Solving Multi-Objective Satellite Data Transmission Scheduling Problems via a Minimum Angle Particle Swarm Optimization

Abstract

With the increasing number of satellites and rising user demands, the volume of satellite data transmissions is growing significantly. Existing scheduling systems suffer from unequal resource allocation and low transmission efficiency. Therefore, effectively addressing the large-scale multi-objective satellite data transmission scheduling problem (SDTSP) within a limited timeframe is crucial. Typically, swarm intelligence algorithms are used to address the SDTSP. While these methods perform well in simple task scenarios, they tend to become stuck in local optima when dealing with complex situations, failing to meet mission requirements. In this context, we propose an improved method based on the minimum angle particle swarm optimization (MAPSO) algorithm. The MAPSO algorithm is encoded as a discrete optimizer to solve discrete scheduling problems. The calculation equation of the sine function is improved according to the problem’s characteristics to deal with complex multi-objective problems. This algorithm employs a minimum angle strategy to select local and global optimal particles, enhancing solution efficiency and avoiding local optima. Additionally, the objective space and solution space exhibit symmetry, where the search within the solution space continuously improves the distribution of fitness values in the objective space. The evaluation of the objective space can guide the search within the solution space. This method can solve multi-objective SDTSPs, meeting the demands of complex scenarios, which our method significantly improves compared to the seven algorithms. Experimental results demonstrate that this algorithm effectively improves the allocation efficiency of satellite and ground station resources and shortens the transmission time of satellite data transmission tasks.

1. Introduction

The rapid development of the space industry has led to a significant increase in the number of satellites in orbit. However, ground station resources remain relatively fixed, resulting in brief communication windows and substantial data transmission demands, posing a global challenge. By the end of 2022, Europe had more than 50 imaging satellites. All types of satellites, including remote sensing, communication, and navigation satellites, rely on stable data transmission to maintain contact with their ground stations, which is essential for their functionality and value. The efficiency of data transmission between satellites and ground stations is a critical factor in the performance of satellite communication systems. Therefore, efficiently scheduling multi-objective satellite data transmission scheduling problems (SDTSPs) has become increasingly urgent [1].

Data transmission scheduling is an NP-hard problem [2]. Also, adding more ground stations cannot address the rapidly growing number of satellites. Research on the conflict between limited ground station resources and the growing number of satellites has been ongoing for over a decade. In the 1990s, Gooley [3] used a mixed-integer programming model to address the scheduling problem of the satellite control network (AFSCN) in the United States. Later, Barbulescu [4] employed heuristic algorithms for the same issue and summarized the evolution of the AFSCN problem. Since then, many scholars have applied various evolutionary algorithms to address these problems [5,6]. For complex SDTSPs, swarm intelligence algorithms have become a primary solution method [7,8,9]. In recent years, new technologies have emerged [10] to solve satellite data transmission issues.

Although early studies have achieved some success in exploring SDTSP, they have not fully considered the cyclical nature of resource competition, the unique aspects of large-scale data transmission missions (SDTT), and the multi-dimensional optimization objectives. For instance, Earth observation satellites operate in fixed orbits, and ground stations have relatively fixed antenna receiving areas. When satellites enter these areas, they periodically compete with ground stations for data transmission, especially during large-scale transmissions. Traditional heuristic algorithms, such as simulated annealing or genetic algorithms, often fail to handle large-scale data transmission problems due to their complexity and time-consuming global search processes. Furthermore, current scheduling algorithms frequently fall into the trap of locally optimal solutions, failing to use the available time window resources fully.

To tackle these challenges effectively, employing suitable algorithms for managing large-scale multi-objective SDTSPs [11] is essential. Multi-objective evolutionary algorithms (MOEAs) have been widely used due to their excellent performance in practical problems [12,13]. The core of these algorithms lies in decision-making methods, typically categorized into Pareto dominance-based [14], index-based [15], and decomposition-based approaches [16]. However, for large-scale multi-objective optimization problems like SDTSP, traditional MOEAs often perform poorly, and research on large-scale multi-objective optimization algorithms remains limited.

Based on the Pareto dominance method, this paper develops a multi-objective SDTSP model to maximize transmission efficiency while minimizing latency. In addition, we enhance the minimum angle selection strategy of the particle swarm optimization (PSO) algorithm. This improvement effectively identifies global and local optimal particles, mitigating the risk of local optima and enhancing the efficiency of solving the SDTSP. To assess the effectiveness of this method, we conducted simulation experiments on tasks of different sizes. The findings show that the proposed method displays robust generalization capabilities and substantially enhances the efficiency of satellite data transmission tasks.

The contributions of this manuscript are as follows:

• The SDTSP was modeled as an optimization problem with two objectives: minimizing the satellite transmission time and maximizing task gain simultaneously;
• We proposed a modified MAPSO algorithm to solve the multi-objective SDTS problem. The proposed algorithm was encoded as a discrete optimizer, and the strategies were modified to enhance its searchability.

Here is the organization of this paper: Section 2 introduces the satellite data transmission problem and establishes its mathematical model. Section 3 describes our MAPSO based on the problem’s key characteristics. Section 4 details the experimental design and evaluates the findings. Finally, Section 5 offers conclusions and suggestions for further research.

2. Problem Description

First, the data transmission problem in satellite communication can be considered a combinatorial optimization challenge. Second, the process includes assigning transmission resources from the ground station when each satellite is in view. This problem can be broken down into three key decision stages:

• Ensuring that the allocation order meets all constraints;
• Determining the order of time window allocation for each satellite;
• Minimizing the satellite transmission time and maximizing task gain.

2.1. Symbol Definition

The relevant symbols are described below:

$T=\left\{t_i\right|1\le i\le n_t\}$ represent the set of data transmission tasks where $\left|T\right|=n_t.$ $S=\left\{s_j\right|1\le j\le n_s\}$ represents the collection of satellites where $\left|S\right|=n_s.$ $G=\left\{g_k\right|1\le k\le n_g\}$ represents the set of ground stations where $\left|G\right|=n_g.$ $A_{s_j}=\{a_{s_{j}m}\mid 1\le m\le m_{s_j}\}$ represents a collection of antennas, a set of antennas. The antennas at the ground station are $g_k$ with $\left|A_{g_k}\right|=d_{g_k}.$ Each antenna $a_{g_k}$ is associated with a frequency band $a{f}_{gkd}$. For each task $i, j, k, m$ $\in$ $R$, the task symbols’ definitions are given in

Table 1. Symbol of task.

Symbol	Explanation
$p_{t_i}$	$\mathrm{The} \mathrm{priority} \mathrm{of} \mathrm{satellite} t_i$
$g_{t_i}$	$\mathrm{The} \mathrm{priority} \mathrm{of} \mathrm{ground} \mathrm{stations} t_i$
s	Satellite
g	Ground stations
$t_i$	The number of tasks
$a_{s_{j}m}$	The number of antennas
$a{f}_{gkd}$	The ground station antennas
$a{f}_{s_{j}m}$	The satellite antennas

.

$W=\{w_{jk}^z\mid 1\le j\le n_s,1\le k\le n_g,1\le z\le n_w\}$ represent the set of visibility time windows between ground stations and satellites, where $\left|W\right|=n_w.$ Definitions of visibility time window symbols are given in

Table 2. Symbol of visibility time window.

Symbol	Explanation
w	The visibility time windows
$s{w}_{jk}^z$	The visibility start-time windows
$e{w}_{jk}^z$	The visibility end-time windows
$ST$	The scheduled start time
$ET$	The scheduled end time
$d_{t_i}$	The minimal transmission duration
$s_{t_i}$	The task start-time window
$e_{t_i}$	The task end-time window
$st{w}_i$	The transmit start-time window
$et{w}_i$	The transmit end-time window

.

Since PSO is typically used for solving continuous problems, and this scheduling problem is discrete, we use a binary encoding method to represent each satellite’s selection state. Among them, we use binary encoding to represent the selection status of each satellite. Specifically, the selected satellites are denoted by 1, and the unselected satellites are denoted by 0. Based on this condition, the decision variables for the data transmission tasks are established. This decision variable ensures the data transmission between the satellite and the ground station matches.

${\xi }_{\mathit{ikj}}=1$ indicates that satellite $S$ transmits data to ground station $G$ during the time window. ${\xi }_{\mathit{ikj}}=0$ indicates that satellite $S$ does not transmit data to ground station $G$ during the $u$ time window. ${\chi }_{\mathit{ikj}}=1$ indicates that ground station $G$ receives data from satellite $S,$ while ${\chi }_{\mathit{ikj}}=0$ indicates that ground station $k$ does not receive data from the satellite.

2.2. Assumptions

In order to develop analyses and research quickly, we make the following assumptions:

(1)

Visibility of Satellites and Ground Stations: Each satellite will maintain visibility with at least one ground station during its orbital period, ensuring that data transmission can be completed within the planned time window.

(2)

Uninterrupted Data Transmission: Communication between satellites and ground stations is assumed to be free from interference or blockages, ensuring the stability and continuity of data transmission.

(3)

Stable Power Supply: All satellites and ground stations are assumed to have a stable power supply. Specifically, satellites are considered to have sufficient power from batteries and solar panels to support long-duration continuous transmission tasks.

(4)

Independence of Tasks: Each data transmission task is independent, with no dependencies between tasks.

(5)

Constant Transmission Rate: The data transmission rate between satellites and ground stations is assumed to be constant, unaffected by environmental changes or equipment performance fluctuations, ensuring the predictability of task planning and execution.

2.3. Constraints

The following constraints are established based on the symbols and assumptions:

Constraint (1) guarantees that each task is scheduled within its designated transmission window:

$$\forall i:s{t}_{t_i}\le st{w}_i<et{w}_i\le e{t}_{t_i}$$

(1)

Constraint (2) ensures that each task is scheduled within the visibility time windows between ground stations and satellites:

$$\forall i,j,k,z:\left({\chi }_{\mathit{ikj}}=1\right)\cap \left({\xi }_{\mathit{ikj}}=1\right)\cap \left(s{w}_{jk}^z\le st{w}_i<et{w}_i\le e{w}_{jk}^z\right)$$

(2)

Constraint (3) indicates that there should be a correspondence in the frequency bands between the satellite and ground station antenna:

$$\forall i,j,k,d,z:\left({\chi }_{\mathit{ikj}}=1\right)\cap \left({\xi }_{\mathit{ikj}}=1\right)\cap \left(a{f}_{s_{j}m}=f_{t_i}\right)\cap \left(a{f}_{gkd}=f_{t_i}\right)$$

(3)

Constraint (4) indicates that a single ground station antenna cannot handle multiple tasks concurrently:

$$\forall i,j,k;\exists i_1,i_2\left(i_1\ne i_2\right):\left({\chi }_{i1\mathit{kj}}\cdot {\chi }_{i2\mathit{kj}}=0\right)\cup \left(T{W}_{i_1}\cap T{W}_{i_2}=\varnothing \right)$$

(4)

Constraint (5) guarantees that each available task is allocated to no more than one ground station antenna:

$$\forall i,j,k,d,z; \exists k_1,k_2, j_1,j_2\left(\left(k_1\ne k_2\right)\cup \left(d_1\ne d_2\right)\right):{\chi }_{\mathit{ik}1j1}+{\chi }_{\mathit{ik}2j2}\le 1$$

(5)

Constraint (6) ensures that each task’s allocated transmission time slot meets the minimum transmission duration criteria:

$$\forall i:du{r}_{t_i}\le et{w}_i-st{w}_i$$

(6)

Constraint (7) specifies that each task is scheduled at most once:

$$\forall i,j, k:\sum _i^{t_i}{\chi }_{\mathit{ikj}}{ \cdot \xi }_{\mathit{ikj}} \le 1$$

(7)

Constraint (8) guarantees that the interval between two tasks should be sufficient to accommodate the antenna transition time. $C$ is a constant:

$$\forall i_1, i_2, j_1,j_2,m_1,m_2,z_1,z_2,k,d:\left({\chi }_{i1\mathit{kj}}\hspace{0.25em}\cdot \hspace{0.25em}{\chi }_{i2k2j}=1\right)\cap \left[max\left(st{w}_{i_1},st{w}_{i_2}\right)-min\left(et{w}_{i_1},et{w}_{i_2}\right)\ge C\right]$$

(8)

Constraints (9) and (10) define the scope of values for the choice variables:

$$\forall i,j,m,k,d,z:x_{ijmkd}^z\in \{\mathrm{0,1}\}$$

(9)

$$\forall i:st{w}_i,et{w}_i\ge 0$$

(10)

2.4. Optimization Objectives

Two objectives equations are based on the characteristics of the data transmission tasks. The objective function $F$ encompasses two objectives, $f_1$ and $f_{2 }$:

• Maximize task gain: The task gain is meant to maximize the product of ground station and satellite priorities, determined by the following Equation (11):

  $$f_1=\mathit{max}\sum _{i=1}^n{p}_{t_i}·g_{k_i}$$

  (11)
• Minimize transmission time: Reduce the total time needed to accomplish the data transmission tasks, as calculated by the following Equation (12):

  $$f_2=\mathit{min} \left(ET-ST\right)$$

  (12)

The comprehensive objective function, denoted as $F$, is formulated as shown in Equation (13):

$$\mathit{max} F=\{f_1,{ -f}_2\}$$

(13)

The proposed model effectively simplifies the complexities of large-scale satellite data transmission scheduling constraints to ensure that the scheduling of satellite data transmissions is efficient.

3. Methodology

3.1. Minimum Angle Particle Swarm Optimization

In multi-objective evolutionary algorithms, the MAPSO algorithm was first proposed by Gong [17] in 2005 as one of the pioneering angle-based evolutionary algorithms. In 2021, Yang [18] improved upon this algorithm by using vector-based angles to select elite particles, addressing the rapid convergence issue of PSO algorithms and finding wide applications in various fields [19,20].

To address the high complexity and large scale of data transmission tasks, we employ and improve the MAPSO algorithm [18] to balance convergence and diversity in the external archive set. By utilizing the minimum angle strategy to select global guides, the algorithm effectively avoids local optima, thereby improving solution quality, reducing solution time, and enhancing the efficiency of satellite data transmission tasks.

3.2. General Framework

The flowchart of the MAPSO algorithm is illustrated in Figure 1. Initially, parameters such as the number of ground stations, satellites, and the population size are input. A set of uniformly distributed weight vectors [21] is then generated. The population $p$ is initialized by generating a group of randomly distributed particles in space with initial velocities $v_i$ (where $\left\{v_i\right|1\le i\le n,i\in Z\}$). An initial archive $A$ is established to store the non-dominated solutions of population $p$, which is updated based on the clone strategy.

After completing the population initialization, we employ the archive update strategy to update the external archive. At the start of each iteration, the contents of archive $A$ are copied to prevent the loss of solution sets. The PSO search strategy selects leaders from the archive to update the population. After updating the population, the archive update strategy is employed again to update the sets $P$ and $A$. The procedure is repeated until the end condition is met. The algorithm comprises three key components: the clone, the PSO search process, and the archive update operation. The upcoming sections will offer a detailed introduction to these three components.

3.3. Clone

As shown in Algorithm 1, the clone steps aim to enhance the diversity and convergence of the external archive $A$, thereby accelerating the PSO convergence speed, particularly for the parameter-independent MAPSO algorithm. The process involves the following steps: initialization, SDE distance calculation, selection and duplication, and generation of offspring population. The clone process is illustrated in Figure 2. Below are the detailed descriptions.

Initialization: First, copy the external archive $A$ to a temporary archive $B$ and initialize a temporary archive $C$ as empty. Make sure that Archive $A$ remains intact in the following steps.

Shift-based density estimation (SDE) distance calculation: SDE is an improved density estimation strategy designed to address the challenges faced by Pareto-based evolutionary multi-objective optimization algorithms when handling multi-objective optimization problems. Traditional density estimation methods only reflect the distribution of individuals in the population, while SDE considers individuals’ distribution and convergence information. Calculate the SDE distance for each solution $b_i$ in $B.$ The SDE distance measures the diversity of the solutions. Refer to the relevant literature for specific calculation steps and equations [22].

Algorithm 1 Clone Process

Input: $\mathrm{External} \mathrm{repository}$

Output: $\mathrm{New} \mathrm{temporary} \mathrm{solution} \mathrm{set} S$

$B \leftarrow A$ $C \leftarrow \varnothing$ $\mathrm{AssignSDE}(B)$ $\mathrm{sort}\left(B\right)$ // Sort B based on the SDE metric if $\mathrm{Length}(B) > \mathit{NC}$ then $B=B[:, \mathit{NC}]$ end if for $i=1 \mathrm{to} \mathrm{Length}\left(B\right)$ do Calculate $n_i$ Create $n_i$solutions for $b_i$ and append to $C$ end for S←modify(C) Return S

Selection and duplication: Next, select $\mathit{NC}$ solutions from the temporary archive $B$ for duplication. The number of duplications $n_i$ for each solution $b_i$ is calculated using the following (14):

$$n_i=⌈N\times {\displaystyle \frac{SDE\left(b_i\right)}{\sum _{i=1}^{NC} SDE\left(b_i\right)}}⌉$$

(14)

where $\mathit{SDE}(b_i)$ presents the SDE distance of the solution $b_i$, and $N$ is the total number of solutions to be duplicated. This step ensures that solutions with more considerable SDE distances obtain more duplication opportunities, thereby enhancing the diversity and convergence of the population.

Generation of offspring population: Each solution $b_i$ is duplicated $n_i$ times and added to the temporary archive $C$, forming a new offspring population $C$. This population then undergoes evolutionary search, applying operations such as crossover and mutation, resulting in a new offspring population $S$. The clone selects superior solutions based on SDE distance, ensuring that the external archive $A$ maintains good diversity and convergence, effectively guiding the evolution of population $P$ in PSO and avoiding a local optimum. The evolutionary search strategy in MAPSO is implemented in the same manner as in NSGA-II [23].

The clone guarantees the population’s diversity and convergence and accelerates MAPSO convergence speed. Through these steps, the external archive $A$ exhibits excellent diversity and convergence, promoting the practical evolution of the MAPSO population $P$, making the algorithm more efficient in handling multi-objective optimization problems.

3.4. Archive Update

The archive update step in Algorithm 2 determines which solutions will be added to archive $A$. The performance of multi-objective algorithms is closely tied to selecting the external archive set. Therefore, choosing solutions that exhibit good convergence and diversity is crucial. The clone process is illustrated in Figure 3. Below are the detailed descriptions.

Algorithm 2 Archive Update Process

$\mathbf{Input}: \mathrm{External} \mathrm{repository} A, \mathrm{population} P, \mathrm{population} \mathrm{size} N, \mathrm{data}$

$\mathbf{Output}: \mathrm{Updated} \mathrm{repository} A$

$C \leftarrow A\cup P$$$ $\mathbf{If} A$$\mathrm{is} \mathrm{not} \mathrm{empty} \mathbf{then}$ $C \leftarrow \mathrm{Normalize}(C)$$$ $C \leftarrow \mathrm{EvaluateFitness}(C)$$$ $\mathit{Cnd} \leftarrow \mathrm{SelectNonDominated}(C)$$// \mathrm{Keep} \mathrm{only} \mathrm{non}-\mathrm{dominated} \mathrm{solutions} Cnd$$\mathrm{in} C$$$ $\mathbf{else}$ $\mathbf{if} \mathrm{size}(Cnd$$) >N$$\mathbf{then}$ $A \leftarrow \mathrm{SelectMaxMinAngle}(\mathit{Cnd}, A)$ $\mathbf{else}$ $A \leftarrow C$ end if Return A

The archive update operation updates the external archive A through the following steps:

• Merge archives: Combine archive $A$ with population $P$, creating a joint population $C$ that includes both $A$ and $P$.
• Normalization: Normalize each solution in the joint population $C$, standardizing the objective values to a range between 0 and 1.
• Calculation fitness: Calculate the fitness value for each solution in $C$ using Equation (15):

  $$\mathit{fit}\left(c_j\right)=\sqrt{\sum _{i=1}^m{f}_i(c_j{)}^2}$$

  (15)
• Non-dominated sorting: Pareto dominance filters out non-dominated solutions from C, forming the non-dominated solution set Cnd.
  • If the number of solutions in Cnd exceeds the population size $N$, use the minimum angle selection strategy to select solutions added to $A$;
  • If the number of solutions in Cnd is less than or equal to N, directly add all solutions in Cnd to $A$. Return updated archive: return the updated archive $A$.

The $\mathit{fit}\left(c_j\right)$ is the fitness value of the $c_j$ solution in the joint population $C$.

Fitness value serves as a standard for evaluating the convergence of the solutions. Subsequently, the non-dominated solution set $\mathit{Cnd}$ is selected from $C$-based Pareto dominance criteria.

Because in MAPSO, most solutions are non-dominated, the minimum angle selection strategy is adopted when the Pareto dominance criterion is no longer adequate. The steps of the minimum angle selection strategy, as detailed in Algorithm 3, can be divided into three main parts:

• Eliminate dominance-resistant solutions (DRS):
  • First, remove DRS from the non-dominated solutions, Cnd. These solutions are highly inferior in at least one objective and difficult to dominate;
  • Calculate the minimum vector angle for each ${\delta }_{\mathit{ij}}$ solution in $\mathit{Cnd}$, representing the smallest angle between the solution and other solutions;
  • Add m extreme solutions to archive $A$. Extreme solutions are those in $\mathit{Cnd}$ that have the smallest angle with the reference vectors (1, 0, …, 0), (0, 1, …, 0), …, (0, 0, …, 1).
• Remove minimum vector angle of solutions:
  • Remove solutions with the smallest vector angles from Cnd until the number of solutions in $Cnd$ reaches N-m;
  • During the removal process, select solutions with lower fitness values, meaning those with fitness values lower than other solutions;
  • Removed solutions are added to the removed solution set $R$.

The angle calculation method directly affects particle distribution and search space exploration. This study uses the following (16) equation for calculating the angle between particles:

$${\delta }_{\mathit{ij}}=\delta \left({\mathit{x}}_{\mathit{i}},{\mathit{x}}_{\mathit{j}}\right)=\mathit{arcsin}{\displaystyle \frac{f\left({\mathit{x}}_{\mathit{i}}\right)·f\left({\mathit{x}}_{\mathit{j}}\right)}{|f\left({\mathit{x}}_{\mathit{i}}\right)|·|f\left({\mathit{x}}_{\mathit{j}}\right)|}}$$

(16)

instead of (17):

$${\delta }_{\mathit{ij}}=\delta \left({\mathit{x}}_{\mathit{i}},{\mathit{x}}_{\mathit{j}}\right)=\mathit{arcos}{\displaystyle \frac{f\left({\mathit{x}}_{\mathit{i}}\right)·f\left({\mathit{x}}_{\mathit{j}}\right)}{|f\left({\mathit{x}}_{\mathit{i}}\right)|·|f\left({\mathit{x}}_{\mathit{j}}\right)|}}$$

(17)

The benefits of using Equation (16) (based on arcsin) are as follows:

(1)

Intuitive geometric interpretation and visualization: The range of arcsinX is closer to actual movement angles, making it easier to interpret and visualize particle movement paths. Due to its smaller range, arcsinX can better describe angle changes in physical systems, preventing significant angle jumps and facilitating fine-tuning.

Assume a satellite is orbiting along a circular path around the Earth. The angle $\theta$ between the satellite and a point on the Earth’s surface can be represented as θ = arcsin(d/h), where $d$ is the distance along the Earth’s surface from the point directly beneath the satellite to the point of interest, and $h$ is the altitude of the satellite.

Algorithm 3 Minimum Angle Selection Process

$\mathbf{while} \mathrm{size}\left(\mathit{Cnd}\right)> N$$\mathrm{do}$ $[\mathrm{MaxVal}, \mathit{Cnd}] \leftarrow \mathrm{FindMax}(\mathit{Cnd})$ $C \leftarrow \mathrm{Normalization}(C)$$// \mathrm{Identify} \mathrm{the} \mathrm{maximum} \mathrm{objective} \mathrm{value} \mathrm{MaxVal}$ $\mathbf{If} \mathrm{MaxVal}>1 \mathbf{then}$ $\mathrm{Remove}\left(\mathit{Cnd}\right) //\mathrm{Eliminate} \mathit{Cnd}$$$ $\mathbf{else}$ $\mathrm{break}$ $\mathbf{end} \mathbf{if}$ $\mathbf{end} \mathbf{while}$ $\mathbf{if} \mathrm{size}(Cnd$$) >N$$\mathrm{then}$ $\mathrm{AddExtremeSolutions}(A, m), i \in 1, 2, \dots , m$$// \mathrm{add} m \mathrm{extreme} \mathrm{solution} E_i$$\mathrm{to} A$ $\mathbf{while} \mathrm{size}(Cnd$$) < \left(0.9\right) · \left(N-m\right)$$\mathbf{do}$ $[c_k,c_j]=\mathrm{findMinAngle}\left(\mathit{Cnd}\right)$$// \mathrm{Identify} \mathrm{two} \mathrm{solutions} c_k$$\mathrm{and} c_j$$\mathrm{with} \mathrm{the} \mathrm{smallest} \mathit{angle} {\delta }_{c_k}$$\mathrm{in} \mathit{Cnd}$ $\mathbf{if} \mathrm{Fitness} \{ c_k \} < \mathrm{Fitness}\{c_j \}$$\mathrm{then}$ $R \mathrm{U} \{ c_j \}$ $\mathrm{Remove}\{c_j \}$$//\mathrm{remove} \mathrm{the} c_j$$\mathrm{from} Cnd$ $\mathbf{else}$ $R \mathrm{U} \{ c_k \}$ $\mathrm{remove} \{{ c}_k \}$$//\mathrm{remove} \mathrm{the} c_k$$\mathrm{from} \mathit{Cnd}$ $\mathbf{end} \mathbf{if}$ $\mathbf{end} \mathbf{while}$ $A=A \mathrm{U} \mathit{Cnd}$ $\mathbf{while} \mathrm{size}\left(A\right)<N$$\mathbf{do}$ $r_k=\mathrm{FindMaxAngle}(A, R)$$// \mathrm{Find} \mathrm{the} \mathrm{solution} r_k$$\mathrm{with} \mathrm{maximum} {\delta }_{r_k}$$, \mathrm{in} R$ $A \mathrm{U} \left\{r_k\right\}$ $\mathrm{remove}\left\{r_k\right\}$$//\mathrm{Eliminate} r_k$$\mathrm{from} R$ $\mathbf{end} \mathbf{while}$ $\mathbf{end} \mathbf{if}$ $\mathrm{Return} A$

We compare it to a linear approximation of the actual movement angles to show that arcsinX provides a closer approximation. The linear approximation can be written as $\theta \approx d/h$, which is only accurate for tiny angles ($d<<h$).

When we expand $\theta =arcsin(d/h)$ with the Taylor series, we can use the general form of the Taylor series $arcsin\left(x\right)$ using Equation (18).

$$arcsin\left(x\right)=x+{\displaystyle \frac{x^3}{6}}+{\displaystyle \frac{3x^5}{40}}+O\left(x^7\right)$$

(18)

For $\theta =arcsin\left({\displaystyle \frac{d}{h}}\right)$, we consider ${\displaystyle \frac{d}{h}}$ as $x.$ Thus, the expansion becomes $\theta =arcsin\left({\displaystyle \frac{d}{h}}\right)={\displaystyle \frac{d}{h}}+{\displaystyle \frac{1}{6}}{\left({\displaystyle \frac{d}{h}}\right)}^3+{\displaystyle \frac{3}{40}}{\left({\displaystyle \frac{d}{h}}\right)}^5+O\left({\left({\displaystyle \frac{d}{h}}\right)}^7\right)$.

This expansion shows that for small values of ${\displaystyle \frac{d}{h}}$, the linear approximation ${\displaystyle \frac{d}{h}}$ is reasonable. However, as ${\displaystyle \frac{d}{h}}$ increases, higher-order terms become significant, making the approximation of $arcsin\left({\displaystyle \frac{d}{h}}\right)$ more accurate.

Figure 4 illustrates a comparison between the calculated actual movement angles$\left(\theta \right)$ using the $arcsin(d/h)$ function and a straightforward linear approximation$(d/h)$. The plot demonstrates that the arcsin function (blue curve) closely follows the accurate movement angles, especially for larger values of d, illustrating its superior accuracy. In contrast, the linear approximation (red dashed line) deviates more as d increases, highlighting the effectiveness of the arcsin approach in accurately representing movement angles.

(2)

Numerical stability: $arcsinX$ varies more smoothly within its range, and its derivative does not tend to infinity, resulting in more stable numerical computations. In contrast, $arccosX$ has larger derivative changes near boundary values, which can cause numerical instability.

(3)

Fine-grained search: For multi-objective optimization problems in satellite data transmission, optimizing transmission paths and resource allocation requires high precision and fine adjustments. $arcsinX$ offers finer search granularity, ensuring better solutions even under complex constraints. The smooth and gradual changes of $arcsinX$ lead to more uniform particle distribution and stable computations.

In summary, Equation (16) based on $arcsinX$ is more suitable for PSO algorithms dealing with complex multi-objective optimization problems, such as satellite data transmission systems, due to its intuitive geometric interpretation, numerical stability, and fine-grained search capability.

3.

Add Solutions with Maximum Vector Angles:

The process of the minimum angle selection operation is illustrated in Figure 5. The triangular point (▲) represents solutions in R, while solid (•) belongs to $A$. R represents the set of solutions that were removed in step (2). $A$ well-distributed solution $r_i$ is removed, indicating that some well-distributed solutions might be eliminated. MAPSO allows solutions removed from R to be re-added to $A$ if the number of solutions in $Cnd$ is less than N−m. Find the solution in $R$ with the maximum vector angle relative to solutions in $A$ using the following (19):

$${\delta }_{r_i }=\stackrel{\left|A\right|}{\underset{j=1}{\mathit{min}}} \left(\mathit{angle}\right(r_i,{ a}_j\left)\right), i=1, 2, \dots , \left|R\right|$$

(19)

Add this solution to $A,$ and remove it from $R$. Repeat this process until the number of solutions in $A$ reaches $N$. Merge $C$ and $R$, removing solutions already in $A$ to form the new combined population $C$. Return the updated archive $A$.

3.5. PSO Search

The procedure for PSO is detailed in Algorithm 4. The PSO method is employed to update the population $P$. Before updating the solutions’ position information, it is crucial to determine pbest and gbest. In multi-objective optimization problems, solutions cannot be directly compared using a single objective value; however, Pareto dominance can be utilized. In contrast, for many-objective optimization problems, most solutions are non-dominated, which complicates the selection of suitable solutions (pbest and gbest) for updating the population. Therefore, pbest and gbest are chosen based on the particle angle size. Similar to conventional PSO, this process includes selecting pbest and gbest and updating the velocity and position (lines 2–3 of Algorithm 4). The specific steps are as follows:

Algorithm 4 PSO Algorithm

$\mathbf{Input}: \mathrm{Population} P, \mathrm{external} \mathrm{repository} A, \mathrm{population} \mathrm{size} N$

$\mathbf{Output}: \mathrm{new} \mathrm{population} P$

$\mathbf{for} I=1 \mathrm{to} N$$\mathbf{do}$ $\mathrm{Select} {\mathit{pbest}}_i \mathrm{and} {\mathit{gbest}}_i \mathrm{based} \mathrm{on} \mathrm{the} \mathrm{vector} \mathit{angle} \mathrm{in} A$ $\mathrm{Update} \mathrm{the} \mathrm{velocity} v \mathrm{and} \mathrm{position} x$ end for Return P

Choosing pbest and gbest: $A$ penalty factor ${\theta }_i$ balances population diversity and convergence when choosing pbest and gbest. $A$ higher ${\theta }_i$ encourages diversity, while a lower ${\theta }_i$ favors convergence. The equation for calculating ${\theta }_i$ is given in Equation (20):

$${\theta }_i=k\times \angle (p_i,{\lambda }_i)\in 1,2,\dots ,N$$

(20)

$k$ represents a scaling parameter, and $\angle \left(p_i, {\lambda }_i\right)$ denotes the vector angle between $p_i$ and ${\lambda }_i$, with N as the population size. This vector angle is normalized to range between 0° and 90°.

Figure 6 illustrates the selection of $\mathit{pbest}$. The red lines represent the reference vectors ${\lambda }_1$ and ${\lambda }_2$. $A$ smaller vector angle between $p_1$ and ${\lambda }_1$ indicates that $p_1$ has good diversity. By using the equation above to calculate ${\theta }_1$, a particle with a smaller angle can be chosen as pbest₁ to guide $p_1$ rather than another particle with better diversity but worse convergence. Similarly, $p_2$ should be selected as gbest₂ for its diversity rather than $p_1$, which might have better convergence. For MAOP, there is no single optimal solution, so a solution close to ${\lambda }_i$ from A is selected as ${\mathit{gbest}}_i$ to improve diversity. In Figure 6, $p_1$ and $p_2$ will be chosen as gbest₁ and gbest₂, respectively. Sometimes, ${\mathit{pbest}}_i$ and ${\mathit{gbest}}_i$ are equal, indicating that the solution can provide diversity and convergence directions.

Updating particle position and velocity: Once pbest and gbest have been identified, the particle’s velocity $v_i$ is determined using the following Equation (21):

$$v_i\left(t+1\right)=\omega v_i\left(t\right)+c{r}_1\left(x_{{\mathit{pbest}}_i}\left(t\right)-x_i\left(t\right)\right)+\mathit{sgn}\left(\mathit{fit}\right({\mathit{gbest}}_i)-\mathit{fit}(p_i\left)\right)F_i{r}_2(x_{{\mathit{gbest}}_i}\left(t\right)-x_i\left(t\right))$$

(21)

where $x_i$ represents the position of the i-th solution in the population, $r_1$ and $r_2$ are evenly dispersed within the range of [0, 1], $\omega$ denotes the inertia weight, and c represents the learning factor. $F_i$ balances the search direction between pbest_i and gbest_i, calculated as Equation (22):

$$F_i={\displaystyle \frac{\frac{\pi }{2}-\angle ({\mathit{gbest}}_i,{\lambda }_i)}{\frac{\pi }{2}}}+{\displaystyle \frac{\mathit{fit}({\mathit{pbest}}_i)}{\mathit{fit}({\mathit{gbest}}_i)}}$$

(22)

Based on Equation (22), the overall search direction is primarily influenced by pbest_i and gbest_i. The branch search direction of ${\mathit{gbest}}_i$ is governed by $\mathit{fit}({\mathit{gbest}}_i)-\mathit{fit}(p_i)$, which emphasizes diversity rather than convergence. If ${\mathit{gbest}}_i$ fitness is higher than $p_i,$it indicates that $p_i$ is closer to the Pareto front (PF) than ${\mathit{gbest}}_i$. The search direction is from ${\mathit{gbest}}_i$ to $p_i;$ otherwise, it is the opposite. Finally, the position $x_i$ is updated as follows (23):

$$x_i(t+1)=x_i\left(t\right)+v_i(t+1)$$

(23)

If the position exceeds the boundaries in some dimensions, it is set to the boundary value, and the corresponding velocity is reset to zero.

4. Experimental Design and Results Analysis

To validate the effectiveness of the MAPSO algorithm, we created a simulation scenario in Systems Tool Kits. We generated test instances based on data transmission flows from Systems Tool Kits.

4.1. Scenario Design

In the Systems Tool Kits-designed simulation scenario, we selected four ground stations (Hainan, Kashgar, Yunnan, and Xi’an) and various numbers of satellites for data transmission experiments. We designed ten sun-synchronous orbits, each with nine satellites evenly distributed. This section shows 24 scenarios. Each scenario varies in the number of ground stations (one to four) and satellites (10 to 90), with a planning period of 24 h, from 1 March 2024 00:00:00 to 2 March 2024 00:00:00. The parameters are detailed in

Table 3. Experimental scenarios and parameters.

Experiment ID	Number of Satellites	Number of Ground Stations	Experiment ID	Number of Satellites	Number of Ground Stations
$\mathrm{c}1$	10	1	$\mathrm{c}13$	40	3
$\mathrm{c}2$	20	1	$\mathrm{c}14$	50	3
$\mathrm{c}3$	30	1	$\mathrm{c}15$	60	3
$\mathrm{c}4$	40	1	$\mathrm{c}16$	10	4
$\mathrm{c}5$	10	2	$\mathrm{c}17$	20	4
$\mathrm{c}6$	20	2	$\mathrm{c}18$	30	4
$\mathrm{c}7$	30	2	$\mathrm{c}19$	40	4
$\mathrm{c}8$	40	2	$\mathrm{c}20$	50	4
$\mathrm{c}9$	50	2	$\mathrm{c}21$	60	4
$\mathrm{c}10$	10	3	$\mathrm{c}22$	70	4
$\mathrm{c}11$	20	3	$\mathrm{c}23$	80	4
$\mathrm{c}12$	30	3	$\mathrm{c}24$	90	4

.

The essential attributes for the transmission tasks are as follows:

• Each satellite can only transmit data to one ground station;
• The transmission priority of each satellite is randomly assigned using a 1–10 scale, where 10 denotes the highest priority;
• The priorities for the ground stations are Hainan (seven), Kashgar (five), Xi’an (ten), and Yunnan (three);
• The transmission time window must fall within the planning period, and if the window is shorter than 7 min, it must equal the task’s minimum required transmission time;
• All tasks use a standard communication protocol between satellites and ground stations;
• The transmission frequency band of all tasks is set to be S-band.

This section evaluates MAPSO by comparing it with NSGA-II [24], MOEA/D [25], MOGA [26], DRSC-MOAGDE [27], DSC-MOPSO [28], and MO-Ring-PSO-SCD [29]. NSGA-II is Pareto-based, MOGA is indicator-based, and MOEA/D is decomposition-based. DRSC-MOAGDE has a competitive performance on practical MOP, DSC-MOPSO is a multi-objective PSO algorithm with dynamic-switching crowding, MO-Ring-PSO-SCD is a powerful PSO algorithm for solving multi-objective problems. We generated 24 test instances, as described in Section 4.1. Each test instance was run 30 times, and the average value was taken as the final result. Our algorithm was implemented using MATLAB R2022b and tested on a personal computer equipped with an AMD Ryzen 7 4800H CPU with Radeon Graphics (2.90 GHz).

4.2. Experimental Results and Performance Analysis

We tested 24 experimental scenarios recorded in Table 3 using MAPSO, NSGA-II, MOEA/D, and MOGA algorithms. The NSGA-II parameters were set as population size 100, maximum iterations 100, crossover probability 0.9, mutation probability 0.1, and roulette wheel selection method. The optimization parameters of MAPSO were population size 100, inertia weight 0.7, and learning factors c1 = c2 = 1.5. MOEA/D parameters were population size 100, neighborhood size 20, mutation probability 0.02, crossover probability 0.9, and maximum iterations 100. MOGA parameters were population size 100, crossover probability 0.9, mutation probability 0.02, maximum iterations 100, and roulette wheel selection.

Table 4. Algorithm time consumption in different scenarios (s).

Experiment ID	MAPSO	DRSC-MOAGDE	DSC-MOPSO	MO-Ring-PSO-SCD	NSGA-II	MOEA/D	MOGA
$\mathrm{c}1$	68.721	60.018	63.879	101.592	166.0427	207.0557	120.4097
$\mathrm{c}2$	104.373	95.499	99.312	142.554	201.463	243.6916	163.3364
$\mathrm{c}3$	140.589	133.62	132.381	179.064	241.8197	276.7497	196.0603
$\mathrm{c}4$	176.731	171.462	168.063	217.887	275.8042	309.9191	231.333
$\mathrm{c}5$	75.306	71.46	69.561	116.76	168.3898	206.9387	134.2394
$\mathrm{c}6$	101.019	97.326	93.876	144.621	196.8151	238.5583	160.6354
$\mathrm{c}7$	132.663	125.976	128.64	176.328	225.8332	268.8849	186.2558
$\mathrm{c}8$	168.387	159.555	160.029	203.223	266.7544	303.7772	225.4224
$\mathrm{c}9$	201.054	193.992	198.036	236.07	297.1121	339.3375	257.7917
$\mathrm{c}10$	75.714	70.197	68.253	117.27	176.9157	217.1687	135.2166
$\mathrm{c}11$	97.029	89.871	93.825	141.975	194.9647	237.4704	152.9181
$\mathrm{c}12$	120.222	114.033	116.661	172.062	219.9124	258.5598	179.7287
$\mathrm{c}13$	159.045	152.28	153.903	209.04	252.822	295.9644	217.6764
$\mathrm{c}14$	190.179	184.935	181.431	235.839	288.494	326.0212	246.127
$\mathrm{c}15$	232.329	223.425	224.913	282.906	333.6264	369.0082	284.6459
$\mathrm{c}16$	67.447	72.582	74.13	111.975	167.7088	200.2816	124.347
$\mathrm{c}17$	80.087	85.95	87.579	128.79	179.2077	218.7366	139.6489
$\mathrm{c}18$	96.206	101.061	102.966	140.469	192.9514	234.0556	148.4967
$\mathrm{c}19$	119.507	126.933	127.977	167.544	219.457	256.1416	171.2443
$\mathrm{c}20$	153.452	158.637	159.156	211.776	248.7481	286.194	204.6377
$\mathrm{c}21$	186.713	195.546	194.073	238.659	284.4103	324.8533	241.7649
$\mathrm{c}22$	210.807	216.108	218.565	251.862	305.4156	349.3471	263.0285
$\mathrm{c}23$	247.022	253.851	255.438	291.633	343.9177	385.8968	300.5855
$\mathrm{c}24$	279.164	285.867	287.88	324.519	374.9307	419.8057	334.68

records the time consumed by each algorithm under different experimental scenarios, and

Table 5. Algorithm planning time for data transmission tasks in different scenarios (s).

Experiment ID	MAPSO	DRSC-MOAGDE	DSC-MOPSO	MO-Ring-PSO-SCD	NSGA-II	MOEA/D	MOGA
$\mathrm{c}1$	33,266	33,266	33,266	33,266	33,266	33,266	33,266
$\mathrm{c}2$	55,126	55,126	55,126	55,126	55,126	55,126	55,126
$\mathrm{c}3$	60,425	60,425	60,425	60,425	60,425	60,425	60,425
$\mathrm{c}4$	65,052	65,052	65,052	65,052	65,052	65,052	65,052
$\mathrm{c}5$	19,420	19,385	19,433	19,473	19,468	19,344	19,268
$\mathrm{c}6$	29,722	29,706	29,750	29,757	29,795	34,810	35,090
$\mathrm{c}7$	44,272	44,267	44,306	44,349	44,402	46,476	52,374
$\mathrm{c}8$	55,345	55,354	55,393	55,482	55,619	57,923	63,666
$\mathrm{c}9$	61,746	61,763	61,808	61,995	62,671	62,897	62,647
$\mathrm{c}10$	17,299	17,287	17,325	17,372	17,457	17,532	17,740
$\mathrm{c}11$	21,268	21,264	21,297	21,404	21,422	26,604	26,763
$\mathrm{c}12$	28,012	28,034	28,069	28,250	28,562	29,382	34,825
$\mathrm{c}13$	44,455	44,619	44,770	45,216	46,153	48,141	51,127
$\mathrm{c}14$	46,449	46,460	46,518	46,571	46,720	47,884	55,554
$\mathrm{c}15$	48,428	48,491	48,604	48,742	49,619	53,497	53,570
$\mathrm{c}16$	12,597	12,469	12,505	12,519	12,536	12,561	12,630
$\mathrm{c}17$	21,262	21,183	21,208	21,234	21,267	21,794	22,374
$\mathrm{c}18$	23,899	23,925	23,958	24,016	24,087	25,035	25,224
$\mathrm{c}19$	30,815	30,953	31,056	31,257	31,368	33,324	35,193
$\mathrm{c}20$	40,033	40,189	40,246	40,448	40,504	43,510	45,514
$\mathrm{c}21$	43,309	43,852	44,127	45,036	46,362	47,955	50,385
$\mathrm{c}22$	44,965	45,431	45,751	46,460	47,707	50,446	51,806
$\mathrm{c}23$	45,630	46,062	46,359	47,198	48,247	50,707	52,055
$\mathrm{c}24$	47,409	47,736	48,094	48,731	49,974	51,080	52,337

records the time each algorithm takes to schedule data transmission tasks in different scenarios. The MAPSO algorithm improved data transmission efficiency using the minimum angle method to select particles with the best fitness for gbest and pbest. The archive update operation also updated the archive set, making MAPSO the fastest in solving satellite data transmission problems across all scenarios.

We selected the most complex scenarios from Table 1: C4 (one station, 40 satellites), C9 (two stations, 50 satellites), C15 (three stations, 60 satellites), and C24 (four stations, 90 satellites) to plot Pareto fronts. Figure 7 displays the Pareto fronts obtained by the four optimization algorithms under different scenarios. Scenarios C1 to C4 were not chosen because when there is only one ground station, satellites transmit data sequentially within the time window, resulting in fixed values for the shortest transmission time and maximum priority between satellites and ground stations, making it impossible to plot corresponding Pareto fronts. Compared to NSGA-II, MOEA/D, and MOGA, the MAPSO algorithm shows significantly better performance in terms of Pareto front diversity and uniformity.

Figure 8 illustrates the final scheduling results for satellite data transmission tasks under different scenarios. The x-axis represents the scheduling time window, and the y-axis represents the ground stations. Satellites are recorded in the figure by sequence number. If a satellite is within the range of 1 on the y-axis, it indicates that the satellite is transmitting data to the first ground station, and so on. In Figure 8, we present a selected Pareto optimal scheduling result for different configurations of ground stations and satellites. Each subplot illustrates a scheduling outcome under varying conditions. The displayed scheduling result was chosen because it effectively demonstrates the balance between minimizing transmission time and maximizing resource utilization. While multiple Pareto optimal solutions exist, this result provides a representative and clear visualization of our algorithm’s performance. It highlights how our approach handles the complexities of large-scale multi-objective optimization in satellite data transmission scheduling. The results in Figure 8 show that the MAPSO algorithm provides a better scheduling plan for satellite data transmission tasks.

The MAPSO algorithm enhances the efficiency of solving multi-objective SDTSPs, making it more suitable for handling satellite data transmission tasks across various scenarios. MAPSO enhances the system’s service capability, meeting diverse user needs. The scheduling results indicate an improved Pareto front, increasing the efficiency and accuracy of satellite data transmission tasks. The MAPSO algorithm employs a minimum angle particle selection strategy in dynamic scheduling, effectively selecting gbest and pbest particles. Combined with objective functions tailored to satellite data transmission characteristics, this increases the probability of generating better offspring during iterations, enhancing optimization performance.

4.3. Stability Analysis

BIG-O notation, typically employed to characterize the time complexity of algorithms, is not well suited for quantifying the computational complexity of evolutionary algorithms (EAs). This inadequacy stems from EAs not navigating within differentiable spaces but instead seeking global optima in complex, multimodal problem landscapes. Such landscapes often feature many local optima, which can entrap the algorithm and result in early convergence. In these contexts, the algorithm’s internal complexity is of lesser importance. As the search process defies straightforward optimization, the computational complexity, or search time, is potentially infinite. Consequently, assessing the computational complexity of the MAPSO algorithm necessitates stability analysis.

The stability analysis method quantifies evolutionary algorithms’ time taken and success rate in discovering feasible solutions within a controlled simulation environment [30]. For its innovative approach and effectiveness over other metrics, this research utilizes success rate (SR), mean fitness evaluations (MFE), and mean search time (MST) as the criteria for stability analysis. The collective performance of seven algorithms is evaluated by determining the values of SR, MFE, and MST.

The fundamental goal of the MAPSO algorithm is to identify feasible solutions for the SDTSP. Its success rate (SR) is computed using Equation (24), where $aa$ signifies the count of independent trials that successfully yield a feasible solution, and $tt$ denotes the overall number of trials conducted:

$$SR={\displaystyle \frac{aa}{tt}}\times 100$$

(24)

A further aspect of stability assessment involves monitoring the frequency of fitness evaluations executed by the algorithm. In detail, the fitness evaluation counter (FE) is incremented with every invocation of the objective function by the algorithm. The search concludes once the algorithm effectively locates a viable solution, at which point the FE counter’s tally is noted. Following this methodology, the mean number of fitness evaluations (MFE) across successful trials is computed using Equation (25):

$$MFE={\displaystyle \frac{1}{aa}}{\sum }_{i=1}^{aa}{fes}_{\left[i\right]}$$

(25)

Another critical aspect of stability analysis is examining the time required by the algorithm to find a feasible solution. Once a feasible solution is located, the search halts, and the search time (ST) is recorded. The average search time (MST) for successful runs is calculated using Equation (26):

$$MST={\displaystyle \frac{1}{aa}}{\sum }_{i=1}^{aa}{st}_{\left[i\right]}$$

(26)

The methods for calculating SR, MFE, and MST are based on the stability analysis described in [27], with detailed steps outlined in Algorithm 5. The statistical results for the SR, MFE, and MST of seven algorithms across 24 scenarios are presented in

Table 6. SR performance of algorithms.

Experiment ID	MAPSO	DRSC-MOAGDE	DSC-MOPSO	MO-Ring-PSO-SCD	NSGA-II	MOEA/D	MOGA
$\mathrm{c}1$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}2$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}3$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}4$	88.54	90.42	88.51	85.62	79.73	66.93	67.94
$\mathrm{c}5$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}6$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}7$	100.00	100.00	100.00	95.07	88.40	87.64	77.79
$\mathrm{c}8$	93.91	94.33	95.45	88.50	82.20	78.12	75.14
$\mathrm{c}9$	81.70	81.95	82.10	78.86	76.84	70.30	71.44
$\mathrm{c}10$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}11$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}12$	100.00	100.00	100.00	93.73	90.37	84.51	80.48
$\mathrm{c}13$	82.37	81.66	79.66	73.16	70.19	66.96	60.49
$\mathrm{c}14$	76.63	74.38	74.34	71.79	65.05	61.00	56.73
$\mathrm{c}15$	72.44	69.36	69.54	64.48	57.98	56.64	53.14
$\mathrm{c}16$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}17$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}18$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}19$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}20$	100.00	100.00	100.00	100.00	100.00	100.00	100.00
$\mathrm{c}21$	100.00	100.00	97.63	89.01	80.47	68.97	59.31
$\mathrm{c}22$	79.15	77.46	75.71	70.32	60.85	59.02	55.56
$\mathrm{c}23$	78.55	73.65	68.89	65.36	52.13	47.43	30.94
$\mathrm{c}24$	75.62	67.31	66.41	54.77	38.35	28.01	2.59

,

Table 7. MST performance of algorithms.

Experiment ID	MAPSO	DRSC-MOAGDE	DSC-MOPSO	MO-Ring-PSO-SCD	NSGA-II	MOEA/D	MOGA
$\mathrm{c}1$	2.20	2.12	2.23	2.51	3.05	3.27	3.43
$\mathrm{c}2$	2.31	2.07	2.15	2.64	3.26	3.52	3.95
$\mathrm{c}3$	2.44	2.33	2.30	2.96	3.41	3.80	4.56
$\mathrm{c}4$	3.57	3.43	3.47	3.72	4.11	4.61	5.71
$\mathrm{c}5$	2.21	2.17	2.25	2.57	3.17	3.43	3.68
$\mathrm{c}6$	2.85	2.77	2.82	3.03	3.74	4.15	4.53
$\mathrm{c}7$	3.16	3.38	3.48	3.61	4.82	5.34	5.94
$\mathrm{c}8$	3.82	3.66	3.73	3.95	5.27	5.75	6.67
$\mathrm{c}9$	4.47	4.27	4.36	4.62	6.10	6.46	7.49
$\mathrm{c}10$	2.84	2.75	2.82	2.99	3.33	3.57	3.82
$\mathrm{c}11$	3.44	3.36	3.39	3.72	4.72	4.62	5.79
$\mathrm{c}12$	4.31	4.12	4.22	4.86	5.95	5.80	6.51
$\mathrm{c}13$	4.75	4.77	4.81	5.25	6.31	6.33	7.63
$\mathrm{c}14$	5.19	5.26	5.46	5.95	7.32	7.41	8.77
$\mathrm{c}15$	5.43	5.69	5.75	6.54	8.05	8.18	9.43
$\mathrm{c}16$	2.37	2.43	2.50	2.71	3.41	3.57	4.20
$\mathrm{c}17$	2.89	2.92	2.93	3.28	4.05	4.12	5.32
$\mathrm{c}18$	3.36	3.48	3.54	3.83	4.77	4.85	6.34
$\mathrm{c}19$	4.55	4.61	4.78	5.06	5.92	5.89	7.87
$\mathrm{c}20$	5.22	5.37	5.46	6.22	7.18	7.34	9.24
$\mathrm{c}21$	6.38	6.59	6.62	7.47	8.37	8.41	10.32
$\mathrm{c}22$	7.49	7.60	7.57	8.51	9.54	9.66	12.58
$\mathrm{c}23$	8.52	8.73	8.89	9.49	10.79	10.94	14.95
$\mathrm{c}24$	9.86	10.11	10.20	12.77	13.22	13.53	16.62

and

Table 8. MFE performance of algorithms.

Experiment ID	MAPSO	DRSC-MOAGDE	DSC-MOPSO	MO-Ring-PSO-SCD	NSGA-II	MOEA/D	MOGA
$\mathrm{c}1$	2.2907	2.0006	2.1293	3.3864	5.534757	6.901857	4.013657
$\mathrm{c}2$	3.4791	3.1833	3.3104	4.7518	6.715433	8.123053	5.444547
$\mathrm{c}3$	4.6863	4.454	4.4127	5.9688	8.060657	9.22499	6.535343
$\mathrm{c}4$	5.8910	5.7154	5.6021	7.2629	9.193473	10.33064	7.7111
$\mathrm{c}5$	2.5102	2.382	2.3187	3.892	5.612993	6.897957	4.474647
$\mathrm{c}6$	3.3673	3.2442	3.1292	4.8207	6.560503	7.951943	5.354513
$\mathrm{c}7$	4.4221	4.1992	4.288	5.8776	7.527773	8.96283	6.208527
$\mathrm{c}8$	5.6129	5.3185	5.3343	6.7741	8.891813	10.12591	7.51408
$\mathrm{c}9$	6.7018	6.4664	6.6012	7.869	9.903737	11.31125	8.593057
$\mathrm{c}10$	2.5238	2.3399	2.2751	3.909	5.89719	7.238957	4.50722
$\mathrm{c}11$	3.2343	2.9957	3.1275	4.7325	6.498823	7.91568	5.09727
$\mathrm{c}12$	4.0074	3.8011	3.8887	5.7354	7.330413	8.61866	5.990957
$\mathrm{c}13$	5.3015	5.076	5.1301	6.968	8.4274	9.86548	7.25588
$\mathrm{c}14$	6.3393	6.1645	6.0477	7.8613	9.616467	10.86737	8.204233
$\mathrm{c}15$	7.7443	7.4475	7.4971	9.4302	11.12088	12.30027	9.488197
$\mathrm{c}16$	2.2482	2.4194	2.4710	3.7325	5.590293	6.676053	4.1449
$\mathrm{c}17$	2.6695	2.8650	2.9193	4.293	5.97359	7.29122	4.654963
$\mathrm{c}18$	3.2068	3.3687	3.4322	4.6823	6.431713	7.801853	4.94989
$\mathrm{c}19$	3.9835	4.2311	4.2659	5.5848	7.315233	8.538053	5.708143
$\mathrm{c}20$	5.1150	5.2879	5.3052	7.0592	8.291603	9.5398	6.821257
$\mathrm{c}21$	6.2237	6.5182	6.4691	7.9553	9.480343	10.82844	8.05883
$\mathrm{c}22$	7.0269	7.2036	7.2855	8.3954	10.18052	11.6449	8.767617
$\mathrm{c}23$	8.2340	8.4617	8.5146	9.7211	11.46392	12.86323	10.01952
$\mathrm{c}24$	9.3054	9.5289	9.5960	10.8173	12.49769	13.99352	11.156

.

Algorithm 5 Stability Analysis

$\mathbf{Input}: \mathrm{feasible} \mathrm{solution} \mathrm{defined} \mathrm{for} \mathrm{the} \mathrm{problem}$

$\mathbf{Output}: SR, MFE, MST$

$\mathbf{for} i=1 \mathrm{to} \mathrm{run} //\mathrm{total} \mathrm{count} \mathrm{of} \mathrm{independent} \mathrm{runs}$ $\mathbf{While} (fes$$< max FEs$$) \mathrm{do}$ $\mathrm{Start} \mathrm{the} \mathrm{timer}, \mathrm{then} \mathrm{initiate} \mathrm{the} \mathrm{evolutionary} \mathrm{search} \mathrm{steps}$ $\mathbf{If} (fes$$> 0.3 * max FEs$$\mathrm{and} \mathrm{best}\_\mathrm{solution} \le FS$$)$ $\mathrm{Stop} \mathrm{the} \mathrm{timer} \mathrm{and} \mathrm{record} \mathrm{the} \mathrm{values} \mathrm{of} ST\left[i\right]$$\mathrm{and} FE\left[i\right]$ $\mathrm{Increment} \mathrm{the} \mathrm{variable} \mathit{aa} \mathrm{by} 1$ $\mathrm{Exit} \mathrm{the} \mathrm{while} \mathrm{loop}$ $\mathbf{else}$ $\mathrm{Increment} \mathrm{the} \mathrm{fes} \mathrm{count} \mathrm{in} FE\left[i\right]$ end if end While end for $\mathrm{Compute} \mathit{SR}, \mathit{MFE},$$\mathrm{and} \mathit{MST}$ using Formulas (24)–(26)

Table 6 shows that the MAPSO algorithm achieves a high success rate across all scenarios, indicating its ability to find feasible solutions in every case. Table 6 demonstrates that MAPSO possesses strong robustness and reliability, effectively addressing satellite data transmission problems of varying scales and complexities. The DRSC-MOAGDE and DSC-MOPSO algorithms also achieve high success rates in smaller-scale scenarios, surpassing MAPSO. However, their success rates decline as the number of satellites and ground stations increases. Table 6 suggests these algorithms may struggle with more complex scenarios, potentially falling into local optima or exhibiting slower convergence. In contrast, the success rates of MO-Ring-PSO-SCD, NSGA-II, MOEA/D, and MOGA are generally low, particularly in large-scale scenarios. Table 6 implies that these algorithms are more prone to premature convergence or being trapped in local optima when handling complex problems.

As shown in Table 7, the MAPSO algorithm achieves relatively short average search times across all scenarios, indicating its ability to identify feasible solutions quickly. This efficiency is primarily attributed to the minimum angle particle selection strategy employed in MAPSO, which significantly enhances search performance. The DRSC-MOAGDE and DSC-MOPSO algorithms also exhibit relatively short average search times in smaller-scale scenarios. However, as the number of satellites and ground stations increases, their average search times slightly exceed those of MAPSO. Table 7 indicates that while these algorithms maintain relatively high search efficiency, they are still less effective than MAPSO.

In contrast, the average search times of MO-Ring-PSO-SCD, NSGA-II, MOEA/D, and MOGA are generally longer, particularly in large-scale scenarios. Table 7 suggests that these algorithms have lower search efficiency and require more time to find feasible solutions. As shown in Table 8, the MAPSO algorithm demonstrates relatively low average fitness evaluation counts across all scenarios, indicating its ability to find feasible solutions with fewer iterations. This efficiency is primarily attributed to the minimum angle particle selection strategy and the fitness function employed by MAPSO, which effectively enhance search performance and optimization results. Similarly, the DRSC-MOAGDE and DSC-MOPSO algorithms also exhibit low average fitness evaluation counts, though slightly higher than MAPSO in large-scale scenarios. This suggests that while these algorithms maintain relatively high optimization efficiency, they still fall short compared to MAPSO. In contrast, the MO-Ring-PSO-SCD, NSGA-II, MOEA/D, and MOGA algorithms have significantly higher average fitness evaluation counts, particularly in large-scale scenarios, indicating lower optimization efficiency and requiring more iterations to locate feasible solutions.

Based on the analysis of these three indicators, the following conclusions can be drawn: The MAPSO algorithm demonstrates superior stability in solving satellite data transmission problems. It consistently identifies feasible solutions, achieves shorter search times, and requires fewer fitness evaluations, highlighting its ability to address complex problems effectively and efficiently. While the DRSC-MOAGDE and DSC-MOPSO algorithms also exhibit high stability, they are slightly less effective than MAPSO. On the other hand, the MO-Ring-PSO-SCD, NSGA-II, MOEA/D, and MOGA algorithms show poor stability, often becoming trapped in local optima or suffering from premature convergence, making them unsuitable for solving large-scale or complex satellite data transmission problems.

4.4. Hypervolume and Friedman Evaluation Index

This study utilized the hypervolume (HV) and Friedman fraction metrics to evaluate the algorithm’s performance [31] and comprehensively analyze the diversity and convergence of the Pareto front generated by the algorithm. Each solution within the Pareto front corresponds to a scheduling plan that satisfies the dynamic scheduling demands of relay satellites under different preference scenarios. The HV metric is computed by summing the areas of the rectangles formed between each point in the non-dominated solution set and a reference point, typically the maximum individual in the objective space. A higher HV value signifies superior overall algorithm performance. The HV calculation is expressed by the following Formula (27):

$$HV=\sum _{i=1}^n \left({\displaystyle \frac{r_1-f_1\left(x_i\right)}{f_1(x_i{)}^{max}}}\right)\times \left({\displaystyle \frac{f_2\left(x_i\right)-r_2}{f_2(x_i{)}^{max}}}\right)$$

(27)

In this context, $f_1\left(x_i\right)$ and $f_2\left(x_i\right)$ denote the first and second objective values of solution $x_i$, respectively. Similarly, $r_1$ and $r_2$ are the respective coordinates of the reference point. $f_1(x_i{)}^{max}$ represents the maximum value of the first objective function $f_1$ in all solutions $x_i$. $f_2(x_i{)}^{max}$ represents the maximum value of the second objective function $f_2$ in all solutions $x_i$. Cases $c1$ through $c4$ lack a Pareto front, and as a result, they are not considered in the hypervolume (HV) assessment. Table 4 presents the HV values and the reference point coordinates for the 20 experimental scenarios. A normalization process has been implemented to avoid excessively high HV values. This process involves standardizing each objective value by dividing it by its maximum, which maintains uniform scaling among the objectives.

Table 9. Hypervolume value.

Experiment ID	MAPSO	DRSC-MOAGDE	DSC-MOPSO	MO-Ring-PSO-SCD	NSGA-II	MOEA/D	MOGA	$({\mathit{r}}_1,{\mathit{r}}_2)$
$\mathrm{c}5$	1.8643	1.933	1.8781	1.759	1.4546	1.3109	0.4688	(22,000, 350)
$\mathrm{c}6$	2.1385	2.2142	2.164	2.0117	1.5687	1.4493	0.6678	(40,000, 800)
$\mathrm{c}7$	2.4821	2.6056	2.5382	2.3015	1.8887	1.6887	0.9002	(62,000, 1100)
$\mathrm{c}8$	2.5811	2.6837	2.6687	2.384	1.9333	1.7056	0.8436	(68,000, 1500)
$\mathrm{c}9$	2.8036	2.9306	2.9146	2.5939	2.0256	1.7963	1.0452	(69,000, 1700)
$\mathrm{c}10$	1.3491	1.4798	1.4504	1.1346	1.1349	0.8035	0.5479	(23,000, 400)
$\mathrm{c}11$	1.3913	1.4619	1.4554	1.1566	1.1587	0.8288	0.5787	(40,000, 800)
$\mathrm{c}12$	1.5034	1.4123	1.3879	1.2595	1.2201	0.9312	0.6761	(54,000, 1200)
$\mathrm{c}13$	1.6304	1.5168	1.4953	1.3692	1.3274	1.0559	0.7779	(56,000, 1500)
$\mathrm{c}14$	1.7058	1.5845	1.5612	1.4668	1.3804	1.1115	0.808	(57,000, 2000)
$\mathrm{c}15$	1.8316	1.7343	1.7267	1.5777	1.412	1.2063	0.9465	(58,000, 2500)
$\mathrm{c}16$	2.0992	1.9974	1.9621	1.85	1.5967	0.7781	0.412	(14,000, 400)
$\mathrm{c}17$	2.3119	2.1932	2.1501	2.0406	1.7374	0.8672	0.4618	(30,000, 850)
$\mathrm{c}18$	2.6322	2.5113	2.4778	2.3618	2.0116	1.1221	0.6498	(40,000, 1200)
$\mathrm{c}19$	2.9624	2.8154	2.829	2.6767	2.2743	1.3069	0.7946	(50,000, 1500)
$\mathrm{c}20$	3.4458	3.312	3.2611	3.1492	2.7236	1.6766	1.1481	(53,000, 1600)
$\mathrm{c}21$	3.6943	3.5292	3.4659	3.366	2.9313	1.7517	1.3172	(55,000, 2100)
$\mathrm{c}22$	3.9328	3.7508	3.7452	3.55	3.1034	1.7862	1.3576	(60,000, 2500)
$\mathrm{c}23$	4.0047	3.8169	3.8017	3.5916	3.1393	1.6375	1.2498	(55,000, 2800)
$\mathrm{c}24$	4.1859	4.0126	3.9654	3.7728	3.2642	1.8505	1.306	(56,000, 3100)

presents the average HV values obtained by seven algorithms over 30 runs across all experimental scenarios. In solving satellite data transmission problems, the MAPSO algorithm demonstrates a significant advantage, particularly in scenarios 15 to 24, where its HV values are consistently higher than those of other algorithms. Table 9 indicates that MAPSO effectively balances solution diversity and convergence, achieving superior solution sets. In contrast, the HV values of the MOGA algorithm are noticeably lower across all scenarios, reflecting poor solution diversity and limited ability to explore the solution space effectively. The HV values of the DRSC-MOAGDE, DSC-MOPSO, MO-Ring-PSO-SCD, NSGA-II, and MOEA/D algorithms fall between those of MAPSO and MOGA, exhibiting varying performance levels.

5. Conclusions

We have proposed a minimum angle particle swarm optimization (MAPSO) algorithm to solve multi-objective satellite data transmission scheduling problems. The proposed algorithm was encoded as a discrete optimizer, and the calculation equation of the sine function was modified to enhance the PSO algorithm’s ability to deal with complex multi-objective problems. The MAPSO method introduces this new minimum angle selection strategy into the design and effectively selects the global optimal solution by calculating the minimum angle between particles. MAPSO balances convergence and diversity by avoiding local extrema in the global search process.

In the experimental validation section, we tested the MAPSO method on 24-scale task instances, demonstrating its ability to solve multi-objective data transmission scheduling problems and improve data transmission efficiency and accuracy. The experimental results show that the MAPSO method has an advantage in solving data transmission problems and maintains high solution quality across different task scales, indicating its generalization capability and adaptability in practical applications. Future research can further optimize this method, combining it with other intelligent optimization techniques to enhance its application potential in more extensive and complex task environments, promoting the development of satellite communication technology.