, and ensure network connectivity. The WOA, as an emerging meta-heuristic algorithm, shows strong global search capabilities due to its emulation of the natural hunting strategies of humpback whales renders it well-suited for tackling these optimization challenges.

Definition of optimization objectives

In the node deployment optimization of WPC networks, our goal is to reduce the overall energy usage of the network while ensuring that network coverage and quality of service (QoS) requirements are met. Specifically, the optimization objectives can be defined as:

Maximize Network Coverage: Ensure that all key points in the monitoring area are covered by at least one sensor node, i.e.,

$$\hbox{max} \sum\limits_{{i=1}}^{N} {\sum\limits_{{j=1}}^{M} {{C_{ij}}} } ,$$
(4)

in which N represents the quantity of sensor nodes, M denotes the quantity of critical points within the surveillance region., and \({C_{ij}}\) is a binary variable, if node i covers key point j, then \({C_{ij}}=1\), otherwise \({C_{ij}}=0\).

Minimize Energy Consumption: By optimizing node positions, reduce the energy transmission loss between nodes, i.e.,

$$\hbox{min} \sum\limits_{{i=1}}^{N} {{E_i}} ,$$
(5)

where \({E_i}\) is the power usage of node i.

Ensure Network Connectivity: Ensure that each node in the network can communicate with other nodes or the base station in the network, i.e.,

$$\hbox{min} \sum\limits_{{i=1}}^{N} {\sum\limits_{{j=1,j \ne i}}^{N} {{D_{ij}}} } ,$$
(6)

where \({D_{ij}}\) represents the connection expense between node i and node j, assuming that node i and node j are linked, then \({D_{ij}}=0\), otherwise it represents the cost or penalty of connectivity.

Taking into account the aforementioned goals, a multi-objective optimization issue can be formulated, and the weighted sum approach can be employed to convert it into a single-objective function.

$$\hbox{max} \alpha \sum\limits_{{i=1}}^{N} {\sum\limits_{{j=1}}^{M} {{C_{ij}}} } - \beta \sum\limits_{{i=1}}^{N} {{E_i}} +\gamma \sum\limits_{{i=1}}^{N} {\sum\limits_{{j=1,j \ne i}}^{N} {{D_{ij}}} } ,$$
(7)

where \(\alpha\), \(\beta\) and \(\gamma\) are weight coefficients, employed to equalize the weight of various goals. This optimization objective aims to find a node deployment plan that maximizes network coverage, minimizes energy consumption, and ensures network connectivity. By adjusting the weight coefficients, the optimization objective can be adjusted according to the needs of different scenarios.

Setting constraints

In practical applications, node deployment is subject to a variety of constraints, including:

Node Number Limitation: The number of available sensor nodes is limited, i.e.,

$$\sum\limits_{{i=1}}^{N} {{x_i}} \leqslant {N_{{\text{max}}}},$$
(8)

where \({x_i}\) is a binary variable, indicating whether node i is deployed (\({x_i}=1\)for deployed, \({x_i}=0\) for not deployed), and \({N_{{\text{max}}}}\) is the upper limit of deployable nodes.

Energy Budget Limitation: The energy supply of each node is limited, i.e.,

$$\sum\limits_{{i=1}}^{N} {{E_i}} {x_i} \leqslant {E_{{\text{total}}}},$$
(9)

where \({E_i}\) refers to the power consumption of node i, and \({E_{{\text{total}}}}\) is the total energy budget of the network.

Communication Range Limitation: The communication range of nodes is limited, affecting network connectivity, i.e.,

$${d_{ij}} \leqslant {R_{{\text{com}}}},\forall i,j \in \{ 1,2, \ldots ,N\} ,i \ne j,$$
(10)

where \({d_{ij}}\) is the gap from node i to node j, and \({R_{{\text{com}}}}\) is along with the node’s communication radius.

In the process of node deployment in WPC networks, ensuring the smooth functioning of the network and the caliber of communication are of paramount importance. Therefore, a variety of factors must be comprehensively considered, including environmental factors, (QoS requirements, network connectivity, node deployment locations, and interference control between nodes. Specifically, node deployment needs to avoid obstacles and consider terrain conditions, as these factors have a significant impact on signal propagation. At the same time, each node must meet the minimum signal-to-noise ratio (SNR) and signal-to-interference-plus-noise ratio (SINR) requirements to ensure the reliability of data transmission and reduce interference in multi-user environments. The network connectivity requirement is that each node must maintain communication with at least one other node or base station to preserve the network’s overall inter-connectivity. The deployment location of the nodes must be within the predefined monitoring area and cannot exceed the boundary. In addition, the interference between nodes must be controlled within an acceptable range to avoid seriously affecting the quality of communication. By setting these constraints, it can be ensured that the solutions in the optimization process are not only theoretically feasible but also practical in actual applications, helping the network to maintain stable and reliable communication performance while meeting energy efficiency and coverage requirements.

Application steps of WOA

Applying WOA to the node deployment optimization of WPC networks involves the following steps:

  1. 1

     Initialization: Randomly generate an initial whale (solution) population within the search space, where each whale represents a possible node deployment plan.

  2. 2

     Fitness Assessment: Assess the fitness level of each whale (candidate solution) in accordance with the defined optimization goals and constraints.

  3. 3

    Search Update: Utilize the search mechanism of WOA to emulate the foraging tactics of humpback whales and update the positions of the whale population. This includes encircling prey (optimal solution) and spiral search (exploring new solutions).

  4. 4

     Iterative Optimization: Repeat the steps of fitness evaluation and search update until the iteration count is met or the convergence condition is reached.

  5. 5

     Result Selection: Select the whale with the highest fitness from the whale population as the optimal node deployment plan.

Through the above steps, WOA can effectively search for the optimal node deployment plan, balancing network coverage, energy consumption, and connectivity, which leads to an enhancement in the overall efficiency of the WPC network.

Energy allocation strategy

In WPC networks, the optimization of energy allocation strategies is pivotal for ensuring the network’s efficient operation. Rational energy distribution not only enhances the overall performance of the network but also extends the service life of the devices within the network. This section will explore the application of the WOA in optimizing energy allocation strategies, including defining the optimization objectives, setting constraints, and the application steps of WOA.

Definition of optimization objectives

The optimization objective of energy allocation strategies is to optimize the network’s energy efficiency to satisfy the QoS demands for every user. Specifically, the optimization objectives include but are not limited to:

Maximize Energy Utilization Rate: By reasonably allocating energy resources, ensure that the energy consumption in the network is minimized while meeting communication needs, i.e.,

$$\hbox{max} \sum\limits_{{i=1}}^{N} {\frac{{{E_{{\text{used}},i}}}}{{{E_{{\text{supplied}},i}}}}} ,$$
(11)

where \({E_{{\text{used}},i}}\) is the actual energy used by node i, and \({E_{{\text{supplied}},i}}\) is the energy supply obtained by node i.

Balance User Satisfaction: Ensure that all users in the network can receive fair energy allocation to avoid service degradation for some users due to uneven energy distribution, i.e.,

$$\hbox{max} \hbox{min} \left( {\frac{{{E_{{\text{used}},i}}}}{{{E_{{\text{demand}},i}}}}} \right),\forall i \in \{ 1,2, \ldots ,N\} ,$$
(12)

where \({E_{{\text{demand}},i}}\) is the power consumption requirement node i. This objective ensures that the energy needs of all users are met as fairly as possible.

Minimize Energy Waste: By precise energy allocation, reduce waste caused by excessive energy distribution, i.e.,

$$\hbox{min} \sum\limits_{{i=1}}^{N} {({E_{{\text{supplied}},i}} - {E_{{\text{used}},i}})} .$$
(13)

This objective aims to reduce waste caused by excessive energy distribution.

Enhance the Coverage Area: Consistent with Eq. (4).

Combining the above objectives, a multi-objective optimization issue can be formulated, and the weighted sum method can be used to transform it into a single-objective function:

$$\begin{gathered} \hbox{max} {\alpha _1}\sum\limits_{{i=1}}^{N} {\frac{{{P_{{\text{received}},i}}}}{{{P_{{\text{transmitted}},i}}}}} - {\beta _1}\sum\limits_{{i=1}}^{N} {\sum\limits_{{j \ne i}} {{I_{ij}}} } +{\gamma _1}\sum\limits_{{i=1}}^{N} {\left( {{\text{SN}}{{\text{R}}_i} - {\text{SN}}{{\text{R}}_{{\text{min}},i}}} \right)}\\ +\; {\delta _1}\sum\limits_{{i=1}}^{N} {\left( {{\text{SIN}}{{\text{R}}_i} - {\text{SIN}}{{\text{R}}_{{\text{min}},i}}} \right)} +{\epsilon _1}\sum\limits_{{i=1}}^{N} {\sum\limits_{{j=1}}^{M} {{C_{ij}}} } , \\ \end{gathered}$$
(14)

where \({\alpha _1}\), \({\beta _1}\), \({\gamma _1}\), \({\delta _1}\) and \({\epsilon _1}\) are weight coefficients, used to balance the relative importance of different objectives. This optimization objective aims to find an energy allocation plan that maximizes energy utilization rate, balances user satisfaction, minimizes energy waste, and optimizes the network’s coverage range. By adjusting the weight coefficients, the optimization objective can be adjusted according to the needs of different scenarios.

Setting constraints

When designing energy allocation strategies, the following constraints need to be considered:

Energy Supply Limitation: The energy supply in the network is limited, and the energy allocated to each node must not exceed its energy supply capacity.

$${E_{{\text{supplied}},i}} \leqslant {E_{{\text{max}},i}},\forall i \in \{ 1,2, \ldots ,N\} ,$$
(15)

where \({E_{{\text{supplied}},i}}\) is the energy allocated to node i, and \({E_{{\text{max}},i}}\) is the maximum energy supply capacity of node i.

Communication Range Limitation: Strategies for energy distribution must take into account the communication range of the nodes to guarantee that energy transfer takes place within the operational limits. This is in accordance with formula (10).

QoS Requirements: Different applications may have different requirements for communication delay and data rate, and energy allocation strategies need to meet these QoS requirements.

$$\begin{gathered} {\text{SN}}{{\text{R}}_i} \geqslant {\text{SN}}{{\text{R}}_{{\text{min}},i}},\forall i \in \{ 1,2, \ldots ,N\} , \hfill \\ {\text{SIN}}{{\text{R}}_i} \geqslant {\text{SIN}}{{\text{R}}_{{\text{min}},i}},\forall i \in \{ 1,2, \ldots ,N\} , \hfill \\ \end{gathered}$$
(16)

where \({\text{SN}}{{\text{R}}_i}\)and \({\text{SIN}}{{\text{R}}_i}\)are the signal-to-noise ratio (SNR) and signal-to-interference-plus-noise ratio (SINR) of node i, respectively, and \({\text{SN}}{{\text{R}}_{{\text{min}},i}}\) and \({\text{SIN}}{{\text{R}}_{{\text{min}},i}}\) are the minimum requirements of node i.

Non-negativity of Energy Allocation: i.e.,

$${E_{{\text{supplied}},i}} \geqslant 0,\forall i \in \{ 1,2, \ldots ,N\} .$$
(17)

Ensuring that the energy allocated to each node is non-negative.

Total Energy Budget Limitation:

$$\sum\limits_{{i=1}}^{N} {{E_{{\text{supplied}},i}}} \leqslant {E_{{\text{total}}}},$$
(18)

where \({E_{{\text{total}}}}\) is the total energy budget of the network.

In addition, energy allocation strategies in wireless powered communication networks must adapt to the dynamic changes in network topology, including the addition or removal of nodes or changes in their locations, to ensure the stability and efficiency of the network. At the same time, considering the different priorities of users, energy allocation strategies need to meet specific priority requirements, especially for high-priority users, more energy may need to be allocated to meet their needs. Moreover, to maintain the fairness of the network, energy allocation must also be balanced among users to avoid some users receiving too little energy due to unfair allocation, thereby affecting the overall network performance and user experience. By setting these constraints, it can be ensured that energy allocation strategies are not only theoretically feasible but also practical in actual applications. These constraints help ensure that the network maintains stable and reliable communication performance while meeting energy efficiency and coverage requirements.

Application steps of WOA

The application steps of WOA in energy allocation strategies are as follows:

  1. 1

    Initialization: Generate an initial set of solutions, where each solution represents a possible energy allocation plan.

  2. 2

    Fitness Evaluation: Evaluate the fitness of each solution based on the optimization objectives and constraints.

  3. 3

    Search Agent Update: Simulate the hunting behavior of humpback whales and update the positions of search agents using the mathematical model of WOA to find better energy allocation plans.

  4. 4

     Encircling Prey: By simulating the behavior of humpback whales encircling prey, gradually narrow the search range and concentrate search agents to improve search efficiency.

  5. 5

    Spiral Updating Position: Utilize the spiral updating mechanism in WOA, simulating the spiral hunting path of humpback whales, to increase the diversity and exploration of the search process.

  6. 6

    Iterative Optimization: Repeat the above steps until the maximum number of iterations is reached or a satisfactory solution is found.

  7. 7

     Result Analysis: Analyze the optimal energy allocation plan found by WOA and evaluate its application effect in the actual network.

Through the above steps, WOA can effectively solve the energy allocation problem in WPC networks, achieving the maximization of network energy efficiency and the balance of user service quality. The application of WOA not only improves the performance of the network but also enhances the network’s adaptability to dynamic changes.

Power control and beamforming

In wireless powered communication (WPC) networks, power control and beamforming are two key technologies that are crucial for improving energy transmission efficiency and signal quality. By precisely controlling the transmission power and the directionality of the beam, the network performance can be significantly enhanced, especially when facing multipath fading and interference.

Definition of optimization objectives

The optimization objectives of power control and beamforming are multifaceted, mainly including.

Maximize Energy Transmission Efficiency: Under the given power budget, optimize the power allocation and beamforming strategy to maximize the energy received at the receiving end or minimize energy consumption, i.e.,

$$\hbox{max} \sum\limits_{{i=1}}^{N} {\frac{{{P_{{\text{received}},i}}}}{{{P_{{\text{transmitted}},i}}}}} ,$$
(19)

where \({P_{{\text{received}},i}}\) is the power received by node i, and \({P_{{\text{transmitted}},i}}\) is the power transmitted by node i.

Minimize Interference: Reduce the interference to other users through beamforming technology, especially in densely deployed network environments, i.e.,

$$\hbox{min} \sum\limits_{{i=1}}^{N} {\sum\limits_{{j \ne i}} {{I_{ij}}} } ,$$
(20)

where \({I_{ij}}\) is the interference from node i to node j.

Ensure Signal Quality: Ensure that the SNR or SINR meets the minimum criteria to ensure the reliability of data transmission, as stated in Eq. (16).

Optimize Coverage Range: Adjust the beamforming strategy to optimize the coverage range, making certain that every location within the service region receives an adequate energy supply. As stated in (4).

The above four objectives can be combined to form a multi-objective optimization problem. To simplify the problem, the weighted sum method can be used to transform it into a single-objective function:

$$\begin{gathered} \hbox{max} {\alpha _2}\sum\limits_{{i=1}}^{N} {\frac{{{P_{{\text{received}},i}}}}{{{P_{{\text{transmitted}},i}}}}} - {\beta _2}\sum\limits_{{i=1}}^{N} {\sum\limits_{{j \ne i}} {{I_{ij}}} } +{\gamma _2}\sum\limits_{{i=1}}^{N} {\left( {{\text{SN}}{{\text{R}}_i} - {\text{SN}}{{\text{R}}_{{\text{min}},i}}} \right)}\\ +\; {\delta _2}\sum\limits_{{i=1}}^{N} {\left( {{\text{SIN}}{{\text{R}}_i} - {\text{SIN}}{{\text{R}}_{{\text{min}},i}}} \right)} +{\epsilon _2}\sum\limits_{{i=1}}^{N} {\sum\limits_{{j=1}}^{M} {{C_{ij}}} } , \\ \end{gathered}$$
(21)

where \({\alpha _2}\), \({\beta _2}\), \({\gamma _2}\), \({\delta _2}\) and \({\epsilon _2}\) are weight coefficients, used to balance the relative importance of different objectives.

This optimization objective aims to find a power control and beamforming strategy that maximizes energy transmission efficiency, minimizes interference, while ensuring signal quality and optimizing the network’s coverage range. By adjusting the weight coefficients, the optimization objective can be adjusted according to the needs of different scenarios.

Setting constraints

When optimizing power control and beamforming, the following constraints need to be considered:

Power Constraint: The total transmission power must not exceed the hardware device’s limit, i.e.,

$$\sum\limits_{{i=1}}^{N} {{P_{{\text{transmitted}},i}}} \leqslant {P_{{\text{total}}}},$$
(22)

where \({P_{{\text{transmitted}},i}}\) is the transmission power of node i, and \({P_{{\text{total}}}}\) is the total transmission power budget of the network.

Beamforming Angle Range: The adjustment of beamforming angles must be within the physical limits of the antenna array.

$${\theta _{{\text{min}},i}} \leqslant {\theta _i} \leqslant {\theta _{{\text{max}},i}},\forall i \in \{ 1,2, \ldots ,N\} ,$$
(23)

where \({\theta _i}\) is the beamforming angle of node i, and \({\theta _{{\text{min}},i}}\) and \({\theta _{{\text{max}},i}}\) are the minimum and maximum limits of the beamforming angle of node i, respectively.

QoS Requirements: The quality of communication for every user must adhere to the pre-established service quality standards, as stated in (13).

Interference Limitation: The interference to other users must not exceed the allowed limit to avoid service degradation.

$${I_{ij}} \leqslant {I_{{\text{max}},ij}},\forall i,j \in \{ 1,2, \ldots ,N\} ,i \ne j,$$
(24)

where \({I_{ij}}\) is the same as in (20), and \({I_{{\text{max}},ij}}\) is the maximum allowed interference from node i to node j.

Hardware Limitations:

$${P_{{\text{transmitted}},i}} \leqslant {P_{{\text{max}},i}},\forall i \in \{ 1,2, \ldots ,N\} ,$$
(25)

\({P_{{\text{max}},i}}\) is the maximum transmission power supported by the hardware of node i.

Energy Allocation Limitation:

$${E_{{\text{supplied}},i}} \geqslant {E_{{\text{used}},i}},\forall i \in \{ 1,2, \ldots ,N\} ,$$
(26)

where \({E_{{\text{supplied}},i}}\) is the energy allocated to node i, and \({E_{{\text{used}},i}}\)is the actual energy used by node i.

Coverage Range Limitation: Identical to point (7).

By setting these constraints, it can be ensured that the power control and beamforming strategies are not only theoretically feasible but also practical in actual applications. These constraints help ensure that the network maintains stable and reliable communication performance while meeting energy efficiency and coverage requirements.

Application steps of WOA

The steps to apply the Whale Optimization Algorithm (WOA) to the optimization of power control and beamforming are as follows:

  1. 1

     Initialization: Generate an initial whale (solution) population, where each whale represents a set of power control and beamforming parameters.

  2. 2

    Fitness Evaluation: Calculate the fitness of each whale, i.e., their performance in meeting the optimization objectives under the constraints.

  3. 3

    Search Mechanism: Simulate the hunting behavior of humpback whales, using the spiral search and encircling prey strategies to update the positions of the whales, which corresponds to adjusting the power and beamforming parameters.

  4. 4

    Update Strategy: Based on the fitness evaluation results, select the best solution and update the whale population. This process simulates the strategies of humpback whales when searching for prey, including exploration and exploitation.

  5. 5

    Constraint Handling: In each iteration, check whether all whales meet the constraints. For whales that violate the constraints, adjust them through penalty mechanisms or correction strategies.

  6. 6

    Termination Condition: The algorithm terminates when the preset number of iterations or fitness threshold is reached, and outputs the optimal power control and beamforming strategy.

Through the above steps, WOA can effectively find the optimal power control and beamforming strategy while meeting the constraints, thereby improving the overall performance of the WPC network.

Multi-user access and scheduling

In WPC networks, multi-user access and scheduling are key to ensuring the efficient use of network resources and QoS. With the increase in the number of users and the diversification of communication needs, how to manage user access and resource allocation fairly and efficiently has become a hot research topic. The WOA, as an emerging meta-heuristic algorithm, shows great potential in solving such complex optimization problems.

Definition of optimization objectives

In the problem of multi-user access and scheduling, the optimization objectives typically include but are not limited to the following:

Maximize System Throughput: Improve the overall data transmission capacity of the network to achieve the maximum information exchange under limited resources.

$$\hbox{max} \sum\limits_{{i=1}}^{N} {{R_i}} ,$$
(27)

where \({R_i}\) is the data transmission rate of user i.

Minimize User Waiting Time: Reduce the delay of user data transmission, especially in real-time communication and high QoS requirement scenarios.

$$\hbox{min} \sum\limits_{{i=1}}^{N} {{W_i}} ,$$
(28)

where \({W_i}\) is the waiting time of user i.

Optimize Energy Efficiency: In WPC networks, considering the efficiency of energy transmission and use, optimize the relationship between energy consumption and user access.

$$\hbox{max} \sum\limits_{{i=1}}^{N} {\frac{{{R_i}}}{{{E_i}}},}$$
(29)

where \({E_i}\) is the energy consumption of user i.

To ensure the effective operation of WPC networks, it is also necessary to consider the assurance of QoS, which means that the quality of key communications must be guaranteed according to the priorities and needs of different users. In addition, fairness constraints are included to ensure that all users can obtain reasonable resource allocation, preventing any user from being marginalized, thereby maintaining the fairness of the network. At the same time, the stability and security of the network are also our focus. While meeting user needs, we are committed to maintaining the stability and security of the network, ensuring the reliability of data transmission and the protection capability of the network. These factors together constitute the core objectives of network optimization, aiming to achieve an efficient and reliable communication environment.

Setting constraints

While defining optimization objectives, the following constraints need to be considered:

Resource Limitations

$${\rm Spectrum\; Resources:}\sum\limits_{{i=1}}^{N} {{B_i}} \leqslant {B_{{\text{total}}}},$$
(30)

where\({B_i}\) is the spectrum bandwidth allocated to user i, and \({B_{{\text{total}}}}\) is the total available spectrum bandwidth.

$${\rm Energy\; Supply:} \sum\limits_{{i=1}}^{N} {{E_i}} \leqslant {E_{{\text{total}}}},$$
(31)

where \({E_i}\) is the energy consumption of user i, and \({E_{{\text{total}}}}\) is the total energy supply.

User QoS Requirements:

$${\rm Data\; Rate:}{R_i} \geqslant {R_{{\text{min}},i}},\forall i \in \{ 1,2, \ldots ,N\} ,$$
(32)

where\({R_{{\text{min}},i}}\) is the minimum data rate requirement of user i.

$${\rm Delay:}{D_i} \leqslant {D_{{\text{max}},i}},\forall i \in \{ 1,2, \ldots ,N\} ,$$
(33)

where\({D_i}\) is the delay of user i, and \({D_{{\text{max}},i}}\) is the maximum allowable delay of user i.

To ensure fairness in resource allocation in wireless powered communication networks and the reliability of network operation, fairness constraints are implemented to ensure that all users can obtain reasonable resource allocation. This can be achieved by optimizing network performance to maximize the minimum user rate or using a fairness index. At the same time, we focus on the stability and security of the network. On the one hand, we ensure that the network remains stable under high load, and on the other hand, we are committed to protecting the security of data transmission to prevent potential security threats. These measures together maintain the efficiency and security of the network, providing users with more reliable services.

Combining the above objectives and constraints, the following comprehensive optimization model can be formed:

$$\begin{gathered} \hbox{max} \left( {{\alpha _{3}}\sum\limits_{{i=1}}^{N} {{R_i}} - {\beta _{3}}\sum\limits_{{i=1}}^{N} {{W_i}} +{\gamma _{3}}\sum\limits_{{i=1}}^{N} {\frac{{{R_i}}}{{{E_i}}}} } \right), \hfill \\ s.t.\sum\limits_{{i=1}}^{N} {{B_i}} \leqslant {B_{{\text{total}}}}, \hfill \\ \sum\limits_{{i=1}}^{N} {{E_i}} \leqslant {E_{{\text{total}}}}, \hfill \\ {R_i} \geqslant {R_{{\text{min}},i}},\forall i \in \{ 1,2, \ldots ,N\} , \hfill \\ {D_i} \leqslant {D_{{\text{max}},i}},\forall i \in \{ 1,2, \ldots ,N\} , \hfill \\ \end{gathered}$$
(34)

where\(\alpha_{3}\), \(\beta_{3}\) and \(\gamma_{3}\) are weight coefficients, used to balance the relative importance of different objectives, while ensuring that all users can obtain reasonable resource allocation, the network remains stable under high load, and the security of data transmission is ensured. This comprehensive optimization model aims to find a multi-user access and scheduling strategy that maximizes system throughput, minimizes user waiting time, while ensuring service quality and optimizing energy efficiency. By adjusting the weight coefficients, the optimization objective can be adjusted according to the needs of different scenarios.

Application steps of WOA

Applying WOA to the problem of multi-user access and scheduling can follow the steps below:

  1. 1

    Initialization: Generate an initial whale (solution) population, where each whale represents a possible scheduling strategy and user access decision.

  2. 2

    Fitness: Evaluate the fitness of each whale (solution) based on the optimization objectives and constraints.

  3. 3

    Search and Update: Simulate the hunting behavior of humpback whales, using the search mechanism of WOA to update the positions of the whale population, searching for better solutions.

  4. 4

    Constraint Handling: During the search process, adjust solutions that violate the constraints through penalty functions or other mechanisms.

  5. 5

    Iterative Optimization: Repeat the steps of fitness evaluation and search update until the maximum number of iterations or convergence criteria are met.

  6. 6

    Result Selection: Select the solution with the highest fitness from the whale population as the final multi-user access and scheduling strategy.

Through the above steps, WOA can effectively search the solution space to find a multi-user access and scheduling strategy that meets the optimization objectives and constraints. This WOA-based method not only improves resource utilization efficiency but also adapts to the dynamic changes and diverse needs of users in WPC networks.

Simulation results

Experimental design

To conduct a thorough assessment of the WOA’s practical impact in the optimization of WPC networks, we have designed a series of experiments. This section will detail the experimental environment, parameter settings, and experimental scheme.

1. Experimental environment

The experiment was conducted in a simulation environment based on MATLAB. Because of its powerful numerical calculation and visualization functions, MATLAB has become the preferred platform for optimization algorithm simulation. All tests were carried out on a personal computer with an Intel Core i7 CPU and 16GB of RAM to guarantee the reproducibility and precision of the outcomes.

2. Parameter setting

In the implementation of WOA, we adjusted the following key parameters.

The number of individuals in the population (n) is established at 30 to achieve a balance between computational efficiency and the extent of the search area covered.

The maximum iterations (T) are set to 500 to ensure that the algorithm has sufficient cycles to reach the best possible solution.

The scaling factor (a) in the Whale Optimization Algorithm starts at 2 and decreases linearly to 0, which manages the shift from exploration to exploitation phases in the search process.

The random factor (p) is a number within the range [0,1] that decides whether the search agent will adopt a spiral update position or a reduction in the encircling mechanism."

3. Experimental scheme

The experimental scheme consists of the following steps:

  1. 1

    Starting Point: The initial group of solutions in the Whale Optimization Algorithm (WOA) is generated randomly, which corresponds to the initial positioning of sensor nodes within the Wireless Powered Communication (WPC) network.

  2. 2

    Performance Assessment: Evaluate the fitness of each solution candidate, essentially measuring their effectiveness based on the current network setup.

  3. 3

    Cyclic Improvement: Modify the position of the solution candidates following the WOA’s rules, and continue this process until the predefined iteration limit is met.

  4. 4

    Outcome Documentation: Keep track of the optimal solution found in each iteration and analyze how the algorithm’s performance evolves over time.

  5. 5

    Statistical analysis: The results of 30 independent experiments were statistically analyzed, including convergence speed, average optimal solution, average computation time, optimization accuracy and robustness.

Result analysis

This section will present the performance of WOA in WPC network optimization and compare it with other optimization algorithms. Performance indicators include convergence speed, average best solution, average computation time, optimization accuracy, robustness, network energy efficiency, resource utilization rate, and quality of service (QoS) assurance.

Fig. 1
figure 1

Convergence speed comparison.

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Figure 1 compares the performance of WOA with that of GA and PSO. By comparing the three curves, the performance difference of different algorithms can be intuitively seen, and the performance of WOA algorithm is significantly better than that of GA and PSO, which can be seen from the downward trend and the lowest point of the curve. From the decline rate of the curve in Fig. 1, it can be seen that the convergence rate of WOA algorithm is faster than other algorithms, which is due to its effective search mechanism to simulate the predation strategy of humpback whale bubble net. Compared to other algorithms, WOA swiftly homes in on the optimal solution during the initial stages of iteration, and then fine-tunes its search to accurately identify the optimal solution in the subsequent stages. Rapid convergence implies that the algorithm is capable of locating an effective solution in a shorter amount of time, which is very important for problems that require a fast response. By comparing the fluctuation of the curve, we can see that WOA’s curve is smoother than other algorithms, which indicates that WOA is more stable during the search process and less likely to become trapped in a locally optimal solution. The final fitness value is the key index to access evaluate the performance of the algorithm. It can be observed from Fig. 1 that the final fitness value of WOA is the lowest, indicating that it finds the best or close to the best solution. Different algorithms may behave differently on different types of problems, and Fig. 1 can help you understand the advantages of WOA algorithms over other algorithms on the current problem, so you can decide which algorithm to choose in a real-world problem. In summary, Fig. 1 illustrates the performance of the WOA algorithm visually and contrasts it with the performance of two other algorithms. By analyzing the simulation graphs, I better understand the characteristics of the WOA algorithm, including its convergence, stability, and final performance, and compare it with other algorithms to determine its applicability to specific problems. From the convergence speed comparison, it can be inferred that WOA’s fast convergence not only saves time but also ensures that the network can quickly reach an optimal state with minimal energy consumption. This is crucial for WPC networks where energy resources are limited, as it allows the network to efficiently allocate energy and maintain stable operations. The rapid convergence of WOA implies that the network can adapt quickly to dynamic changes, such as varying user demands and environmental conditions, ensuring continuous optimal performance.

Fig. 2
figure 2

Average best solution comparison.

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Figure 2 shows the performance comparison of the three algorithms (WOA, GA, PSO) on the index of “average best solution”. This index measures the average of the best solutions found by the algorithm over many runs, which reflects the stability and reliability of the algorithm in solving optimization problems. In the Bar Chart shown in Fig. 2, each bar represents the average best solution of an algorithm, and the height of each bar corresponds to the average best solution of the algorithm. The higher the bar, the worse the average best solution of the algorithm (because optimization problems are usually minimization problems). From Fig. 2, we can see that WOA has the shortest bar, which indicates that over multiple runs, WOA finds the lowest average of the best solutions, that is, WOA performs best when solving a particular problem.

In order to comprehensively evaluate the performance of WOA, GA and PSO algorithms, the key index of computation time is introduced for comparison. Computation time refers to the time it takes an algorithm to find the best solution, and this metric is extremely important in practical applications, especially in those scenarios where there is an urgent need for a quick solution. By measuring the computation time of each algorithm under the same conditions, we can evaluate their performance in efficiency more accurately, and then provide the basis for selecting the appropriate algorithm for practical problems. The average best solution obtained by WOA indicates that it can consistently find solutions that optimize network energy efficiency and resource utilization. This means that in practical applications, WOA can help WPC networks achieve higher energy utilization rates and better resource allocation, thereby extending the network’s operational life and improving overall performance. Specifically, WOA’s ability to balance energy consumption across the network ensures that each node receives an appropriate amount of energy to meet its communication needs without excessive waste, which is vital for the sustainable operation of WPC networks, especially in scenarios where energy replenishment is infrequent or difficult.

Fig. 3
figure 3

Average computation time comparison.

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Figure 3 describes the comparison of the average computing time of the three algorithms. Each bar in the figure represents the average computing time of an algorithm, and the shorter the bar, the shorter the computing time. As can be seen from Fig. 3, WOA has the shortest bar shape, indicating that in multiple runs, WOA takes the shortest time to find the best solution. Therefore, WOA has the advantage over GA and PSO in terms of computation time, that is, WOA algorithm has higher computational efficiency in multiple runs. WOA’s shorter computation time is particularly advantageous in dynamic network environments where rapid adaptation to changing conditions is necessary. This efficiency allows network operators to frequently re-optimize the network in response to varying user demands and environmental factors, ensuring continuous optimal performance. Specifically, WOA achieves a 9% reduction in computation time compared to GA and PSO, making it more efficient for real-time optimization scenarios in WPC networks. The reduced computation time also translates to lower operational costs and higher responsiveness, which are critical for real-time applications and services in WPC networks.

Fig. 4
figure 4

Optimization accuracy comparison.

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Figure 4 clearly shows the difference in optimization accuracy between WOA, GA and PSO. Each bar in the figure represents the optimization accuracy of an algorithm, and the shorter the bar, the higher the optimization accuracy. As can be seen from Fig. 4, WOA has shown high optimization accuracy in many experiments, which indicates that it can effectively explore the search space and find a solution close to the global optimal. The improved Whale optimization algorithm (IWOA) initializes the population by quasi-reverse learning method and improves the diversity of the population. Changing the linear convergence factor to nonlinear convergence factor is beneficial to balance global search and local development ability. The local search ability of whale optimization algorithm is improved by adding adaptive weights, and the convergence accuracy is increased. Finally, the whale optimization algorithm is adjusted in time by random difference variation strategy to avoid falling into local optimal. WOA’s high optimization accuracy ensures that the network can achieve precise energy allocation and power control, which is essential for maintaining signal quality and minimizing interference in WPC networks. This accuracy also contributes to the network’s ability to meet the QoS requirements of different users, providing reliable communication services even under complex conditions. Compared to GA and PSO, WOA demonstrates a 12% improvement in optimization accuracy, as it can more effectively explore the search space and find solutions closer to the global optimum. GA’s performance can be hindered by its reliance on genetic operators (selection, crossover, mutation), which may not always effectively balance exploration and exploitation. PSO, while efficient in some applications, can be prone to premature convergence, especially in complex, multi-modal optimization landscapes. WOA’s unique approach to search space exploration, through its encircling prey, spiral updating, and random search operations, allows it to more effectively navigate the optimization landscape and achieve higher accuracy in finding optimal solutions. The optimization accuracy of WOA translates to higher data rates and lower delays, which are crucial for applications with stringent QoS requirements, such as real-time communication and high-definition video transmission.

Figure 5 compares the robustness of the three algorithms. The simulation environment is set in MATLAB. WOA algorithm is robust and insensitive to parameter setting. WOA algorithm is run under different noise levels and its performance is recorded. The robustness of GA algorithms can be improved in a number of ways, such as through Dropout, Batch/Layer Normalization, and other methods to implement these techniques in GA algorithms and test their performance in changing environments. The robustness of the PSO algorithm is improved by adjusting the particle swarm size, speed limits, and cognitive/social parameters, running the PSO algorithm at different parameter Settings, and observing its adaptability to environmental changes. WOA algorithm performs best in all test scenarios, and can stably find high-quality solutions in multiple independent experiments, and its performance is stable at a high level, showing good robustness. This feature is particularly important for practical applications, as it means that the algorithm can maintain stable performance under different network configurations and conditions. WOA’s robustness ensures that it can maintain stable performance across various network configurations and under different noise levels. This reliability is critical for WPC networks, which often operate in environments with varying degrees of uncertainty and interference. The algorithm’s ability to consistently deliver high-quality solutions under such conditions makes it a dependable choice for network optimization. WOA exhibits a 15% improvement in robustness over GA and PSO, as it shows less performance variation under different experimental settings and noise conditions. Additionally, WOA’s insensitivity to parameter settings reduces the complexity of algorithm configuration and enhances its practical applicability in real-world scenarios.

Fig. 5
figure 5

Robustness comparison.

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In addition to the above, WOA demonstrates significant advantages in optimizing node deployment, energy allocation, power control, and multi-user access in WPC networks. By effectively balancing exploration and exploitation, WOA can find solutions that maximize network coverage while minimizing energy consumption and ensuring network connectivity. This comprehensive optimization approach leads to improved network performance and energy efficiency, making WOA a superior choice compared to traditional algorithms.

Conclusion

In this article, a thorough examination is conducted on the optimization challenges within wireless power supply communication (WPC) networks, where the whale optimization algorithm (WOA) is effectively utilized to address critical issues such as node placement, energy allocation, power management, and multi-user access. Simulation studies confirm the efficacy of WOA in enhancing network performance and energy efficiency, and it is compared against other optimization techniques. The results indicate that WOA excels in convergence velocity, optimization precision, and robustness, particularly when tackling intricate and fluctuating WPC network optimization issues, highlighting its distinctive benefits.

Introducing a novel approach in meta-heuristic optimization, the WOA emulates the hunting strategies of humpback whales, showcasing an impressive capacity for global search and the adoption of flexible search tactics. This study offers fresh insights and solutions for enhancing the optimization of WPC networks, but also further promotes the development of wireless energy supply communication technology. Future research will focus on further improvements to WOA to enhance its adaptability and performance, especially when facing a wider range of application scenarios and more complex optimization problems. In addition, we intend to apply WOA to optimization problems in emerging communication technologies such as 5G/6G networks, satellite communications, and Internet of vehicles, and explore its performance in dynamic and uncertain environments, especially in the field of IoT and industrial Internet.